A Unified Approach to Linear and Nonlinear Normal Forms For

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J. Symbolic Computation (1999) 27, 49–131 Article No. jsco.1998.0244 Available online at http://www.idealibrary.com on

A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems†

R. C. CHURCHILL‡‡§AND MARTIN KUMMER§

‡Department of Mathematics, Hunter College and Graduate Center, CUNY, 695 Park Avenue, NY 10021, U.S.A. § Department of Mathematics, University of Toldeo, Toledo, OH 43606, U.S.A.

We prove that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function. Calculations are emphasized; the methods are illustrated with detailed examples, and for the sake of completeness the exceptional case is also reviewed and illustrated. Alternate methods are also discussed, along with detailed examples. c 1999 Academic Press

1. Introduction A linear operator T : V → V on a real nite-dimensional symplectic space V =(V,ω)is Hamiltonian,orinnitesimally symplectic,if ω(Tu,v)+ω(u, T v)=0,u,v∈ V. (1.1) The linear Hamiltonian normal form problem is that of nding simple matrix representations for such operators relative to symplectic bases of V . The theoretical solution was provided by Williamson (1936) (also see Wall (1963)), but calculation of normal forms following his ideas proves dicult in practice. Simplications have since appeared, but the methods are still quite involved, (see, e.g. Burgoyne and Cushman (1974, 1977a, 1977b), Djokovic et al. (1983), Laub and Meyer (1974), Springer and Steinberg (1970) and, for an introduction to the Russian literature, Bruno (1994) and Milnor (1969)). In this paper we show that in all but one instance Williamson normal forms can be cal- culated from modied Jordan forms (as in (1.2)) using Lie series methods in formally the same manner as one calculates nonlinear normal forms for Hamiltonian functions (e.g. as in Baider and Sanders (1991)), vector elds (e.g. see Baider (1989)), and PDEs (e.g. see Wittenberg and Holmes (1997)). Moreover, in the exceptional case we show that our method applies to the complexication. It will be evident that the technique can be computer implemented using standard symbolic manipulation packages. For the sake of completeness we include an algorithm for the exceptional case (with no claim to originality) based on more classical approaches.

†In fond memory of Martin Kummer. ‡E-mail: [email protected] §Research partially supported by NSF Grant DMS-9508415

0747–7171/99/010049 + 83 $30.00/0 c 1999 Academic Press 50 R. C. Churchill and M. Kummer

The basic idea, ignoring the exception, is as follows. If T as in (1.1) is “indecomposable”, i.e. if V admits no proper T -invariant nondegenerate subspace, then one can construct a T -invariant Lagrangian plane L V with the property that the Jordan form of T |L is an elementary Jordan block. (Here “plane” means “subspace”. For a very simple proof of the existence of an invariant Lagrangian plane see Duistermaat (1973, Theorem 3.4.5).) Extending the Jordan basis of L to a symplectic basis of V one then obtains a matrix representation of T of the form AS , (1.2) 0 A where A is upper triangular (in fact an elementary Jordan block), S is symmetric, and denotes transposition. But the collection of all such 2n 2n matrices has the structure of a graded Lie algebra w.r.t. the usual matrix commutator [M,N]:=MN NM (see Examples 8.2(b) and (c)), and this observation enables us to employ Baider’s modication of the usual Lie series method (Baider, 1989) to calculate unique normal forms. When S is appropriately normalized we show that these normal forms agree with the Williamson forms. Indeed, our methods provide an alternate proof of the existence of these forms, and from that point it is not dicult, by exploiting the uniqueness of the Jordan form, to establish uniqueness under the full action of Sp(V ). (Baider’s methods apply to a proper subgroup thereof.) But the uniqueness argument follows a fairly standard format, and for this reason has been omitted. In the exceptional case there is no invariant Lagrangian plane, and achieving the form (1.2) is therefore impossible. Instead there is a Lagrangian splitting with summands permuted by the semisimple portion of the Jordan decomposition of T , and invariant under the nilpotent portion. Williamson’s normal form clearly reects this splitting. The graded Lie algebra structure we use with matrices of the form (1.2) will hardly be a surprise to matrix analysts: the central contribution of this work is in relating that structure to Baider’s methods. A naive approach to computing the normal form of the matrix H of (1.2) might proceed as follows: look for an n n symmetric matrix X satisfying S = AX+ XA ; the IX A 0 symplectic matrix P := then has the property that PHP1 = 0 I 0 A is in Williamson form (when A is in Jordan form). In fact this works when the matrix equation S = AX + XA has a solution, but this is not always the case. (The equation is a particular instance of a classical operator equation with an extensive literature, e.g. see Bhatia and Rosenthal (1997), and in particular Roth’s Theorem on p. 3 of that reference.) What Baider’s methods guarantee is that (when H is appropriately normalized) there BY is always a symplectic matrix P of the form , with B upper triangular, 0(B )1 which conjugates H to the Williamson form. But once this is understood it is often a simple matter to calculate P directly, and to emphasize this point we work out explicit examples in all cases before introducing Baider’s techniques. Indeed, in Section 6 we show that in the traditionally “easy” cases such a conjugating (i.e. transition) matrix can be constructed in an even simpler way. As is well-known, the minimal polynomial m(t) of an indecomposable T must have one of the following forms: (i) m(t)=tr (the “nilpotent case”); (ii) m(t)=(t2 + b2)r, where 0

0 =6 p and p2 4q<0. In cases (iii) and (iv) the conjugacy class of T is classied by the Jordan form, but in cases (i) and (ii) one also needs an associated “sign characteristic”. (More precisely, the sign characteristic is needed for one of two subcases of case (i). For an interesting historical note on this characteristic see Gohberg et al. (1983, p. 141).) For convenience the classical results are summarized in Section 2. In particular, all of the Williamson normal forms are listed and the sign characteristic is explained in that section. This also serves to establish our notational conventions, which unfortunately vary from author to author in this subject (e.g. see Bruno (1994, p. 49)). Ramications for real linear Hamiltonian systems are detailed in Section 3, but these results are not used elsewhere. In fact enough information is given in Section 2 to enable one to write down the Williamson normal form of any given real or complex Hamiltonian matrix, and with sucient computer power one can then uncover a conjugating symplectic matrix using undetermined coecients. But for dim(V ) large this procedure is unrealistic. For the case of indecomposables our Lie series method is far more systematic, and exploits implemented algorithms for computing the Jordan form and the attendant conjugating matrix. Moreover, each (sub-)step of the normal form calculation involves solving a very small number of equations, as will be evident from Example 10.10, thereby allowing normal form calculations without restrictions on dim(V ). On the other hand, when dim(V ) is small, or when there is a Lagrangian splitting, alternatives to and/or modications of the normal form method seem preferable; these are discussed in Section 4 and Section 6. In those last mentioned sections we indicate how the Jordan form can be used to construct a decomposition when indecomposability is not the case. (The treatment in Section 4 is a variation on standard folklore.) Our algorithms for constructing Williamson forms of indecomposables, along with detailed examples, are presented in Section 6 and Section 7; the elementary matrix preliminaries are covered in Section 5. In Section 8 we develop the necessary Lie algebraic background for understanding Baider’s methods, which we then introduce in Section 9. With the exception of the proof of Proposition 8.7 our exposition of these methods is self-contained, and complete for our purposes, but far less general than what can be found in Baider (1989) (see Remark 9.20). In Section 10 we use Baider’s ideas to establish the algorithms in Section 7. The many examples in the paper are intended to be nontrivial without being over- whelming. Most were worked out with MAPLE, and verication would be tedious, if not simply unrealistic, without access to such a symbolic manipulation package. These examples are central to the work, and, in combination with the detailed description of the Williamson normal forms given in Section 2, contribute signicantly to its length. Readers should be aware of the fact that normal forms for symplectic matrices can be derived from normal forms for the indecomposable Hamiltonian cases (e.g. see Laub and Meyer (1974)). In our exposition of normal forms in Sections 8 and 9 we work for simplicity over elds of characteristic 0, but it will be clear that the proofs hold verbatim for prime characteristics provided the prime is suciently large relative to the dimension of V . It is expected that the ideas can be applied to other linear Lie algebras.

2. A Summary of the Williamson Normal Forms This section can be viewed as a summary of the results of Burgoyne and Cushman (1977b), although here the emphasis on Jordan forms is far greater. Readers are referred 52 R. C. Churchill and M. Kummer to that reference, as well as Burgoyne and Cushman (1977a), for proofs; Laub and Meyer (1974) is also highly recommended. (It is hoped that our adjustments in terminology and notation will not be a problem in this regard.) Here K denotes the real eld R or the complex eld C, V =(V,ω) is a nite- dimensional symplectic space over K, end(V ) is the collection of linear operators T : V → V , and Lω denotes the collection of Hamiltonian or innitesimally symplectic operators on V , i.e. those T ∈ end(V ) satisfying ω(Tu,v)+ω(u, T v)=0,u,v∈ V. (2.1)

Note that Lω is a Lie algebra w.r.t. the usual commutator [T1,T2]=T1T2 T2T1. The spectrum (i.e. collection of eigenvalues) of an element T ∈ end(V ) (not necessarily in Lω) will be denoted (T ). { }2n Recall that a basis e = ei i=1 of V is symplectic if for all 1 i, j n we have ω(ei,ej)=ω(en+i,en+j)=0andω(ei,en+j)=ij (the Kronecker delta). Equivalently: 0 I the e-matrix of ω must be the 2n2n “canonical” matrix J = .Byasymplectic I 0 basis representation of T ∈ end(V ) we mean a matrix which represents T relative to a symplectic basis. Williamson’s Theorem is concerned with a symplectic basis representation of a typical element of Lω in analogy with the Jordan matrix representation of a typical element of end(V ). The counterparts of the elementary Jordan blocks are called elementary Williamson blocks, which we now introduce. For any integer n>0 let MK (n) denote the collection of matrices of “size” n, i.e. n n matrices over K. Matrices of the following two types are called elementary Williamson blocks, real or complex according as K = R or K = C: those square matrices of the form A(n)0 ∈M K (2n), with n odd if =0, (2.2) 0 (A(n)) where A(n) is a standard elementary Jordan block, i.e.   10 0 00    0 10 00      . . .  ∈M ∈ A(n)= . .. .  K (n),K, (2.3)       00 1 00 0 and the denotes transposition; and those square matrices of the form A0(n) BK (n) ∈M K (2n)(n can be odd or even), (2.4) 0 (A0(n)) where A0(n) ∈MK (n) is as in (2.3) (with = 0) and   00 00      . . .   . .. .  BK (n):=  ∈MK (n), (2.5)     00 00 00 0(1)n1 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 53 with BK (1) = (1). Note that both elementary Williamson blocks have the form (1.2). When K = C (but not when K = R) these constitute all the elementary Williamson blocks, and in that instance a matrix in complex Williamson form,oracomplex William- son matrix, refers to a (necessarily even) square matrix over C of the form       0 W1 0 0 W1 0 0     0     0 W2 0   0 W2 0        0   00 Ws  00 Ws    000  , (2.6)  00 0 W1 0 0    00 0   0 W 000 0       2   000 00 0 00 Ws where each 0 Wj Wj 000 (2.7) 0 Wj is an elementary complex Williamson block. It proves convenient to refer to the matrices (2.7) as the blocks of (2.6), j =1, 2,...,s. A Williamson matrix (resp. elementary Williamson block) representation of an operator T ∈Lω refers to a symplectic basis representation of T which is a Williamson matrix (resp. elementary Williamson block). The representation is real or complex according as K = R or C. Any symplectic basis resulting in such a matrix representation is a Williamson basis for T .

Theorem 2.8. (Williamson) Any Hamiltonian operator on a nite dimensional complex symplectic vector space V admits a complex Williamson matrix representation, and the matrix is unique up to permutations of the blocks. Moreover, two Hamiltonian operators on V have the same complex Williamson form i they have the same Jordan form.

The initial statement also holds in the real case, but the nal assertion does not: the real Jordan form is not sucient to determine the real Williamson form. An alternate formulation of Williamson’s Theorem is more easily adapted to the real case, and for that reason will now be introduced. The ideas are from Burgoyne and Cushman (1977b), but the terminology has been altered. When (W, ωW ) is a (nite-dimensional) symplectic vector space (over K) and TW ∈ L W ωW we refer to := (W, ωW ,TW )asa(Hamiltonian) triple (over K). Moreover, we associate with such a triple W terminology and notation generally associated with W , ωW and TW , e.g. the dimension dim(W)ofW (over K) means the dimension dim(W ) of W (over K), W is said to be semisimple (resp. nilpotent) when this is the case for TW , a matrix representation of W refers to such a representation of TW , and the Jordan form of W means that of TW . W is real or complex according as K = R or C. It is clear from Williamson’s Theorem that for each T ∈Lω there is a direct sum decomposition V = V1 Vs of V into nondegenerate T -invariant subspaces Vj, and | | ∈L that for ωj := ω (Vj Vj)wehaveTj := T Vj ωj . We express this fact by writing

V = V1 + + Vs, where V := (V,ω,T) and Vj := (Vj,ωj,Tj) for j =1, 2,...,s, and refer to V1 + + Vs as a decomposition of V; the triples Vj are the summands of the decomposition. A triple 54 R. C. Churchill and M. Kummer

is decomposable if it admits a decomposition with at least two summands (i.e. if s>1); otherwise it is indecomposable. A decomposition V = V1 + + Vs is irreducible when each Vj is indecomposable. We note that the decomposability of a triple V is equivalent to the existence of a proper nondegenerate T -invariant subspace W V . Indeed, in the introduction we used this equivalent formulation.

Proposition 2.9. A complex Hamiltonian triple V =(V,ω,T) of dimension 2n is indecomposable i:

(ai) the Jordan form is A0(2n);or A (n)0 (aii) the Jordan form is 0 with n odd; or 0 A0(n) A (n)0 (b) the Jordan form is for some 0 =6 ∈ C, and in this last 0 A(n) instance the T -invariant subspace decomposition corresponding to the elementary Jordan blocks is a Lagrangian splitting.

Moreover, in the three respective cases, V has the following elementary Williamson block representation: 0 A0(n) BC(n) (a i) ; 0 (A0(n)) 0 A0(n)0 (a ii) (n must be odd); and 0 (A0(n)) 0 A(n)0 (b ) . 0 (A(n))

Finally, the minimal polynomials m(t) ∈ C[t] corresponding to the two cases of (a) have the form m(t)=tr, and that corresponding to (b) has the form m(t)=((t )(t + ))r.

In particular, the Jordan form of an indecomposable complex Hamiltonian triple determines the Williamson form, and vice versa. (ai) and (aii) are the complex nilpotent cases, and (b) is the complex linear case. The matrices in (a0ii) and (b0) reect Lagrangian splittings of V ; that of (aii) need not. The 0 matrix A0(n) occupying the upper left-hand corner in (a i) corresponds to a T -invariant Lagrangian subspace. Two triples W =(W, ωW ,TW ) and Y =(Y,ωY ,TY ) are equivalent, and we write W 1 Y, if there is a symplectic isomorphism :(W, ωW ) → (Y,ωY ) such that TW = TY . Decompositions W = W1 + + Ws and Y = Y1 + + Ys0 of equivalent triples W and Y are equivalent if s = s0, and if for some permutation : {1, 2,...,s}→{1, 2,...,s} we have Wj W(j) for j =1, 2,...,s. In these denitions the triples can be real or complex. Williamson’s Theorem has the following formulation in terms of complex Hamiltonian triples.

Theorem 2.10. For any complex Hamiltonian triple V the following statements hold:

(a) V admits an irreducible decomposition V = V1 + + Vs; (b) up to equivalence there is only one irreducible decomposition of V; A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 55

(c) a Hamiltonian triple is indecomposable i it is represented by an elementary Will- iamson block w.r.t. some symplectic basis, and that elementary Williamson block (but not the corresponding Williamson basis) is unique;

(d) the equivalence class of an indecomposable triple is uniquely determined by the elementary Williamson matrix of (c); and

(e) the equivalence class of an indecomposable triple is also uniquely determined by the Jordan form of the triple.

Notice that (a) is an automatic consequence of the denitions. The importance of the theorem lies in the remaining assertions. The real formulation of Theorem 2.10 diers only in the statement of (e). To see why some modication is necessary, endow R2 with the usual symplectic structure and 01 let T1,T2 be the Hamiltonian operators with usual basis representations and 00 0 1 . These are both elementary Williamson matrices (this will be a consequence 00 01 of the denition), and both have the same real Jordan form, i.e. . But it is easily 00 01 0 1 seen that there is no real symplectic matrix conjugating to , and as 00 00 a result the (real) triples associated with T1 and T2 cannot be equivalent. To dene the real elementary Williamson blocks rst assume n>0 is even and let Aa+ib(n) denote the usual real n n elementary Jordan block with complex eigenvalues a ib, where b =6 0, i.e. set   ab 10 00 00    ba01 00 00     00 ab 00     00 ba 00    A (n):=  . . ∈M (n). a+ib  .. .  R    ab 10   ...   ba01    00 00 ab 00 00 ba (2.11) (For an account of real Jordan forms see, e.g., Gohberg et al. (1986).) Next, again for an even n>0 set   00 00    00 00   . . .   . .. .   ∈M CR(n):= . 00 00  R(n), (2.12)  .   00 00    00 00 10 (1)(n/2)1 00 00 01 56 R. C. Churchill and M. Kummer and for an odd n>0 set   0 01  .   .. 0 10    . . .  DR(n):= . . .  . (2.13)  .   .  0 10 .. 0 10 00 The (real) elementary Williamson blocks are those square real matrices of the following forms: A(n)0 ∈M 6 ∈ R(2n), 0 = R; (2.14) 0 (A(n)) where A(n) is a standard real elementary Jordan block as in (2.3); Aa+ib(n)0 ∈M 6 R(2n),ab=0,neven,b>0; (2.15) 0 (Aa+ib(n)) where Aa+ib(n) is a standard real elementary Jordan block as in (2.11); A0(n) BR(n) ∈M R(2n), (2.16) 0 (A0(n)) 0 1 which for n = 1 is simply ; 00 A0(n)0 ∈M R(2n),nodd, (2.17) 0 (A0(n)) 00 which for n = 1 is the zero matrix ; 00 Aib(n) CR(n) ∈M R(2n),neven,b>0, (2.18) 0 (Aib(n)) where Aib(n) is a standard elementary real Jordan block as in (2.11) (with a = 0); and A0(n) bDR(n) ∈M R(2n),nodd,b>0 (2.19) bDR(n) (A0(n)) (note that the eigenvalues of this last matrix are ib), which for n = 1 is simply 0 b . Note that all but (2.19), which is the exceptional case, have the form (1.2). b 0 An even square matrix over R having the form       0 W1 0 0 W1 0 0     0     0 W2 0   0 W2 0        0    00 Ws   00 Ws    00 000  , (2.20)  W1 0 0 W1 0 0    0 W 00 0   0 W 000 0     2   2   00 000 00 Ws 00 Ws A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 57 where each 0 Wj Wj 00 000 (2.21) Wj Wj is an elementary real Williamson block, is said to be in real Williamson form,ortobea real Williamson matrix.

Theorem 2.22. (Williamson) Any Hamiltonian operator on a nite dimensional real symplectic vector space V admits a real Williamson matrix representation, and the matrix is unique up to permutations of the blocks.

The next proposition is the basis for the choice of the sign in the elementary real Williamson matrices (2.16), (2.18) and (2.19). For the statement recall that any T ∈ end(V ) can be written uniquely as T = S + N, where S ∈ end(V ) is semisimple, N ∈ end(V ) is nilpotent, and SN = NS (e.g. see Homan and Kunze (1961, pp. 267–8)). We refer to T = S + N as the Jordan decomposition of T . By the denition of “nilpotent” there is a unique integer m 0 such that N m =0=6 N m+1: m is the height of T ; m +1 is the index of nilpotency of N. When V =(V,ω) is symplectic we have T ∈Lω i S, N ∈Lω; the height of a Hamiltonian triple V =(V,ω,T = S + N) refers to that of N.

Remark 2.23. At the matrix level the Jordan decomposition T = S + N of any (not 1 1 necessarily symplectic) square matrix T is given by S = P JorSP and N = P JorN P , where Jor = JorS +JorN is the (obvious) Jordan decomposition of the (complex) Jordan form Jor = PTP1 of T . (In this paper the convention PTP1 proves more convenient than P 1TP: see Example 8.9(a).) Note from the uniqueness of the Jordan decomposition that S and N must be real when this is the case for T , even if this is not true of P and Jor.

Proposition 2.24. Let V =(V,ω,T) be a real Hamiltonian triple of dimension 2n and height m, and let T = S + N be the Jordan decomposition of T . Then the following statements hold.

(a) Suppose the real Jordan form of T is A0(2n), or, equivalently, that T = N and m = dim(V ) 1. Then ω(N mv, v) =06 for all v/∈ NV and the sign is independent of v (i.e. of v/∈ NV). (Note that m =2n 1 must be odd.) (b) Suppose the real Jordan form of T is Aib(2n), with 0

Moreover, in all three cases T is indecomposable.

Any indecomposable real triple V having one of the three forms of Proposition 2.24 is said to be signed, and when this is the case we dene the sign characteristic sgn(V)of V to be +1 (resp. 1) if the relevant sign is positive (resp. negative). Any other triple is unsigned. 58 R. C. Churchill and M. Kummer

Examples are more easily presented using matrix notation. A denition is required. A matrix is Hamiltonian,orinnitesimally symplectic, if it is a symplectic basis representation of some Hamiltonian operator. Equivalently: M ∈MK (2n) is Hamil- tonian i any (and therefore all) of the following conditions holds: JM is symmetric; AB M J + JM =0;M = JS, where S ∈M (2n) is symmetric; M = , where K CD A, B, C, D ∈MK (n), D = A , and B and C are symmetric. In particular, any matrix as in (1.2) is Hamiltonian, and all Williamson matrices are Hamiltonian.

Examples 2.25. In all three of the following examples V denotes R2n, ω is the usual { }2n symplectic structure, and e denotes the usual basis, i.e. e = ej j=1, where the 1 occurs in ej =(0, 0,...,0, 1, 0,...,0) in slot j. We follow the standard convention of using this basis to identify vectors with column vectors. The bilinear form ω, evaluated on vectors u, v ∈ V, is thus written ω(u, v)=hu, Jvi, where (as always) J denotes the canonical 0 I matrix and h, i is the usual inner product. I 0 Jordan decompositions are computed as in Remark 2.23. Our methods will eventually be applied to each of these examples. (a) The Hamiltonian matrix   0100 1210   00102024    00011260   0000 0401  N =   00000000   00001000 0000 0100 0000 0 010

(which we note has the form (1.2)) has Jordan form A0(8), and is therefore nilpotent of 7 height m = 7. The usual basis vector e5 satises N e5 = e1 =6 0, hence cannot be in 7 NV, and from ω(N e5,e5)=ω(e1,e5) = 1 we conclude that sgn(V) = 1, where V is the associated Hamiltonian triple. (b) The Hamiltonian matrix   11323  12334 T =   28 15 11 12  15 16 3 3 A (2) 0 has complex Jordan form 2i , hence height m = 1, and Jordan decom- 0 A2i(2) position T = S + N, where S and N are given by     17 12 13 61530  31231  15 903   and    59 90 17 3  87 75 6 15  90 86 12 12 75 102 15 9 respectively. From ω(Ne1,e1)=hNe1,Je1i = 87 we conclude that sgn(V)=1. A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 59

(c) The Hamiltonian matrix   8 297 418 24165    228 464 321 41 71 43     314 272 72 65 43 18  T =    154 1595 2061 8 228 314  1595 3071 2023 297 464 272  2061 2023 150 418 321 72 A (3) 0 has Jordan form 3i , and therefore height m = 2. The Jordan decom- 0 A3i(0) position is T = S + N, where   12 264 402 63663    216 480 360 36 75 48     318 288 60 63 48 18  S =    66 1512 2079 12 216 318  1512 3099 2172 264 480 288  2079 2172 6 402 360 60 and   20 33 16452    12 16 39 5 45    41612 250 N =   . 88 83 18 20 12 4  83 28 149 33 16 16  18 149 156 16 39 12 2 2 2 From TN e1 =(21, 24, 60, 75, 174, 381) one has ω(TN e1,e1)=hTN e1,Je1i = 75, and therefore sgn(V)=1.

A more conceptual denition of the sign characteristic for signed Hamiltonian triples is seen from the following result.

Proposition 2.26. Assume the notation of Proposition 2.24 and dene L ∈ end(V ) to be N m in cases (a) and (b) of that statement and TNm in case (c). Then a quadratic form is well-dened on V/NV by [v] 7→ ω(Lv, v), where [v] denotes the equivalence class of v in V/NV, and this quadratic form is either positive denite or negative denite.

In this (obviously equivalent) approach the sign characteristic is dened to be +1 if the quadratic form of the proposition is positive denite; 1 if it is negative denite.

Proposition 2.27. A real Hamiltonian triple V =(V,ω,T) of height m and dimension 2n is indecomposable i:

(ai) the real Jordan form is A0(2n), in which case m =2n 1 must be odd (note that n can be odd or even);or A (n)0 (aii) the real Jordan form is 0 with n odd, in which case m = n 1 0 A0(n) must be even; or 60 R. C. Churchill and M. Kummer A (n)0 (b) the real Jordan form is for some 0 =6 ∈ R, in which case n = 0 A(n) m +1 and the T -invariant subspace decomposition corresponding to the elementary Jordan blocks is a Lagrangian splitting; or A (n)0 (c) the real and complex Jordan forms are a+ib and 0 A(a+ib)(n)   Aa+ib(n/2) 0 0 0    0 Aaib(n/2) 0 0  00Aaib(n/2) 0 00 0Aa+ib(n/2) respectively, for some a + ib ∈ C with ab =06 and b>0, in which case n =2(m +1) must be even. Moreover, the T -invariant subspaces corresponding to the elementary Jordan blocks give a Lagrangian splitting in the real case, and an isotropic splitting in the complex case. Or; Aib(n)0 (di) the real and complex Jordan forms are Aib(2n) and respec- 0 Aib(n) tively, for some 0

Moreover, V is signed in cases (aii), (di) and (dii), and in the six respective cases V has the following elementary Williamson block representation, where m denotes the height of V: V 0 A0(n) sgn( )BR(n) (a i) , m odd, m =2n 1; 0 (A0(n)) 0 A0(n)0 (a ii) , m even, m = n 1; 0 (A0(n)) 0 A(n)0 6 ∈ (b ) , 0 = R, n = m +1 with no restriction on m; 0 (A(n)) 0 Aa+ib(n)0 6 (c ) , ab =0, b>0, n =2(m +1) 0 (Aa+ib(n)) with no restriction on m; 0 Aib(n) sgn(V)CR(n) (d i) , m odd, n = m +1; and 0 (Aib(n)) 0 A0(n) sgn(V)bDR(n) (d ii) , m even, n = m +1. sgn(V)bDR(n) (A0(n))

Finally, the minimimal polynomials m(t) ∈ R[t] corresponding to the two cases of (a) have the form m(t)=tr, that corresponding to (b) has the form m(t)=((t )(t + ))r, that corresponding to (c) has the form m(t)=((t2 + pt + q)(t2 pt + q))r, where p, q ∈ R satisfy 0

(ai) and (aii) are the real nilpotent cases, (b) is the real linear case, (c) is the case A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 61 of complex eigenvalues, and (di) and (dii) are the cases of purely imaginary eigenvalues. (Note that the spectrum in the nal case (i.e. (d0ii)) is {ib, ib}.) The matrices in (a00ii), (b0) and (c0) reect Lagrangian splittings of V ; that of (aii) need not. The upper left corner matrices in (a0i) and (d0i) reect T -invariant Lagrangian subspaces.

Examples 2.28. Proposition 2.27 applies to Examples 2.25 as follows.

(a) The Williamson normal form of the matrix N of Example 2.25(a) is A0(4) BR(4) , i.e. 0 (A0(4))   01000000   00100000   00010000   00000001    . 00000000   00001000 0000 0100 0000 0 010

(b) The Williamson normal form of the matrix T of Example 2.25(b) is A2i(2) CR(2) , i.e. 0 (A2i(2))   02 10 20 01   .  00 02 0020

(c) The Williamson normal form of the matrix T of Example 2.25(c) is A0(3) 3DR(3) , i.e. 3DR(3) (A0(3))   010003   001030   000300   . 003000 0 30100 300010

Theorem 2.29. For any real Hamiltonian triple V the following statements hold:

(a) V admits an irreducible decomposition V = V1 + + Vs; (b) up to equivalence there is only one irreducible decomposition of V; (c) a Hamiltonian triple is indecomposable i it is represented by an elementary Will- iamson matrix w.r.t. some symplectic basis, and that elementary Williamson matrix (but not the basis) is unique; (d) the equivalence class of an indecomposable triple is uniquely determined by the elementary Williamson matrix of (c); and 62 R. C. Churchill and M. Kummer

(e) the equivalence class of an indecomposable triple is also uniquely determined by the Jordan form of the triple and the sign characteristic when the triple is signed, and by the Jordan form alone when the triple is unsigned.

The “Williamson basis” terminology is also used in the real case, i.e. a Williamson basis for a real Hamiltonian triple V =(V,ω,T), or simply for T ∈Lω, is a symplectic basis of V which results in a Williamson matrix representation for T .

3. Consequences for Linear Hamiltonian Systems To begin K denotes the real or complex eld, n>0 is an integer, I denotes the n n 0 I identity matrix, and J, as always, denotes the 2n 2n matrix . Recall that I 0 the usual symplectic structure ω on K2n can be expressed as ω(u, v)=hu, Jvi, where to the right of the equal sign u and v are identied with the column vectors they determine relative to the usual basis and h, i is the usual (non Hermitian) inner product. Next recall that a square matrix is symplectic if it is a symplectic basis representation of a symplectic automorphism of some symplectic vector space V . Equivalently: M ∈MK (2n) is symplectic i any (and therefore all) of the following conditions holds: M JM = J; M 1J = JM ; M is a transition matrix relating symplectic bases e and eˆ of some 2n-dimensional symplectic vector space (V,ω ) overK, i.e. column j of M consists AB of the e-coordinates of the jth-element of eˆ; M = , where A, B, C, D ∈M (n), CD K A C and B D are symmetric, and A D C B = I; and M =(GH), where the 2n n ∈M ∈M matricesG, H satisfyG JG = H JH =0 K (n) and G JH= I K (n). In par- AB IB ticular, is symplectic i A1B is symmetric, and is symplectic 0(A )1 0 I i B is symmetric. Note that M ∈MK (2) is symplectic i det(M) = 1. The collection of 2n 2n symplectic matrices is a group Sp(n, K), obviously isomorphic to the group Sp(V ) of symplectic automorphisms of any 2n-dimensional symplectic space (V,ω) over K. The matrix interpretation of Williamson’s result is now evident: any linear Hamiltonian matrix is canonically conjugate (i.e. conjugate by means of a symplectic matrix) to a unique Hamiltonian matrix in Williamson normal form (where “unique” means “unique up to permutations of the elementary Williamson blocks”). 2n Henceforth K = R, V = R with the usual symplectic structure ω, JS ∈MR(2n) is Hamiltonian, T ∈Lω is the linear operator with usual basis matrix JS, and V is the Hamiltonian triple (R2n,ω,T). We let m denote the height of V. 1 h i 2n Hamilton’s equations for the quadratic Hamiltonian H(x)= 2 Sx,x on R are x˙ = JSx, and any such equation is called a linear Hamiltonian system.Ifx = Py is a 1 linear canonical transformation, i.e. if P ∈MR(2n) is symplectic, theny ˙ = P JSTy = ˆ 1 h i JP SPy, which constitute Hamilton’s equations for H(y)= 2 P SPy,y = H(Py). We conclude that linear canonical transformations preserve linear Hamiltonian systems, and that the Hamiltonian of a transformed system is simply the composition of the original Hamiltonian with the transformation. In this context Williamson’s result is: any linear Hamiltonian system is canonically equivalent, i.e. may be transformed using a linear canonical transformation, to a unique linear Hamiltonian system x˙ = JSx in which A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 63

JS is a Williamson matrix. (Here “unique” means “unique up to permutations of the Williamson blocks”.) It is immediate from Williamson’s Theorem that any linear Hamiltonian can be written 1 h i as a sum of linear Hamiltonians in linear normal form, i.e. Hamiltonians H(x)= 2 Sx,x having the property that JS is an elementary Williamson block. For such an H the Hamiltonian triple V must be indecomposable, and we see from Proposition 2.27 that the possibilities for JS are: V A0(n) sgn( )BR(n) 1 (ai) , m odd, n = 2 (m + 1); 0 (A0(n)) A0(n)0 (aii) , m even, n = m +1; 0 (A0(n)) A(n)0 6 ∈ (b) ,0= R, no restriction on m, n = m +1; 0 (A(n)) Aa+ib(n)0 6 (c) , ab =0,b>0, no restriction on m, n =2(m + 1); 0 (Aa+ib(n)) Aib(n) sgn(V)CR(n) (di) , m odd, n = m + 1; and 0 (Aib(n)) A0(n) sgn(V)bDR(n) (dii) , m even, n = m +1. sgn(V)bDR(n) (A0(n))

The associated symmetric matrices S = J(JS) are then, in the same order: 0 0(A0(n)) (a i) V ; A0(n) sgn( )BR(n) 0(A (n)) (a0ii) 0 ; A0(n)0 0(A (n)) (b0i) ,0=6 ∈ R; A(n)0 0(A (n)) (c0i) a+ib , ab =06 ,b>0; Aa+ib(n)0 0 0(Aib(n)) (d i) V ; and Aib(n) sgn( )CR(n) sgn(V)bD (n)(A (n)) (d0ii) R 0 . A0(n) sgn(V)bDR(n)

Now write x =(q, p)=(q1,...,qn,p1,...,pn). Then the corresponding normal forms 1 h i for the linear Hamiltonians H(x)= 2 Sx,x , in the same order, are: 00 h i 1 V h i (a i) H(x)=PA0(n)q, p + 2 sgn( ) BR(n)p, p n1 1 n1 V 2 = j=1 pjqj+1 + 2 ( 1) sgn( )pn; 00 h i (a ii) H(x)=PA0(n)q, p n1 = j=1 pjqj+1; 00 h i (b ) H(x)= A(n)q, pP h i n1 = q, p + j=1 pjqj+1; 00 h i (c ) H(x)= Aa+ib(n)q,P p P h i n/2 n2 = a q, p + b j=1(q2jp2j1 q2j1p2j)+ j=1 qj+2pj; 64 R. C. Churchill and M. Kummer

00 h i 1 V h i (d i) H(x)= APib(n)q, p + 2 sgn( ) CR(n)p, p n/2 1 V (n/2) 1 2 2 = b j=1(q2jp2j 1 q2j 1p2j)+ 2 sgn( )( 1) (pn1 + pn); 00 h i 1 V h i h i (d ii) H(x)=PA0(n)q, p + 2 b sgn( )( DPR(n)q, q + DR(n)p, p ) n1 1 n V j 1 = j=1 pjqj+1 + 2 b sgn( ) j=1( 1) (qjqn+1 j + pjpn+1 j). Note that (d00ii) includes the linear harmonic oscillator Hamiltonian ω 2 2 H(x)= 2 (q + p ) with frequency ω>0: take n = 1 and b = ω. (The use of ω to denote frequency is standard practice, and for that reason it appears here. However, unless specically stated to the contrary, in all other occurrences in this paper ω refers to the symplectic form.) We now turn to the development of algorithms for computing Williamson normal forms for indecomposable triples.

4. On the Construction of Indecomposables Here we address the problem of constructing an irreducible decomposition of a real or complex Hamiltonian triple V =(V,ω,T). For ease of reference we recall the following standard fact, already mentioned immediately before the statement of Proposition 2.9.

Proposition 4.1. V is decomposable i V admits a nondegenerate T -invariant subspace W , and when this is the case V = W W ⊥ corresponds to a decomposition. In particular, when W is nondegenerate and T -invariant W ⊥ will also have these properties.

We also need a few standard preliminaries on the primary decomposition of V (i.e. of T ), and to this end we express the dening condition (2.1) for T ∈Lω in the equivalent form ω(T ku, v)=(1)kω(u, T kv),u,v∈ V,1 k ∈ Z. (4.2) Let m(t) ∈ k[t] denote the minimal polynomial of T . From (4.2) we then have ω(m(T )u, v)=ω(u, m(T )v),u,v∈ V, (4.3) where for a monic polynomial p(t) ∈ K[t]weletp(t):=(1)degree(p(t))p(t), which we note is again monic. p(t)isthedual of p(t), and the latter is self-dual when p(t)=p(t).

Proposition 4.4. The minimal polynomial m(t) of T is self-dual. In particular, the nonzero eigenvalues of T occur in pairs, i.e. for any 0 =6 ∈ C we have ∈ (T ) i ∈ (T ).

Note how the last assertion is reected in the structure of each elementary Williamson block with nonzero eigenvalues.

Proof. We have m(T ) = 0, whence from (4.3) and the nondegeneracy of ω we have m(T ) = 0. Since m(t) and m(t) are monic of the same degree, the assertion now follows from the uniqueness of the minimal polynomial. 2 Q s V rj Now write the minimal polynomial m(t)of as m(t)= j=1(pj(t)) , where the polynomials pj(t) ∈ K[t] are monic, distinct and irreducible, and each rj is a positive A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 65 integer. From the Primary Decomposition Theorem (e.g. see Homan and Kunze (1961, p.220)) we then have

rj V = V1 Vs,Vj = ker((pj(T )) ), (4.5) with each Vj being T -invariant. Note from Proposition 4.4 that for each 1 i s there is a unique 1 j s such that pi (t)=pj(t); we write j as i .

Proposition 4.6. For any 1 i =6 j s the following statements hold.

(a) i = i ⇒ Vi is a nondegenerate T -invariant subspace of V . (b) i =6 i ⇒ Vi is an isotropic T -invariant subspace of V . Moreover, Vi Vi is a nondegenerate T -invariant subspace of V and the splitting is Lagrangian. (c) The decomposition V = V + + V 0 of V dened by 1 s (Vi Vi ,ω|(Vi Vi Vi Vi ),T|(Vi Vi )) in the nonself-dual case, Vi = (Vi,ω|(Vi Vi),T|Vi) in the self-dual case,

is characterized by the following two properties: distinct Vi have relatively prime minimal polynomials; and each of these polynomials has one of the two forms (i) (p(t))r, where p(t) ∈ K[t] is irreducible and self-dual, or (ii) (p(t)p(t))r, where p(t) ∈ K[t] is irreducible and not self-dual.

The decomposition V = V1 ++Vs0 of (d) is the primary decomposition of the Hamil- tonian triple V.

Proof. See, e.g. Laub and Meyer (1974, Section 2) or the proof of Lemma 3.1 in Burgoyne and Cushman (1977b). (A proof is also easily constructed from arguments in Milnor (1969).) 2

The result reduces the search for indecomposables to the cases in which the minimal polynomial has one of the two forms in Proposition 4.6(c). In this section we will only be concerned with the self-dual case; the nonself-dual, case will be handled in Section 6. In the complex case the only possibility for a self-dual minimal polynomial is m(t)=tk (the nilpotent case). In the real case there is an additional possibility, i.e. m(t)=(t2+b2)k, where b>0 (the case of purely imaginary eigenvalues). We rst examine the nilpotent case. We make constant use of the following observations, often without reference: (a) ω(N iu, N j,v)=(1)jω(N i+ju, v),u,v ∈ V,0 i, j ∈ Z; (b) k>m⇒ ω(N ku, v)=0,u,v ∈ V ; (c) i + j>m⇒ ω(N iu, N jv)=0,u,v ∈ V ; (4.7) (d) j>0 ⇒ ω(N mu, N jv)=0,u,v ∈ V ; and (e) k 0even ⇒ ω(N kv, v)=0,v ∈ V. (a) is immediate from (4.2), (b) from the height assumption on V, (c) and (d) from (a) and (b), and (e) from ω(N kv, v)=(1)kω(v, Nkv)=(1)k+1ω(N kv, v)=ω(N kv, v). 66 R. C. Churchill and M. Kummer

the first nilpotent case: odd height

Until further notice the integer m is assumed to be odd.

Proposition 4.8. Suppose e ∈ V has the following two properties: 1 if K = R (i) ω(N me, e)=ε := ; and 1 if K = C (ii) ω(N je, e)=0for j =0, 1,...,m 1.

m Then the span W V of eJ = {N e,...,Ne,e} is N-invariant and nondegenerate. Moreover, the following assertions hold for W := (W, ωW ,N|W ):

(a) eJ a Jordan basis for W, and the corresponding Jordan form is A0(m +1); (b) W is indecomposable; and W { }m+1 (c) a symplectic basis for is given by ej j=1 , where m+1j ej = N e 0 1 j1 j1 j =1,...,m := 2 (m +1), em0+j = ε(1) N e, 0 0 A0(m ) εBK (m ) and the corresponding Williamson matrix is 0 . 0 (A0(m ))

When W = V we conclude that V is already indecomposable; otherwise we see from Proposition 4.1 that W W ⊥ provides a decomposition.

Proof. The proof is a straightforward verication. 2

Algorithm 4.9. When the height m is odd a vector e ∈ V satisfying (i)–(iii) of Propo- sition 4.8 can be determined as follows. Step I: Choose any symplectic basis e of V and let H ∈MK (2n)bethee-matrix of V. 1 Step II: Choose Q ∈MK (2n) such that B = QHQ is in Jordan form. Step III: Locate a copy of A0(m + 1) within B, let j be that column of B containing the right column of that copy, and lete ˆ ∈ V be the vector having the jth-column of Q1 as e-coordinates. If ω(N me,ˆ eˆ) =6 0 rescale this vector, if necessary, so that for the resulting e the normalization in (i) of Proposition 4.8 holds, and then proceed to Step V. Otherwise proceed withe ˆ to Step IV. Step IV: Locate a second copy of A0(m + 1) within B having the following property: if the right column of this second copy is within column k of B, then the vectore ˆ0 ∈ V having the kth-column of Q1 as e-coordinates satises ω(N me,ˆ eˆ0) =6 0. (The existence m 0 0 of such a copy of A0(m + 1) will be established momentarily.) If ω(N eˆ , eˆ ) =6 0 proceed to Step V using e :=e ˆ0 in place ofe ˆ; if not proceed to that step using e :=e ˆ+ê0 in place ofe ˆ after noting from the oddness assumption on m that ω(N me, e)=2ω(N me,ˆ eˆ0) =0.6 (The existence of the required alternative copy of A0(m + 1) asserted in the previous paragraph can be established as follows. By nondegeneracy there is a vector v ∈ V such that ω(N me,ˆ v) =6 0. Moreover, since e identies the columns of Q1 with a basis of V we may assume v has the form N ieˆ0, where thee ˆ0 is obtained in the same manner A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 67 as wase ˆ although using a dierent elementary Jordan block within B,sayA0(t). Now note the denition of height that t m + 1. Using 0 =6 ω(N me,ˆ v)=ω(N me,ˆ N ieˆ0)= (1)iω(N m+ie,ˆ eˆ0) we conclude that i = 0, and therefore v =ê0. Moreover, from 0 =6 ω(N me,ˆ eˆ0)=(1)mω(ê, N meˆ0) we see that N meˆ0 =6 0, which in turn gives t m +1. The equalities t = m + 1 and A0(t)=A0(m + 1) follow immediately.) Step V: If there is a 0 k

The use of the algorithm is illustrated in the next example.

Examples 4.10. We work in R2n with the usual symplectic structure, and we use the usual basis (which is symplectic) to identify operators with matrices. 1 (a) The matrix N of Example 2.25(a) has Jordan form A0(8) = QNQ , where   101402011   01 0140 120   001010210   00010400 Q 1 =   . 00000001   00000010 00000100 00001000 We can begin at Step III. Step III: We take e =(1, 0, 0, 0, 1, 0, 0, 0) and note that N 7e =(1, 0, 0, 0, 0, 0, 0, 0) , hence ω(N 7e, e) = 1. In particular, the initial normalization is not required. Step IV: Here this step is skipped. Step V: ω(N 5e, e)=14, and we therefore replace e by e7N 2e =(1, 7, 14, 28, 1, 0, 7, 0) , which becomes our new e. But now ω(N 3e, e)=145, so this new e appears 145 4 only briey; it must be replaced by e + 2 N e =(1, 1001/2, 14, 89/2, 1, 0, 7, 0) , which now carries the label e. The new e is eliminated with equal speed since ω(Ne,e) = 659. 659 6 The nal e is e 2 N e =(1, 171, 14, 89/2, 1, 0, 7, 0) . The symplectic basis of Proposition 4.8(c) is then {N 7e, N 6e, N 5e, N 4e, e, Ne,N2, Ne,N2e, N 3e, N 3e}, and is reected in the columns of the symplectic matrix   47 1070 1 165 0 2  49  0107171 2 2 0   7  0010 14 2 2 3   89  0001 0 30 P 1 =  2  . 0000 1 0 0 0   0000 0 1 0 0 0000 7010 0000 0 701 68 R. C. Churchill and M. Kummer

One can now check that PNP1 is the Williamson form already given in Example 2.28(a). (b) The Hamiltonian matrix   0 12 8 52 3 38   1 10 14 3 68 44    0 14 2 38 44 50  N =   000010 0 4 0 12 10 14  0048142 has Jordan form A (4) 0 QNQ 1 = 0 , 0 A0(2) where   4 120127 3  5   12 1010 46 2    1  8 1400 90 Q =   , 4 000 10  0 400 0 1 0 000 140 and is therefore nilpotent of height 3. Following Algorithm 4.9 we rst select e =(1, 0, 0, 0, 0, 0) , but since ω(N 3e, e)=4 we must replace this with e =(1/2, 0, 0, 0, 0, 0) . Our (ad hoc) symplectic basis of W ⊥, where W is the span of {N 3e, N 2e, e, Ne}, is reected in the columns of the symplectic matrix   2 60 1 02  2   6 50 0 1 7   2 2  1 4 71000 P1 =   , 200000  020000 000001 which is such that   010000   000010   1 002001  N1 = P1HP1 =   . 000000 00 0100 004002 2 1 We are thus reduced to nding a symplectic matrix Pˆ which converts Nˆ := 1 42 1 to Williamson form, and working through the algorithm once again results in PˆNˆ1Pˆ = A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 69 0 1 1 1 , where Pˆ1 = 2 . In other words, for the symplectic matrix 00 20   10 000 0   01 000 0  1  1 00 100 2  P2 =   00 010 0 00 001 0 00 200 0 we have   010000   000 010   1 000001  P2N1P2 =   , 000000 000100 000000 which is in Williamson normal form. A symplectic matrix conjugating the original N to this form is therefore   1 2 642 00  1   6 5 70 2 0   1  1 1 1  4 7 10 0 2  P = P1 P2 =   . 20000 0  02000 0 00200 0

the second nilpotent case: even height

Until further notice the integer m is assumed even.

Proposition 4.11. Suppose distinct vectors e, e0 ∈ V have the following three properties:

(i) ω(N me, e0)=1; (ii) ω(N je, e0)=0for all 0 j

m m 0 0 Then the span W V of eJ = {N e,...,Ne,e,N e ,...,e} is N-invariant and nondegenerate. Moreover, the following assertions hold for the nilpotent Hamiltonian triple W := (W, ωW ,N|W ):

(a) e is a Jordan basis for W, and the corresponding Jordan form is J A (m +1) 0 0 ; 0 A0(m +1) (b) W is indecomposable ; and 2(m+1) (c) a symplectic basis for W is given by {ej} , where j=1 m+1j ej = N e j1 j1 0 j =1,...,m+1, em+1+j =(1) N e 70 R. C. Churchill and M. Kummer

and the corresponding Williamson matrix is A0(m +1) 0 . 0 (A0(m + 1))

Proof. This is elementary. 2

We need the following analogue of Algorithm 4.9.

Algorithm 4.12. When the height m is even vectors e, e0 ∈ V satisfying (i), (ii) and (iii) of Proposition 4.11 can be determined as follows: Step I: Choose any symplectic basis e of V and let H ∈MK (2n)bethee-matrix of V. 1 Step II: Choose Q ∈MK (2n) such that B = QHQ is in Jordan form. 1 Step III: Locate a copy of A0(m + 1) within B, let j be that column of Q containing the right column of that copy, and let e ∈ V be the vector having the jth-column of Q1 as e-coordinates. Note from the evenness assumption on m that ω(N me, e)=0 (see (4.7e)). Note that the e-coordinates of N ie are given by column k i of Q1 for i =0,...,m. Step IV: Locate a second copy of A0(m + 1) within B having the following property: if the right column of this second copy is within column k of B, then the vectore ˆ0 having the kth-column of Q1 as e-coordinates satises ω(N me, eˆ0) =6 0. (For existence use the argument ending Step IV of Algorithm 4.9.) Rescalee ˆ0, if necessary, so that the resulting vector e0 satises ω(N me, e0)=1. Step V: If there is a 0 k

Example 4.13. We work in R6 with the usual symplectic structure ω, and we use the usual basis to identify operators with matrices and vectors with column vectors. The Hamiltonian matrix  16 73 12 264 361 143    52 36 13 361 82 95     10 6 48 143 95 248  N =    0 10 41652 10  10 8 8 73 36 6  4 8412 13 48 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 71 A (3) 0 has Jordan form B = QNQ1 = 0 , and therefore height 2, where 0 A0(3)   10 16 1 0 264 0   30 52 0 223 361 0     50 10 0 387 143 0  Q 1 =   .  00010 16 1   0 10 0 4 73 0  10 40 73 12 0

We use Algorithm 4.12 and Proposition 4.11 to compute a symplectic matrix which conjugates N to Williamson form. We begin with Step III. Step III: We take e =(1, 0, 0, 0, 0, 0) , i.e. column 3 of Q1. Step IV: The initial choice for e0 is column 6 of Q1, but normalization to e0 = (0, 0, 0, 1/10, 0, 0) is necessary to achieve ω(N 2e, e0)=1. Step V: This step is skipped since ω(N je, e)=0forj =0, 1, 2. 0 0 66 0 Step VI: Since ω(Ne ,e)= 25 we must replace e with 33 e0 + Ne =(528/25, 1716/25, 66/5, 1/10, 66/5, 132/25) , 25 which we henceforth call e0. 0 8 Step VII: We have ω(Ne,e )= 5 , and we therefore replace e by 8 e Ne = (133/5, 416/5, 16, 0, 16, 32/5) . 5 0 133 133 2 But for this new e we nd that ω(e, e )= 50 , and so e now becomes e + 50 N e = (0, 163, 117, 0, 16, 33) . The symplectic basis {N 2e,Ne,e,e0, Ne0,N2e0} of Proposition 4.11(c) is now reected in the columns of the symplectic matrix   10 0 0 528 66 0  25 5   1716 757 223   30 100 163 25 10 10     50 70 117 66 517 387  P 1 =  5 10 10  ,  1 8   00 0 1   10 5   66 73 2  0 10 16 5 10 5 132 72 73 10 20 33 25 5 10 and one can check that   010000   001000   1 000000 A0(3) 0 PNP =   = 000000 0 (A0(3)) 000100 000 010 is an elementary Williamson block representation of V. 72 R. C. Churchill and M. Kummer

the case of purely imaginary eigenvalues

For the remainder of the section we assume V is real. We need to establish our conventions with regard to the complexication VC of V . Specically, we regard VC as the Cartesian product V V with addition dened compo- nentwise and complex multiplication dened by (a+ib)(u, v):=(aubv, av+bu),a+ib ∈ C,u,v ∈ V . Following custom we use the injection v 7→ (v, 0) to identify V with the real subspace V {0} of VC, whereupon the identity (u, v)=(u, 0) + i(v, 0) justies writing (u, v)asu + iv and VC as V + iV . Note that the complexication TC : VC → VC of a linear operator T : V → V can then be expressed by TC(u + iv):=Tu + iT v, and the complexication C : VC VC → C of a bilinear form : V V → R by C(u + iv, x + iy):=(u, x) (v, y)+i((u, y)+(v, x)). The conjugate w of a vector w = u + iv ∈ VC is dened to be u iv ∈ VC, and the conjugate W of a complex subspace W VC (which is again a complex subspace) is {w : w ∈ W }. Henceforth standard properties of complexications will be used without comment. Let T = S + N denote the Jordan decomposition of T . We dene the complexication of (this or any real Hamiltonian triple) V to be VC := (VC,ωC,TC), which we note is a complex Hamiltonian triple. By Proposition 4.6(b) we have a TC-invariant Lagrangian r r splitting VC = Wˆ Yˆ , where Wˆ = ker((TC ib) ) and Yˆ = ker((TC + ib) ). The following observation proves useful: For any vector e ∈ V , C real if j is odd, and ω ((N )je, e)is (4.14) C C purely imaginary if j is even. Indeed, this is immediate from j j j j j+1 j ω((NC) ,e,e)=ω((NC) e, e)=(1) ω(e, (NC) e)=(1) ω((NC) e, e).

Proposition 4.15. Suppose e ∈ VC is any vector satisfying the following three properties: 2 if m is odd, (i) ω ((N )me, e)= C C 2i if m is even; j (ii) ωC((NC) e, e)=0,j=0,...,m 1; and j (iii) ωC((NC) e, e)=0,j=0,...,m.

m 1 m Set := (ωC((NC) e, e)) and let W VC denote the span of {(NC) e,...,NCe, e}. Then the following statements hold.

(a) Y := W W is a nondegenerate TC-invariant subspace of VC and the complex Jordan form of T |Y is C A (m +1) 0 ib . 0 Aib(m +1)

(b) YR is a nondegenerate T -invariant 2(m +1)-dimensional subspace of V , and the real Jordan form of T |YR is Aib(2(m + 1)). m m m (c) {(NC) e,...,NCe, e, e, (1)NCe,...,(1) (NC) e} is a symplectic basis of Y , and the associated matrix on T |Y is the elementary complex Williamson matrix C Aib(m +1) 0 . 0 (Aib(m + 1)) A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 73

(d) The complex triple (Y,ωC|(Y Y ),TC|Y ) is indecomposable, and the elementary Williamson block of (c) is the unique Williamson matrix representation of this Hamiltonian triple.

In particular, when dim(V)=2n the following statements are equivalent: V is indecomposable; n = m+1; the real Jordan form is Aib(2n); the complex Jordan form is A (n)0 ib . 0 Aib(n)

Proof. The assertions are veried by straightforward calculation. 2

Algorithm 4.16. A vector e ∈ VC satisfying (i), (ii) and (iii) of Proposition 4.15 can be determined as follows: Step I: Choose any symplectic basis e of V and let H ∈MR(2n) be the e-matrix of V. 1 Step II: Choose Q ∈MC(2n) such that B = QHQ is in complex Jordan form. 1 Step III: Locate a copy of Aib(m+1) within B, let j be that column of Q containing 1 the right column of that copy, and let e ∈ VC be the vector having the jth-column of Q as e-coordinates (where e is now considered a basis of VC). Note that the e-coordinates i 1 m of (NC) e are given by column j i of Q for i =0,...,m.IfωC((NC) e, e) =6 0 then by rescaling, if necessary, we see from (4.14) that we can achieve (i) of Proposition 4.15. In this case proceed with this rescaled e to Step V; otherwise go on to Step IV. Note m from Proposition 4.6(c) that the span of {(NC) e,...,e} is isotropic, as required in (iii). Step IV: Locate an elementary Jordan block of the form Aib(m + 1) within B having the following property: if the right column of Aib(m + 1) is within column k of 0 1 B, then the vector e ∈ VC having the kth-column of Q as e-coordinates satises m 0 ωC((NC) e, e ) =6 0. (The argument for the existence of such a block is analogous to m 0 0 that in the bracketed paragraph of Step IV of Algorithm 4.9.) If ωC((NC) e , e ) =06 rescale e0, if necessary, to obtain a vector e satisfying (i) of Proposition 4.15 and proceed 0 m 0 with this vector to Step V. Otherwise rescale e , if necessary, so that ωC((NC) e, e )is real if m is odd and purely imaginary if m is even; then set e00 := e + e0 and note that m 00 00 00 ωC((NC) e , e ) =6 0. Rescale e in turn, if necessary, to obtain a vector e satisfying (i) of Proposition 4.15 and then proceed to Step V with this vector. Note that in either m case the constructed e has the property that (NC) e is an eigenvalue of S, and so from m Proposition 4.6 the span of {(NC) e,...,e} must be isotropic, as required in (iii). k Step V: Suppose there is an integer 0 k

Theorem 4.17. Let V =(V,ω,T) be a real 2n-dimensional indecomposable Hamilto- nian triple with height m = n 1 and minimal polynomial m(t)=(t2 + b2)r, where 74 R. C. Churchill and M. Kummer

R I b>0.Lete ∈ VC be as in the statement of Proposition 4.15, and write e = e + ie in accordance with the identication VC ' V + iV . Then m R R m I I 1 m V ω(N e ,e )=ω(N e ,e )= 2 ωC((NC) e, e) for m odd, sgn( )= m R I i m ω(N e ,e )= 2 ωC((NC) e, e) for m even. Moreover, depending on the parity of m we have the following additional results.

(a) (The First Purely Imaginary Eigenvalue Case) for m odd (equivalently: for n even) 0 1 1 0 set m := 2 (m +1)= 2 n. Then the span Y of 0 0 m m(m 1) m(m 1) m {(NC) e,...,(NC) e, (NC) e,...,(NC) e} 0 0 is a TC-invariant n-dimensional Lagrangian subspace of VC satisfying Y = Y .In 0 0 particular, (Y )R := Y ∩ V is a T -invariant n-dimensional Lagrangian plane in V with basis 0 0 {N meR,NmeI ,Nm1eR,Nm1eI ,...,Nm(m 1)eR,Nm(m 1)eI }. (b) (The Second Purely Imaginary Eigenvalue Case) For m even (equivalently: For n odd) dene ni R ei := N e i =1,...,n, j1 j1 I en+j := (1) sgn(V)N e j =1,...,n.

Then {e1,...,en,en+1,...,e2n} is a symplectic basis of V , and the associated matrix of T is the elementary Williamson block A0(n) sgn(V)bDR(n) . sgn(V)bDR(n) (A0(n)) Moreover, this is the unique elementary Williamson block representing T .

Notice that the theorem does not give the existence of an elementary Williamson block representation in the rst purely imaginary eigenvalue case. However, that result now becomes a corollary of Theorem 10.9.

Proof. By direct calculation we nd that ω(N jeR,eR)+ω(N jeI ,eI )ifj is odd, ω ((N )je, e)= (i) C C 2iω(N jeR,eI )ifj is even, as well as ω(N jeR,eR) ω(N jeI ,eI )+2iω(N jeR,eI )ifj is odd, ω ((N )je, e)= (ii) C C ω(N jeR,eR) ω(N jeI ,eI )ifj is even. From (i)–(iii) of Proposition 4.15 and (4.7e) it follows that 2ω(N jeR,eR)=2ω(N jeI ,eI )ifj is odd, ω ((N )je, e)= (iii) C C 2iω(N jeR,eI )ifj is even, j =0,...,m, that ω(N jeR,eR)=ω(N jeI ,eI )=ω(N jeR,eI )=0 for j =0,...,m 1, (iv) A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 75 and that   ω(N meR,eR)=ω(N meI ,eI )=1ifm is odd; m R R m I I  ω(N e ,e )=ω(N e ,e )=0 ifm is even; and (v) ω(N meR,eI )=1ifm is even. For odd m we can compute sgn(V) as the sign of ω(N meR,eR), and in this case we see from (v) that sgn(V)=ω(N meR,eR)=ω(N meI ,eI ) as asserted. For even m we rst m m use the eigenvalue property TC(NC) e = ib(NC) e to see that TNmeR = bN meI , TNmeI = bN meR. In this case sgn(V) is the sign of ω(TNmeR,eR)=bω(N meI ,eR)=bω(N meR,eI ), and since b>0 it follows from (v) that sgn(V)=ω(N meR,eI ), again as asserted. (Note this is the rst use of the assumption b>0.) The remaining equalities in the expression for sgn(V) are immediate from the case j = m of (iii). The proof of (a) is a straightforward verication. To prove (b) rst use (iv) and (v) to check that the given basis is symplectic; then use j1 j1 I (1) sgn(V)en+j = N e and TeR = NeR beI TeI = NeI + beR

(i.e. Tce = ibe + NCe) to check that i1 Tei = ei1 b(1) sgn(V)e2n(i1) j1 Ten+j = en+(j+1) + b(1) sgn(V)en(j1), where e2n+1 :=0=:e0. This gives the asserted Williamson matrix representation. Now suppose A0(n) bDR(n) ,= 1, (vi) bDR(n) (A0(n)) is an elementary Williamson block representation of T relative to some symplectic basis { }2n m m eî i=1 of V . Then from T eˆn =ên2 beˆn+1 and N eˆn =ê1 we see that ω(TN eˆn, eˆn) = bω(ê1, eˆn+1)=b, and therefore = sgn(V). Since an elementary Williamson block representation as in (b) is the only possibility when m is even, this establishes uniqueness. 2

Examples 4.23. We consider a Hamiltonian triple V =(R2n,ω,T), where ω is the usual symplectic structure, and we employ the usual basis to identify operators with matrices. In particular, we view T as the matrix H indicated in each of the following three examples. (a) (n = 2) The Hamiltonian matrix   11323  12334 H =   28 15 11 12  15 16 3 3 76 R. C. Churchill and M. Kummer A (2) 0 has complex Jordan form B = 2i ;thusm = 1 and, by Proposition 0 A2i(2) 4.15, V is indecomposable. Because m is odd the best we can do with the methods developed thus far is to construct a symplectic matrix which conjugates H in such a way as to reveal an invariant Lagrangian plane. We begin with Step II of Algorithm 4.16. Step II: We have B = QHQ1, where   15 1 17 15 1 17  3+ 2 i 2 + 4 i 3 2 i 2 4 i     15 3 15 3   2 +3i 4 i 2 3i 4 i  Q 1 =   .  87 59 87 59   2 4 i 2 4 i  75 57 45 75 57 45 2 2 i 2 i 2 + 2 i 2 i

1 17 3 59 45 Step III: We rst choose e =(2 4 i, 4 i, 4 i, 2√i) , but to achieve the normalization√ ωC((NC)e, e) = 2 we must scalar multiply by 2 87/87, obtaining e = 87(1/87 17i/174,i/58, 59i/174, 15i/29) . Step IV: This step is skipped in this example. Step V: We have ω(e, e)=59√ i/87, hence = 59i/(2 87), = 59i/(4 87), and our 179 217 59 121 59 1121 3745 new e is therefore e + Ne = 87( 10092 2523 i, 5046 10092 i, 348 i, 10092 + 10092 i) . According to Theorem 4.17(a) a basis for a T -invariant Lagrangian plane in R4 is given by {NeR,NeI }, which we note can be obtained from column three of Q1 by applying the rst normalization√ (the second has no eect) and taking real and imaginary parts. Specically, Ne = 87(2/29 5i/29, 5/29 2i/29, 1, 25/29+19i/29) , and so the rst two columns of the symplectic matrix   2 5  29 29 00   √  5 2   29 29 00 P 1 = 87   ,  2 5   1087 87  25 19 5 2 29 29 87 87

(this matrix was constructed using Proposition 5.2(a)) give that basis. Note that the conjugation   16 83  02 29 27     83 42   20 87 29  PHP 1 =      0002 00 20 reveals an invariant Lagrangian plane. (This example is designed to illustrate Algorithm 4.16, but it should be noted that because of the indecomposability the structure indicated for PHP1 is more easily achieved using arguments as in Example 7.20.) A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 77

(b) (n = 3) The complex Jordan form of the Hamiltonian matrix    8 297 418 24165       228 464 321 41 71 43       314 272 72 65 43 18  H =      154 1595 2061 8 228 314       1595 3071 2023 297 464 272  2061 2023 150 418 321 72

A (3) 0 is B = 3i , from which we see that V is indecomposable and m =2. 0 A3i(0) We give a quick summary of the results obtained from the algorithm. Step II: We have B = QHQ1, where   1 7 1 1 7 1  2 2 i 10 i 2 2i 2 + 2 i 10 + i 2 +2i     37 37   3+4i 6+ 2 i 36i 3 4i 6 2 i 36i     15 15   2 10i 2 34i 53i 2 +10i 2+34i 53i  Q 1 =   .  25 25   2 i 44 11i 2 i 44 11i     83 83   28 29i 2 142i 252i 28+29i 2 + 142i 252i  89 127 693 89 127 693 2 + 2 i 9 214i 2 i 2 2 i 9 214i 2 i

Using the initial statement of Theorem 4.17 one computes that sgn(V)=1. Step III: Our rst choice for e is column 6 of Q1, but scalar multiplication by 2/5is 2 necessary to achieve ωC((NC) e, e)=2i. Step IV can again be skipped. Step V: We have ωC((NC)e, e) = 176/25 ⇒ =88i/25 ⇒ =44i/25, so e is replaced 44i by e + 25 Ne. Unfortunately, for the resulting e we have ωC(e, e) = 10516i/625, and a 2629 2 second use of this step is required: e must now be replaced by e 625 N e, i.e. by   4204 6097  3125 + 3125 i     24926 10768   3125 3125 i     7073 3614   625 + 625 i  e :=   .  693   125 i     165176 71218   3125 + 3125 i  236819 119042 3125 3125 i

The symplectic basis {N 2eR,NeR,eR, eI ,NeI , N 2eI } given in Theorem 4.17(b) is 78 R. C. Churchill and M. Kummer displayed (in the same order) as the columns of the symplectic matrix   1 192 4204 6097 94 7  5 125 3125 3125 125 5     6 652 24926 10768 661 8   5 125 3125 3125 125 5     156 7073 3614 208   3 25 625 625 25 4  P 1 =    44 693   0 5 0 125 05    56 4627 165176 71218 4636 58   5 125 3125 3125 125 5  89 5138 236819 119042 6784 127 5 125 3125 3125 125 5 , and one can check that   010003      001030     000300 PHP 1 =   ,   003000     0 30 100 300010 which is an elementary Williamson block. (c) (n = 4) In the rst two examples, Step IV of Algorithm 4.16 was not needed. Here we sketch a simple example where it does play a role. The Hamiltonian matrix    05 00 0 10 0 15       50 00 10 0 15 0       00 05 0 150 0      00 5 0 15 0 0 0  H :=      00 00 0 5 0 0      00 00 5000      00 00 0 0 0 5 00 00 0 050 has Jordan form

B = QHQ1 = diag([5i, 5i, 5i, 5i, 5i, 5i, 5i, 5i]), A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 79 where   1 1  2 2 000000    1 1 3 3   2 i 2 i 00 ii2 i 2 i     1 1   00 2 2 0000    1 1 3 3   00 2 i 2 i 2 i 2 i 00 Q 1 =   .  1 1   00 00 2 2 00    1 1   00 00 2 i 2 i 00    1 1   00 00 0 0 2 2  1 1 00 00 0 0 2 i 2 i Note that m =0. 1 Step III: We choose e as column 2 of Q , but must move to Step IV since ωC(e, e)=0. 0 1 0 0 Step IV: We take e as column 6 of Q , rst noting that ωC(e , e ) = 0, and subse- 0 0 0 0 quently that ωC(e, e )=1/2. We then replace e by ie , which we rename e ,soasto 0 00 0 have a purely√ imaginary value, here i/2, for ωC(e, e ). We then introduce e = e + e and 00 set e := 2 e so as to achieve the normalization ωC(e, e)=2i. Step V: In this example this step is skipped. One can now check that a symplectic basis for the nondegenerate subspace YR of Proposition 4.17(b) is given by {eR, eI }, where √ eR = 2 (1/2, 1, 0, 3/2, 0, 1/2, 0, 0) , √ eI = 2 (0, 1/2, 0, 0, 1/2, 0, 0, 0) .

⊥ One now proceeds to the invariant nondegenerate subspace (YR) . Details are left to the reader.

5. Matrix Preliminaries Here V is a (nite-dimensional) vector space over K = R or C. It is well-known that any basis of a Lagrangian subspace of V can be extended to a symplectic basis of the ambient space (see, e.g., Weinstein (1977, p. 7)). We need to be able to compute this extension mechanically when the elements of the initial basis are expressed as column vectors relative to a symplectic basis e V . Accordingly, we must rst be able to recognize when such column vectors span a Lagrangian plane.

Proposition 5.1. Let e be a symplectic basis of V .

∈ (a) Suppose 1 k n and v1,...,v k V are represented relative to e by the columns A of a 2n k matrix F = , where A and C are n k matrices. The span W of C {v1,...,vk} is isotropic i the k k matrix A C is symmetric, which is the case i F JF is the zero matrix of MK (n). A (b) Suppose k = n in (a), that the columns of are linearly independent, and C that A C is symmetric. Moreover, suppose B,D ∈MK (n) are such that M := 80 R. C. Churchill and M. Kummer AB is symplectic. Then the columns of M provide a symplectic basis of V CD extending the basis {v1,...,vn} of the Lagrangian plane W .

Proof. Assertion (a) is an easy consequence of the identity ω(u, v)=hu, Jvi, where on the right u, v ∈ V are identied with the associated column vectors dened by e. Assertion (b) is then immediate from the transition matrix interpretation of symplectic matrices mentioned at the beginning of Section 3. 2

Proposition 5.2. Suppose a 2nn matrix F with entries in K has rank nand satises E F JF =0∈M (n). Then there is a matrix P ∈M (2n) such that PF = , where K K 0 det(E) =06 . Suppose for any such P we dene 2n n matrices D and G by (E )1 D := JP and G := ( 1 FD J + I)D; 0 2 ∈M then the matrix M =(FG) K (2n) is symplectic. A In particular, suppose F is expressed in the form F = , where A, C ∈M (n). C K Then: 6 I 0 A 0 (a) for det(A) =0the choice P = gives M = 1 ; and 00 C (A ) 0 I A (C )1 (b) for det(C) =06 the choice P = gives M = . 00 C 0

Proof. The initial assertion is elementary linear algebra: some n n minor of G does not vanish. The remaining assertions are veried with straightforward calculations based on the following observations: M =(FG) is symplectic i F JF = G JG =0∈MK (n) and F JG = I ∈MK (n); the construction implies F JD = D JF = I ∈MK (n). 2

The case det(A)=det(C) = 0 of Proposition 5.2 warrants an example.

Example 5.3. For   0010   1130   1100   A 1010 F = =   C 0228 3   5  0 101  0001 0100 we have F JF = 0 and det(A)=det(C) = 0, and because rows 3, 4, 7 and 8 dene a A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 81 nonzero minor we choose   00100000   00010000   00000010   00000001 P =   . 00000000   00000000 00000000 00000000 Straightforward calculation then gives     1100 1010    1010 00 10 E =   , det(E)=1, (E ) 1 =   , 0001  00 01 0100 11 10     1 1 0 1  2 2 2  0000  3 3     2 2 0 1  0000      000 1  0001   2     1 1 1  1 1 10  0  D =   , and G =  2 2 2  . 0000  43 14 1 14     10 5 2 5  0000      1 00 0 1010  2   1 1  0010 2 0 2 0 1 002 0 The 8 8 matrix (FG) is therefore symplectic, as the reader is invited to verify.

6. A Simple Algorithm for the Case of a Lagrangian Splitting In this section K = R or C and V =(V,ω,T) is a Hamiltonian triple over K of dimension 2n. When dealing with matrices we assume the standard symplectic structure ω on the space K2n of column vectors of length 2n, i.e. ω(x, y)=hx, Jyi, where h, i is the usual (non Hermitian) inner product. The existence of a T -invariant Lagrangian splitting V = W Y reduces the computation of Williamson normal forms to a triviality. Moreover, in this case one is not restricted to indecomposable triples.

Proposition 6.1. Assume F and G are 2n n matrices having entries in K with columns spanning complementary Lagrangian subspaces of K2n, i.e. suppose F JF = 1 G JG =0and Q := (FG) ∈MK (2n) is nonsingular. Then the following statements hold.

(a) L := F JG is nonsingular. (b) P 1 := (FGL1) is symplectic. (c) Suppose H ∈M (2n) is Hamiltonian and K A 0 QHQ 1 = ,A,B∈M (n). 0 B K 82 R. C. Churchill and M. Kummer

Then A 0 B = L 1A L and PHP 1 = . 0 A

Proof. F 0 L (a) By (JF JG)= . G L 0 F 0 I (b) By (JF JGL1)= . (L )1G I 0 (c) By assumption we have HF = FA and HG = GB, and therefore F H = A F , which in turn gives F H JG = A L. Since H is assumed to be Hamiltonian this last identity can be written in the form F JHG = A L,orF JGB = A L,orB = L 1A L. Moreover, we then have HGL1 = GBL 1 = G(L 1A L)L 1 = GL 1A , A 0 and therefore H(FGL1)=(FGL1) . 2 0 A

The following statement includes our algorithm for computing the Williamson form of V together with a symplectic matrix which conjugates a given symplectic matrix representation to that form. The matrix Q of the statement is easily constructed with existing computer algebra packages.

Proposition 6.2. Suppose V =(V,ω,T) has symplectic matrix representation H, that H is nonsingular, and no eigenvalues are purely imaginary if K = R. Then there is a A 0 matrix Q ∈M (2n) such that QHQ1 = ∈M (2n) is in Jordan form and K 0 B K the eigenvalue of each elementary block of A has non-negative real part (hence is positive if K = R). Given any such Q construct a symplectic matrix P ∈MK (2n) as in the statement of Proposition 6.1; then PHP1 will be in Williamson form.

Proof. By Propositions 2.9 and 2.27 we can decompose V into a direct sum of T - invariant subspaces of the form Vi Vi with the property that the splitting is Lagrangian and is an eigenvalue of T |Vi i is an eigenvalue of T |Vi . The existence of a matrix Q with the required properties is then evident, whereupon the remaining assertion is immediate from Proposition 6.1. 2

Examples 6.3. Let ω denote the usual symplectic structure on R2n, and let e denote the usual basis. We consider the Hamiltonian triple V =(R2n,ω,T) with e-matrix H as given. (a) The Jordan form of the Hamiltonian matrix   73 114 198 10 11 16    59 215 208 11 4 30    190 270 503 16 30 42  H =    628 1293 1824 73 59 190   1293 1746 3390 114 215 270  1824 3390 5144 198 208 503 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 83 is   210000   020000   003000 QHQ 1 =   , 000300 000 021 000002 where   12 13 0 4 310      0 1201210    0034 14 620 1   Q =   .  12 49 68 68 12 49     36 27 204 84 45 147  24 98 238 176 42 160 In the notation of Proposition 6.3 we have     12 13 0 4 310      0 12 0   12 1 0       0034   14 620 F =   ,G=   ,  12 49 68   68 12 49   36 27 204   84 45 147  24 98 238 176 42 160 whereupon direct calculation gives   0012 L =  01249  68 0 0 and   12 13 0 3 1 1  16 4 17   49 1 3   0 12 0 100 12 17     0034 3 1 7  1  8 2 34  P =   .  12 49 68 0 1 1     49 15 21  36 27 204 16 4 17 23 7 44 24 98 238 24 2 17 Now check that P 1 is symplectic, and that   210000     020000   003000 1   PHP =   000200   0001 20 000003 84 R. C. Churchill and M. Kummer is in Williamson form. From the Williamson (or Jordan) form we see that V is decomposable. (b) The Hamiltonian matrix   125 210 870 1410     210 335 1410 2280  H =    20 30 125 210  30 50 210 335 has Jordan form   5000     0500 QHQ 1 =   , 0050 00 05 where   45 39 2 93  7 7 7 7     0 21 0 21  Q 1 =   .  3 17 10 24  7 7 7 7 6 15 6 15 7 7 7 7 In the notation of Proposition 6.2 we therefore have     45 39 2 93  7 7   7 7       0 21   021 F =   ,G=   ,  3 17   10 24  7 7 7 7 6 15 6 15 7 7 7 7 whereupon direct calculation gives   45 39 54 47    7 7 7 7    456 477  263 76   49 49  1  0 21 21 7  L = and P =   . 526 645  3 17  49 49  7 7 11 6 15 40 11 7 7 21 7 One veries easily that P 1 is symplectic, and that   5000     0500 PHP 1 =     00 50 00 05 is in Williamson form. The triple of this example is also decomposable. A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 85

   183 159 221 474 2560 267 3347 848       111 179 107 22 267 706 57 76       288 122 248 679 3347 57 4594 1165       76 30 9 276 848 76 1165 128  H :=      4 82048 183 111 288 76       844306 159 179 122 30       20 30 2 61 221 107 248 9  48 6 61 40 474 22 679 276 has real Jordan form

  A3+7i(2) I 00  0 A (2) 0 0  B := QHQ 1 =  3+7i  , 00A3+7i(2) I 00 0A3+7i(2)

where I = I2 is the 2 2 identity matrix and

  96 81 194 8 65 60 389 89  2 2 2   99 101 83 27 49 101 167   2 84 2 2 2 2 2 2    108 183 296 8 195 90 296 33   2 2 2   87 33 91   42 132 4 57 132  Q 1 =  2 2 2  .  21 3 19 3001911   2 2 2     21 27 11 13 00 11 19   2 2 2 2 2   21 1 11 59 25  2 9 29 2 19 2 2 2 9 25 11 25 59 2 6 2 5 2 19 2 2

Here

  00 195 45  4   195   00 45  L =  4   195 227 11  4 45 2 2 195 11 277 45 4 2 2 86 R. C. Churchill and M. Kummer and the symplectic matrix P 1 =(FGL1)is

  81 22 14 4  96 2 194 8 45 45 3 0     99 101 83 10 10 2 2   2 84 2 2 9 9 5 15     183 4 2   108 2 296 8 15 15 20    87 4 2 4 2  1  42 2 132 4 15 15 5 5  P =   .  21 3 4 2   2 2 19 3 15 15 00    21 27 11 13 2 4   2 2 2 2 15 15 00    21 1 2 2 4 2   2 9 29 2 45 15 15 15  9 25 2 2 2 4 2 6 2 5 15 45 15 15

Now check that

   37100000      73010000      00370000      00 730000 PHP 1 =   ,    00 00 3700      00 00 7 300      00 00 1037 00 00 01 7 3 which is the Williamson normal form. From Proposition 2.27(c) we see that V is indecomposable. (d) The Hamiltonian matrix

   8 297 418 24165       228 464 321 41 71 43       314 272 72 65 43 18  H =      154 1595 2061 8 228 314       1595 3071 2023 297 464 272  2061 2023 150 418 321 72 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 87 A (3) 0 has complex Jordan form QHQ1 = 3i , where 0 A3i(0)   1 + 7 i 10 + i 1 +2i 1 7 i 10 i 1 2i  2 2 2 2 2 2   37 37   3 4i 6 2 i 36i 3+4i 6+ 2 i 36i     15 +10i 2+34i 53i 15 10i 2 34i 53i  Q1 =  2 2  .  25 25   i 44 11i i 44 11i   2 2   83 83  28+29i 2 + 142i 252i 28 29i 2 142i 252i 89 127 693 89 127 693 2 2 i 9 214i 2 i 2 + 2 i 9 + 214i 2 i Here the analogous calculations result in P 1 =

  1 7 1 3858 10969 138 116 7 1  2 + 2 i 10 + i 2 +2i 15625 + 15625 i 625 625 i 25 25 i     37 38952 38236 397 1004 8 6   3 4i 6 2 i 36i 15625 15625 i 625 + 625 i 25 25 i     15 13766 8338 76 332 4 3   2 +10i 2+34i 53i 3125 + 3125 i 125 125 i 5 + 5 i    ,  25   2 i 44 11i 001    83 284852 221936 2172 7179 58 56   28+29i 2 + 142i 252i 15625 + 15625 i 625 625 i 25 + 25 i  89 127 693 440913 301334 2868 10726 127 89 2 2 i 9 214i 2 i 15625 15625 i 625 + 625 i 25 25 i and one can now check that   3i 10000      03i 1000      003i 000 PHP 1 =   ,    000 3i 00      000 1 3i 0  000 01 3i which is the complex Williamson form. Once again we have indecomposability.

7. Algorithms and Examples for the Remaining Cases Here V =(V,ω,T) is an indecomposable 2n-dimensional Hamiltonian triple over K = R or C and H ∈MK (2n) is a symplectic matrix representation of V. The standard symplectic structure is assumed on K2n, i.e. the symplectic form applied to column vectors x, y is hx, Jyi, where h, i is the usual inner product and J is the canonical matrix. We will assume the Jordan form characterizations of indecomposables given in Propo- sitions 2.9 and 2.27 and explain in each case how one can construct a symplectic matrix conjugating H to Williamson form. The proofs behind the algorithms are established in the remaining sections of the paper. There are always three steps: rst one constructs a 88 R. C. Churchill and M. Kummer symplectic matrix conjugating H to the “pre-normal form” AS Hˆ = , (7.1) 0 A wherein A is an elementary Jordan block and S is symmetric; then one “normalizes” S (if necessary); then one constructs a symplectic matrix conjugating the correspondingly normalized Hˆ to the Williamson normal form. (The Lie series method will eventually be seen to justify the methods we oer for the third step, although in some instances the results are not dicult to prove directly.)

the first nilpotent case: odd height

1 The Jordan form of H is A0 = A0(2n), and if Q ∈MK is such that QHQ = A0 then the rst n columns of Q1 will span a Lagrangian subspace of K2n. By using these columns as the matrix F of Proposition 5.2 the resulting symplectic matrix P 1 (i.e. the matrix M of that proposition) will then conjugate H to pre-normal form, i.e. 1 Hˆ := MHM will be as in (7.1) with A = A0(n). Moreover, the nn-entry snn of the symmetric matrix S ∈MK (n) will be nonzero, and in the real case this entity can be used to determine the sign characteristic through the formula

n1 sgn(V)=(1) (snn/|snn|). (7.2) By choosing the appropriate nonzero constant k ∈ K and conjugating the matrix Hˆ by the symplectic matrix kI 0 Pˆ = , (7.3) 0 k1I i.e. by computing H˜ := PˆHˆ Pˆ1, (7.4) we may then achieve the normalization n 1ifK = R s = , (7.5) nn 1ifK = C in which case (7.2) reduces to

n1 sgn(V)=(1) snn. (7.6)

Example 7.7. The Hamiltonian matrix   157 68 327 13  4 7 16 4   5 13   7 1  H =  2 4   31 157  75 2 4 7 31 5 67 5 2 2 8 2 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 89 has Jordan form QHQ1 = A (4), where 0   2 1 157 1  2 4  0 2 70 Q 1 :=   . 4075 0  3 31 1 2 2 0 By Proposition 5.2(a) (and relabeling M in that statement as P 1) the symplectic matrix   1 2 2 00   1  0 200 P :=  1  402 0 3 1 1 1 2 8 2 is then such that Hˆ := PHP1 will be in the form (7.1). Indeed, direct calculation gives   75 7  01 16 8     7 1   00 8 4  Hˆ =   ,    00 0 0 0010 and since n = 2 we conclude from (7.2) that sgn(V) = 1. By choosing k = 2 in the matrix ˆ P we can then achieve the normalization  75 7  01 4 2     7   00 2 1  H˜ = PˆHˆ Pˆ 1 =   .    0000 0010

For the odd height nilpotent case under consideration an algorithm for converting the matrix Hˆ of (7.1) to Williamson form is provided by the following proposition.

0 Proposition 7.8. Write A0(n) as A, assume the normalization (7.5), and let S ∈ MK (n) denote the symmetric matrix with entry snn in the nn position and 0’s elsewhere. (i.e. S0 has +1 or 1 in the lower right hand corner and 0 in every other position.) Then ∈M there are matrices B,T K (n) satisfying the four conditions listed below, and the BT AS0 symplectic matrix P˜ := will be such that P˜H˜ P˜1 = is the 0(B )1 0 A unique elementary Williamson block representing V:

(a) B ∈MK (n) is upper triangular with all diagonal entries equal to 1; (b) [A, B]=0, i.e. A and B commute; (c) B1T is symmetric; and (d) BS S0(B )1 = AT + TA . 90 R. C. Churchill and M. Kummer

The proof (see Corollary 10.8) involves a systematic method for constructing P˜ directly, and that method could be implemented without dicultly. But in practice, when dealing with small sized matrices, it is generally easier to construct P˜ in this indirect manner by solving (a)-(d) for the matrices B and T using undetermined coecients. Examples 7.9. (a) For the matrix   01 75 7  4 2   7  00 1  H˜ =  2  0000 0010 of Example 7.7 we have ! 75 7 00 01 4 2 0 A = ,S= 7 , and S = . 00 2 1 0 1

Writing 1 b t t B = 12 and T := 11 12 01 t21 t22 the four conditions of Proposition 7.8 then generate a set of equations for b12 and the 1 tij, e.g. the symmetry condition for B T is equivalent to

t21 t12 + b12t22 =0. 75 7 The complete system has many solutions, including b12 = t11 =0,t12 = t21 = 8 ,t22 = 2 . We conclude that the matrix  75  10 0 8  01 75 7  P˜ =  8 2  00 1 0 00 0 1 is symplectic and such that P˜H˜ P˜1 will be in Williamson form. Indeed, one easily computes that   0100   00 01 P˜H˜ P˜ 1 =   . 0000 0010 By combining this work with that of Example 7.2 one sees that the symplectic matrix   1 233 75 75  32 32 8   75 49 41 7    P := P˜PPˆ =  4 16 4 2   1  2 2 10 3 1 0 4 4 1 is such that PHP1 is the Williamson matrix given above. A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 91

(b) (See Example 2.25(a)) The Hamiltonian matrix   0100 1210   00102024    00011260   0000 0401  H =   00000000   00001000 0000 0100 0000 0 010

(which we note has the form (7.1), with s44 already normalized) has Jordan form A0(8), and Proposition 7.8 therefore applies. In this instance we have     1 21 0 000 0     2024 0 000 0 A = A (4),S=   , and S =   . 0  1260 000 0 0 401 0001 We write   1 b12 b13 b14  01b23 b24  B =   ,T=(tij), 00 1b34 0011 substitute in (a)–(d) of the proposition, and then pick a solution for the resulting equations in bij and tij (there are many solutions). Our choice is reected in the symplectic matrix   1070 1171 0 89  2  01 07 165 16 7 0   2  001014 49 23  2  BT 0001 47 03 0 P = =  2  , 0(B ) 1 00001 00 0   00000 10 0 00007 01 0 00000 70 1 and one can check that PHP1 is the Williamson form   01000000   00100000   00010000   00000001    . 00000000   00001000 0000 0100 0000 0 010

Note from (7.6) that sgn(V) = 1, in agreement with the calculation in Example 2.25(a). 92 R. C. Churchill and M. Kummer

the second nilpotent case: even height

Working in this case can be more tedious than in the previous one for the following 1 reason: when Q ∈MK is such that QHQ is in Jordan form it is not generally true that the rst (or last) n columns of Q1 span a Lagrangian subspace of K2n. In practical terms this means added work in achieving the pre-normal form ˆ A0 S H = . (7.10) 0 A0 In this last regard the following well-known result (or some analogue thereof) can be useful.

Proposition 7.11. In the notation of the previous paragraph let e and e0 ∈ K2n denote 1 ∈ the nth and 2nth columns of Q . Then there are constantsP 1,...,n1 K, with j =0 n1 nj 0 when j is even, having the property that for eˆ = e + j=1 njH e the columns of the 2n n matrix F := (Hn1e,ˆ Hn2e,...,ˆ eˆ) span a Lagrangian subspace of K2n. The nj constants j can be determined by solving the system of equations {hH e,ˆ eˆi =0}j.

The proof is straightforward. Assuming the proposition has been applied to determinee ˆ and the corresponding matrix F let P 1 be the symplectic matrix constructed from F as in Proposition 5.2 (i.e. P 1 is the matrix M of that statement). Then from the form of F it is clear that PHP1 will be in the pre-normal form (7.10).

Example 7.12. The Hamiltonian matrix   16 73 12 264 361 143    52 36 13 361 82 95     10 6 48 143 95 248  H =    0 10 41652 10  10 8 8 73 36 6  4 8412 13 48 has Jordan form 1 A0(3) 0 HJ = QHQ = , 0 A0(3) where   9727 104019 20881 11177 92419 19431  145 725 1450 145 725 1450     3154 37518 49738 1196 182 49738   145 725 725 145 725 725     1496 1443 1913 46 7 1913   29 145 145 29 145 145  Q 1 =   .    10 16 1 10 16 1     858 1913 592 1913   4 145 145 4 145 145  858 41 3826 592 141 3826 145 25 725 145 25 725 As the reader can check, neither the rst nor last three columns of Q1 span a Lagrangian A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 93 subspace, and we therefore use Proposition 7.11 with     20881 19431  1450   1450       49738   49738   725   725       1913   1913   145  0  145  e =   and e =   .      1   1       1913   1913   145   145  3826 3826 725 725

Here we have   20881 92419  1450 725 1     49738 182   725 725 1     1913 7  0  145 145 1  eˆ = e + 1He =   ,    1+161     1913 592   145 + 145 1  3826 141 725 + 25 1 nj and for this example the family of equations {hH e,ˆ eˆi =0}j reduces to the single 16638 8319 equation 201 725 = 0; hence 1 = 7250 . The resulting matrix F is seen in the rst three columns of the symplectic matrix P 1 displayed below, which is constructed from F using Proposition 5.2:   9727 57846087 346570018  145 1051250 2628125 00 0    3154 32175312 181057279   145 525625 2628125 00 0    1496 1237512 13927483   29 105125 1051250 00 0 P 1=   .  3281 70177 2928253300750 867289660000   10 725 3625 0 504237723553977 168079241184659     38088 9397049 1 214634492075 131681315000   4 3625 525625 290 24011320169237 24011320169237  858 1600399 2129479 13 5708451979210 739153268750 145 525625 181250 725 504237723553977 168079241184659 As the reader is invited to check, one has   3281 123573356859452 5982846732025 010 36250 2521188617769885 168079241184659     123573356859452 11965693464050 7328363618750  001 2521188617769885 168079241184659 168079241184659     5982846732025 7328363618750  000 168079241184659 168079241184659 0  Hˆ = PHP 1 =   .   000 0 0 0      000 100 000 0 10 94 R. C. Churchill and M. Kummer

Proposition 7.13. In the case under consideration the snn entry of the symmetric ma- ∈M trix S of (7.10) must be 0, and there must be a symmetric matrix T K (n) satisfying IT S = A0T +(A0T ) . The symplectic matrix Pˆ := will then have the property 0 I ˆ ˆ ˆ1 A0 0 that P HP = is in Williamson form. 0 A0

For the proof see Corollary 10.8. In practice the matrix T is easily determined by writing T =(tij) and solving the equations which result from comparing entries of matrix equation S = A0T +(A0T ) .

Example 7.14. Example 7.12 was designed to illustrate Proposition 7.11; a more judi- cious choice of the matrix Q1 vastly simplies the calculations. Indeed, note that   10 16 1 0 264 0   30 52 0 223 361 0     50 10 0 387 143 0  Q 1 =    00010 16 1   0 10 0 4 73 0  10 40 73 12 0

1 1 also satises QHQ = HJ , and in this case the rst three columns of Q span a Lagrangian subspace. Using Proposition 5.2(a) we construct a symplectic matrix   10 16 1 0 0 0      30 52 0 0 0 0     50 10 0 0 0 0  1   P =    000001    1 1 7  0 10 0 290 58 29 13 3 10 10 40 725 290 29 with the property that Hˆ := PHP1 has the form (7.10). Indeed,   3281 361 23  010 36250 14500 1450     361 23 2309   001 14500 725 290     23 2309   000 1450 290 0  Hˆ =   ,    000 0 0 0       000 100 000 0 10 and therefore   3281 361 23  36250 14500 1450    S =  361 23 2309  .  14500 725 290  23 2309 1450 290 0 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 95

We now set T =(tij) ∈MR(3) (assuming tij = tji) and solve the equations resulting from S = A0T +(A0T ) . Our solution (there are many) is reected in the upper right corner of the symplectic matrix   3281 361 100 0 72500 14500     3281 23  010 72500 0 1450     361 23 2309  001 14500 1450 290  Pˆ :=   ,   000100     000010 000001 and one can now verify directly that   010000     001000   ˆ ˆ ˆ1   P HP = 000000 ,     000 100 000 010 which is the Williamson normal form. We conclude that the symplectic matrix   57 2956 50691 42653 3281 0 125 18125 72500 18125 7250     1 1678 1698 9843 3281  0 58 3625 3625 7250 1450     2 953 5769 2279 63  1 5 1450 725 1450 58  P˜ := PPˆ =     00 10 10 30 50      010416 52 10  00 01 0 0 is such that PH˜ P˜1 is in Williamson normal form.

the first purely imaginary eigenvalue case: odd height

Here K = R and the real Jordan form of H is Aib(2n), where b>0, n 2iseven, and (therefore) m = n 1isodd. 1 1 If Q =(FG) ∈MR(2n) is such that QHQ = Aib(2n) then the real 2nn matrix F will satisfy the hypothesis of Proposition 5.2 and we can use that result to construct 1 a symplectic matrix P ∈MR(2n) (i.e. the matrix M of the proposition statement) such that PHP1 is in the pre-normal form ˆ Aib(n) S H = . (7.15) 0 (Aib(n)) 96 R. C. Churchill and M. Kummer

Now write the symmetric matrix S in (7.15) as (sij) and let s s := trace n 1,n 1 n 1,n . sn,n1 sn,n Then this quantity is not zero, and determines the sign characteristic through the formula sgn(V)=(1)(n/2)1ε, ε := /||. (7.16) p (It was noted above that n must be even.) Now dene k := 2/||. Conjugating Hˆ by the symplectic matrix kI 0 Pˆ = , (7.17) 0 k1I i.e. computing ˜ ˜ ˆ ˆ ˆ1 Aib S H := P HP = , (7.18) 0 (Aib) then achieves the normalization s s n 1,n 1 n 1,n = εI + ,ε∈{1, 1}, (7.19) sn,n1 snn where the sij are now assumed to be elements of the real symmetric matrix S˜.

Example 7.20. The Hamiltonian matrix   11323  12334 H =   28 15 11 12  15 16 3 3 has real Jordan form   02 10   20 01 QHQ 1 =   ,  00 02 0020 where   15 1 17  3 2 2 4     15 3   2 30 4  Q 1 =   .  87 59   2 00 4  75 57 45 2 2 0 2 Taking the rst two columns of this last matrix as the F of Proposition 5.2, and writing P 1 for the symplectic matrix M which results from (a) of that statement, we nd that   15  3 2 00    15   2 30 0 P 1 =   ,  87 4 10   2 0 87 87  75 57 10 4 2 2 87 87 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 97 and that   64 332  02 2523 7569     332 56   20 7569 841  Hˆ = PHP 1 =   .    0002 00 20 Here we have =64/2523 + 56/841=8/87, and therefore ε = 1. Moreover, since V n = 2√ we conclude from (7.16) that sgn( ) = 1. The constant k of (7.7) is here given by k = 1 87, and by direct calculation we nd that 2   16 83  02 29 87     83 42   20 87 29  H˜ = PˆHˆ Pˆ 1 =   .    0002 00 20

In particular, in the notation of (7.19) we have ! 13 83 ε = 1 and = 29 87 . 83 13 87 29

Proposition 7.21. Write A for Aib(n), assume the normalization (7.19) in the matrix S˜ of (7.18), and set   0 00    . .. .   . . .  00   S :=  000 ∈MR(n).  .   . 0 ε 0  0 00ε ∈M Then there are matrices B,T R(n) satisfying the four conditions listed below, and BT AS00 the symplectic matrix P˜ = will be such that P˜H˜ P˜1 = is 0(B )1 0 A the unique elementary Williamson block representing V:

(a) B ∈MR(n) is upper triangular with all diagonal entries equal to 1; (b) [A, B]=0, i.e. A := Aib(n) and B commute; (c) B1T is symmetric; and (d) BS S00(B )1 = AT + TA .

For the proof see Corollary 10.11. The remarks following the statement of Proposition 7.8 also apply here.

Examples 7.22. 98 R. C. Churchill and M. Kummer

(a) For the matrix H˜ which ends Example 7.20 we have ! 16 83 00 10 S = 29 87 and S = . 83 42 01 87 29 We write 1 b t t B = 12 ,T= 11 12 , 01 t21 t22 and choose the solution to the resulting equations reected in the symplectic matrix   10 83 13  174 116   13  01 0  P˜ 1 =  116  . 0010 0001

(The equations have many solutions, although b12 must vanish and T must be symmetric.) One can now check that   02 10   20 01 P˜H˜ P˜ 1 =   ,  00 02 0020 which is the Williamson normal form. A symplectic matrix conjugating the original matrix H of Example 7.20 to this Williamson matrix is thus seen to be   79 943 137 227  174 3364 10092 2523    √  59 743 13 65   348 10092 1682 3364  P˜PPˆ = 87   .  25 2 5   1 29 29 29  19 5 2 0 29 29 29 (b) For an example involving purely imaginary eigenvalues with n>2 consider the Hamiltonian matrix    02100301      20013000      00 02 0 0 14      00 201045  H = Hˆ :=   ,    00000200      00 00 2000      00 00 1002 00 00 01 20 which by examining the Jordan form is seen to be indecomposable with spectrum√ {2i, 2i}. Here = 6, hence sgn(V) = 1, and in (7.17) we must take k = 3/3. A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 99 kI 0 Conjugating by the symplectic matrix Pˆ = then results in the Hamilto- 0 k1I nian   1  02 10 0 1 0 3       20 01 1 0 0 0    1 4   00 02 0 0 3 3     1 4 5   00 20 3 0 3 3  H˜ := PˆHˆ Pˆ 1 =      00 00 0 2 0 0      00 00 2000      00 00 1002 00 00 01 20 to which Proposition 7.21 applies. In the notation of that result we have     1  02 10  01 0 3           20 01  1000 A =   ,S=      1 4   00 02  00 3 3  1 4 5 00 22 3 0 3 3 and   0000     00 0000 S =   .   00 10 00 01

Writing     1 b12 b13 b14  t11 t12 t13 t14          01b23 b24  t21 t22 t23 t24  B =   and T =       001b34  t31 t32 t33 t34 

0001 t41 t42 t43 t44

(not assumed symmetric) our solutions (there are many) for the equations in bij,tij dened by (i)–(iv) of Proposition 7.21 are reected in the columns of the symplectic 100 R. C. Churchill and M. Kummer matrix   10 1 1 7 29 1 1  3 6 18 432 8 18   1 1 5 1 1  01 6 3 432 0 3 24    00 1 0 5 1 2 1   72 6 3 6   5 1  00 0 1 0 0  P˜ =  72 6  ,   00 0 0 1 0 0 0   00 0 0 0 1 0 0    1 1  00 0 0 3 6 10 1 1 00 0 0 6 3 01 and one can check that    02100000      20010000      00 02 0 0 10      00 200001  P˜H˜ P˜ 1 =      00000200      00 00 2000      00 00 1002 00 00 01 20 is in Williamson normal form.

the second purely imaginary eigenvalue case: even height

This is the exceptional case, i.e. Baider’s methods are not used. Our treatment is patterned after that in Burgoyne and Cushman (1974) (which has dierent notational conventions) and is included only for the sake of completeness. Here K = R and the real Jordan form of H is Aib(2n), where b>0 and n>0isodd. We let H = S + N denote the Jordan decomposition of H. 1 1 Choose a complex matrix Q ∈MC(2n) such that QHQ is in complex Jordan 1 form and let e denote column n of Q . Then from N ∈Lω, nondegeneracy, and the fact that {N me,...,Ne,e,Nme,...,e}V is a basis one sees that hN me, Jei6= 0, and we may assume, by rescaling if necessary, that hN me, Jei∈{2i, 2i}. (7.23)

Proposition 7.24. 0 P There are complex constants 1,...,m1 such that for e = e + m1 mj h mj 0 0i j=0 mjN e we have N e ,Je =0for j =0,..., m 1, and the j are uniquely determined by this system of equations.

Note from N m+1 = 0 and the denition of e0 that hN me0,Je0i = hN me, Jei∈{2i, 2i}. The proof is elementary. Indeed, using the fact that N ∈Lω one sees that the system is “triangular” in the sense that the equation corresponding to m j involves only i A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 101

(and i) for 1 i j. One can therefore proceed by solving the j = 1 equation for 1, then the j = 2 equation for 2, etc.

Example 7.25. The complex Williamson form of the Hamiltonian matrix   8 297 418 24165    228 464 321 41 71 43     314 272 72 65 43 18  H =    154 1595 2061 8 228 314  1595 3071 2023 297 464 272  2061 2023 150 418 321 72 was computed in Example 6.3(d). Using the Q1 of that example together with Remark 2.23 (wherein Q now takes the place of P ) we nd that the Jordan decomposition H = S + N is given by   12 264 402 63663    216 480 360 36 75 48     318 288 60 63 48 18  S =    66 1512 2079 12 216 318  1512 3099 2172 264 480 288  2079 2172 6 402 360 60 and   20 33 16452    12 16 39 5 45    41612 250 N =   . 88 83 18 20 12 4  83 28 149 33 16 16  18 149 156 16 39 12

We initially choose e as the third column of Q1, but the normalization (7.23) requires 2 multiplying by 5 . That is, we take   1 4  5 + 5 i     7   25 i     106   5 i  e =   .  22   5 i     504   5 i  693 5 i 102 R. C. Churchill and M. Kummer

This gives   1 4 2 1 7  5 + 5 i +(4+ 5 i)1 +(5 + 5 i)2     72 12 37 6 8   5 i +( 5 5 i)1 +(5 5 i)2     106 4 68  2  5 i +(5 + 5 i)1 +( 3+4i)2  e + 1Ne+ 2N e =   ,  22 88   5 i 5 1 5i2     504 83 284 56 58   5 i +( 5 + 5 i)1 +( 5 + 5 i)2  693 18 428 89 127 5 i +( 5 5 i)1 +( 5 5 i)2 the resulting equations are

176 25 +2i1 2i1 =0, 44 176 176 | |2 25 i + 25 1 25 1 +2i2 +2i2 2i 1 =0, 44 2629 and the unique solution is 1 = 25 i and 2 = 625 . Substituting these values in the initial expression for e0 then gives   4204 6097  3125 + 3125 i     24926 10768   3125 3125 i     7073 3614  0  625 + 625 i  e =   .  693   125 i     165176 71218   3125 + 3125 i  236819 119042 3125 3125 i

Proposition 7.26. In the notation of Proposition 7.24 write e0 in terms of real and imaginary parts, i.e. e0 = eR + ieI . Then sgn(V)=hN meR,JeRi = hN meI ,JeI i. Moreover, the matrix P 1 =(N n1eR,Nn2eR,...,eR,sgn(V)eI , sgn(V)NeI ,...,(1)n1sgn(V)N n1eI ) (the indicated entries represent column vectors) is symplectic, and V 1 A0(n) sgn( )bDR(n) PHP = sgn(V)bDR(n) (A0(n)) is the elementary Williamson block representation of V.

Proof. This is easily established by direct calculation. 2

Example 7.27. For e0 as in Example 7.25 we compute easily that sgn(V)=1, and so A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 103 here   1 192 4204 6097 94 7  5 125 3125 3125 125 5   6 652 24926 10768 661 8   5 125 3125 3125 125 5     3 156 7073 3614 208 4  P 1 =  25 625 625 25  .  44 693   0 0 05  5 125   56 4627 165176 71218 4636 58  5 125 3125 3125 125 5 89 5138 236819 119042 6784 127 5 125 3125 3125 125 5 By direct calculation one can now check that   010003   001030   000300 PHP 1 =   , 003000 0 30100 300010 which is an elementary Williamson block.

We will eventually see that all but the nal algorithm of this section are straightforward applications of Baider’s methods (Baider, 1989).

8. Preliminaries on Graded Lie Algebras

Let {Lj}j0 be a family of vector spaces over a eld K of characteristic 0, let j0Lj denote their (external) direct sum, and let L =j0Lj denote their complete direct sum (e.g. see Hungerford (1974, pp. 59–60)). Elements of L are treated as (formal) innite series ` = `0+`1+`2+with `j ∈Lj. Elements ` ∈Lof the form ` =0++0+ì+0+ are homogeneous (of order i), and in such cases we simply write ` = ì, thereby viewing Li as a subspace of L. Subscripting always indicates homogeneity of the corresponding order. Similarly, we identify j0Lj with the subspace of L consisting of those ` = `0 + `1 + `2 + satisfying ì =6 0 for at most nitely many i. Throughout the section we assume L =i0Li is a graded Lie algebra, i.e. a Lie algebra over K with bracket [, ] satisfying

[Li, Lj] Li+j. (8.1)

As before, MK (n) denotes the ring of n n matrices with entries in K.

Examples 8.2. The central examples for this paper will be (b) and (c), which appeal to (a) for their introduction. Examples (d) and (e) recall the standard contexts for nonlinear normal form theory, e.g. as presented in Baider (1989), Baider and Sanders (1991, 1992), and Churchill et al. (1983). (a) Fix n>0 and for each 0 i n 1 let Ti = Ti(n, K) consist of those n n matrices M =(mpq) ∈MK (n) satisfying mpq = 0 whenever q p =6 i. (In other words, the nonzero elements of M ∈Ti, if any, can only be on the ith-“superdiagonal”.) For i n let Ti denote the n n matrix of zeros. Then for Mi ∈Ti and Mj ∈Tj the usual matrix commutator [ , ] satises [Mi,Mj]:=MiMj MjMi ∈Ti+j, and therefore denes a graded Lie algebra structure on the collection T = T (n, K):=i0Ti = i0Ti 104 R. C. Churchill and M. Kummer of upper triangular n n matrices over K. (Note that “upper triangular” includes the possibility of nonzero elements on the diagonal.) (b) Fix n 0 and let H(2n)=H(2n, K) consist of all 2n 2n matrices over K of the form AS , 0 A where A ∈T(n, K) (see Example (a)) and S ∈MK (n) is symmetric. The notation H reects the fact that all such matrices are Hamiltonian (see Section 2). For reasons of space we denote the displayed matrix by (A : S); the commutator (as in Example (a)) of (A : S), (B : T ) ∈H(2n) then becomes [(A : S), (B : T )]=([A, B]:AT +(AT ) (BS +(BS) )).

Now check that a grading is imposed on the Lie algebra H(2n) by dening Hi = Hi(2n), for i 0, as follows:

For 0 i n 1 let Hi consist of those (A :0)∈H(2n) with A ∈Ti. (Here 0 denotes the n n matrix of zeros in MK (n), and Ti = Ti(n, K) is as in Example (a).) For n i 3n 2 let Hi consist of those (0 : S) ∈H(2n) with S =(suv) satisfying suv = 0 whenever u + v =36 n i. (In other words, for n i 2n 1 the nonzero entries of S =(suv), if any, can only be on the “skew-diagonal” terminating at the (2ni, n)-entry of S, and for 2n i 3n2 they can only be on the skew-diagonal terminating at the (1, 3n 1 i)-entry of S.) For i 3n 1 let Hi denote the 2n 2n matrix of zeros.

Two elementary examples should make the ideas transparent. The grading is illustrated by writing     310 527 H = 031 : 2 36 ∈H(6) 003 762 as H = ` + ` + ` + ` + ` + ` + ` + ` , where 0 1 2 3 4 5 6 7   300 010       `0 = 030 :0 ,`1 = 001 :0 ,`2 =(0:0), 003 000          000 000 007          `3 = 0: 000 ,`4 = 0: 006 ,`5 = 0: 0 30 , 002 060 700       020 500       `6 = 0: 200 , and `7 = 0: 000 . 000 000

The special case [H1, H5] H6 of (8.1) is seen from          0 a 0 00c 0 ad + bc 0  00b  :0 , 0: 0 d 0  = 0: ad + bc 00 . 000 c 00 000

Notice that H(2n)=i0Hi = i0Hi. A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 105

(c) Let Hˆ(4n) consist of those 4n 4n real matrices Aˆ Sˆ (Aˆ : Sˆ):= , 0 Aˆ with A,ˆ Sˆ ∈MR(2n) and Sˆ symmetric, which can be constructed in the following manner from a matrix (A : S)∈H(2n) consisting only of 0’s and 1’s: each entry of 0 in (A : S) 00 is replaced with 0:=ˆ , and each entry of 1 is replaced by a nonzero 2 2 real 00 matrix. (Note that if the ij-entry of S is replaced by the 22 matrix M, then to insure the symmetry of Sˆ the ji-entry must be replaced by M .) By the entries of (Aˆ : Sˆ) ∈ Hˆ(4n) we will mean these 2 2 matrices, and the index of an entry will be that of the element of (A : S) which it replaces. As an example consider the matrix   02100301   20013000    00 02 0 014    00201045  Hˆ =(Aˆ : Sˆ)=  ,  00000200    00 002000  00 001002 00 00 01 20 which we claim is in Hˆ(8). Indeed, the corresponding matrix in H(4) is   1111  0111 (A : S)=  . 0010 0011 14 The 2, 4-entry of (Aˆ : Sˆ) is the 2 2 matrix , and the 3, 1-entry is 4 5 00 0=ˆ , etc. 00

Note that each (A : S) is associated with many (Aˆ : Sˆ), i.e. the mapping (Aˆ : Sˆ) 7→ (A : S) is not injective. Hˆ(4n) is a Lie algebra when the bracket is taken to be the usual commutator, and becomes a graded Lie algebra when for i 0weletHˆi = Hˆi(4n) consist of those (Aˆ : Sˆ) for which (A : S) ∈Hi(2n). To illustrate the grading note that the matrix Hˆ introduced above can be written Hˆ = `0 + `1 + `2 + `3 + `4, where       02 00 0010 20 00   0001  ` =   :0 ,`=   :0 , 0  00 02  1 0000 0020 0000 106 R. C. Churchill and M. Kummer       0000 0001  0000   0000 ` = 0:  ,`= 0:  , 2  0014 3 0000 00 45 1000    0300   3000 ` = 0:  . 4 0000 0000

The special case [Hˆ1, Hˆ3] Hˆ4 of (8.1) is seen from       00ab 00ef  00cd    00gh   :0 , 0:  0000 eg00 0000 fh00    2(ae + bf) ag + bh + ce + df 00   ag + bh + ce + df 2(cg + dh)00 = 0:  . 0000 0000

(d) Let K = R or C, let x1,...,xn,y1,...,yn be indeterminants, and for each i 0 let Hamlt i = Hamlt i(2n) denote the space of homogeneous polynomials in K[x1,...,xn,y1, ...,yn] of degree i + 2. Then for Hi ∈ Hamlt i and Hj ∈ Hamlt j the Poisson bracket { , } satises {Hi,Hj}∈Hamlt i+j, and therefore induces a graded Lie algebra structure on Hamlt = Hamlt(2n):=i0Hamlt i. The (formal) Poisson bracket on Hamlt(2n) is also denoted { , }. (e) For K = R or C and xed n>0,i 0 let Vect i(n)=Vect i(n, K) consist of all n ∂ n vector elds pk on K in which pk is a homogeneous polynomial of degree i +1 k=1 ∂xk in the variables x1,...,xn. Then for Xi ∈ Vect i(n) and Xj ∈ Vect j(n) the usual Lie bracket [ , ] satises [Xi,Xj] ∈ Vect i+j(n), and therefore induces a graded Lie algebra structure on Vect (n)=Vect (n, K):=i0Vect i(n). The induced (formal) Lie bracket on Vect (n) is also denoted [ , ].

For any m ∈Lthe adjoint mapping ad(m):L→Lis dened by ` 7→ [m, `]. Note from (8.1) that

m ∈Li ⇒ ad(m)|Lj : Lj →Li+j. (8.3) It will be important to understand the adjoint mapping in particular instances of Exam- ples 8.2(b) and (c). Example 8.2(a) is useful for contrast.

Examples 8.4.

(a) For any ∈ K we have I ∈T0(n, K), where I ∈Mn denotes the identity matrix. Since I commutes with all n n matrices, ad(I)|Tj : Tj →Tj is trivial, i.e. ad(I):`j 7→ 0 for all j 0. (b) Let `0 =(I :0)∈H(2n), where I is as in (a). Then from Example 8.2(b) we see that

ad(`0)((B : T ))=(0:2T ).

In particular, ad(`0)|Hj : Hj →Hj is: A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 107

(i) trivial for 0 j n 1; (ii) trivial for all j 0i = 0; and (iii) surjective for all j n i =0.6

Now let   01 0 0    0010 0   . . . .   ......   . .  ∈M N =  . . . .  K (n)  ......    0 01 0 00 k and for k =1,...,n 1 let ek := (N :0)∈Hk. Then for any (0 : T ) ∈Hj we see that 0 0 ad(ek)((0 : T ))=(0:T ) ∈Hj+k, where T is obtained from T by pushing the bottom n k rows of T to the top, lling in the last k rows with zeros, and then “symmetrizing” the result by adding the transpose. For example,    00 00   . .       .. .  00 00   0 .    . .    00   . .. 00         . .. .    .  ad(ek) 0: . . .  = 0: .01 ,      .  00   ..    00 00 01   .  00 . 0 010 0 where the 1’s in the right matrix occupy the n, nk and nk, n positions. More generally, for n i 3n 2 and 1 u, v n let E(i; u, v) denote the element (0 : S) ∈Hi with 1 in the uv and vu-entries of S (in the uu-entry alone if u = v) and zeros elsewhere. (Note the symmetry E(i; u, v)=E(i; v, u).) Then (iv) ad(ek)(E(i; u, u)) = E(i + k; u k, u), and for u

For n i 3n 2 an element (0 : Sˆ) ∈ Hˆ(4n)isinHˆi i all nonzero entriess ˆuv of Sˆ satisfy u + v =3n i. Fix such a pair (u, v) with u

every other entry 0.ˆ An example might be helpful:    011 1   3 6    1 1    10  0: 6 3  = Eˆ(1)(2; 2, 2)   1 1  3 6 10 1 1 6 3 0 1 1 1 + Eˆ(3)(3; 1, 2) + Eˆ(4)(3; 1, 2) + Eˆ(2)(4; 1, 1). 3 6 Now let    Eˆ(4) 0ˆ 0ˆ 0ˆ  (4)    0ˆ Eˆ 0ˆ 0ˆ    . . .    . ˆ .. .   ∈ Hˆ `0 :=  . 0 .  :0 0  .   . Eˆ(4) 0ˆ 0ˆ 0ˆ Eˆ(4) and check the following observations.

(j) (i) {Eˆ (i; u, v)}j,uv is a basis of Hˆi for n i 3n 2. (ii) ad(`0)(Hˆn) consists of those (0 : Sˆ) ∈ Hˆn of the form

  0ˆ 0ˆ  . . .   . .. .    Sˆ =  .  .  . 0ˆ 0ˆ  0ˆ 0ˆ (iii) For n i 3n 2wehave   0ifj =1 2Eˆ(3)(i; u, v)ifj =2 ad(` )(Eˆ(j)(i; u, v)) = 0  2Eˆ(2)(i; u, v)ifj =3 0ifj = 4 and u =6 v.

In particular, Hî = im(ad(`0)|Hî) ker(ad(`0)|Hî). (iv) Let    0ˆ Iˆ 0ˆ 0ˆ  ˆ ˆ ˆ    0 0 I    . . .   ∈ Hˆ `1 :=  . .. .  :0 1    0ˆ Iˆ 0ˆ 0ˆ 0ˆ ∈ Hˆ ˆ p (4) and for p =1, 2 letm ˆ p p be the result of replacing each I in `1 with E . Then for n i 3n 2 with i even we have (1) (4) ad(ˆm1)(Eˆ (i; u, u)) = Eˆ (i +1;u 1,u), and for n i 3n 2 with i oddwehave (1) (4) ad(ˆm2)(Eˆ (i; u, u)) = Eˆ (i +2;u 2,u), A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 109 with the understanding that Eˆ(4)(p; u, v) represents the zero matrix when that expression is not dened. (v) For n i<2n 1wehave (j) (j) ad(`1)(Eˆ (i; u, u)) = Eˆ (i +1;u 1,u), and for the same n and u0. (In particular, ba det(Eˆ)=a2 + b2 =6 0.) Then for    Eˆ 0ˆ 0ˆ 0ˆ     0ˆ Eˆ 0ˆ 0ˆ    . . .    . ˆ .. .   `0 =  . 0 .  :0  .   . Eˆ 0ˆ 0ˆ 0ˆ Eˆ one veries easily that

Hˆ Hˆsym ad(`0)( j)= j for all j n, (i) Hˆsym Hˆ where j consists of those elements of j with all symmetric entries. (Recall that “entries” of an element of Hˆ(4n) refer to 2 2 matrices.)

For each j 1 set

Fj := ijLi. (8.5)

Then for m ∈ Fi we see from (8.1) that ad(m)|Fj : Fj → Fi+j, and that for all k 0we k have (ad(m)) |Fj : Fj → Fki+j. It follows easily that for m ∈ F1 and any ` ∈Lthe right hand side of X∞ 1 exp(ad(m))` := (ad(m))n` (8.6) n! n=0 contributes only nite sums to each Lj, and therefore denes a linear operator exp(ad(m)) : L→L. (The characteristic zero assumption on K is needed in this paper precisely because of the factors 1/n! in (8.6). If one assumes (ad(m)))n = 0 for all n suciently large, as in the matrix examples, then one could also assume characteristic p for suciently large p.) Set := exp ad : F1 → end(L). Let Aut(L) denote the collection of Lie algebra automorphisms : L→Lsatisfying (Fj) Fj for all j 1 (“ltration preserving automorphisms”).

Proposition 8.7.

(a) For all m ∈ F1 we have (m) ∈ Aut(L). (b) The binary operation 1 1 ` m := ` + m + 2 [`, m]+ 12 [`, [`, m]] + 110 R. C. Churchill and M. Kummer

dened on F1 by the Campbell–Hausdor formula gives F1 the structure of a group, with identity 0 ∈ F1 L, and : F1 → Aut(L) the structure of a group representation.

Proof. (a) For a proof in the nite-dimensional context (which is all we require) see, e.g. Jacobson (1979, pp. 8–9); for the general case see Baider and Churchill (1988a, Proposition 4.4). (b) See Baider and Churchill (1988a, Section 2 and Proposition 4.4).2

The image G := expad(F1) Aut(L) will be called the group of inner automorphisms of L. (The terminology will be justied in Example 8.9(a).) The theory of normal forms is concerned with the linear action of G on L, or, equivalently, with the action of F1 on L induced by .

Corollary 8.8. Suppose m(j) ∈ F1 for j =1,...,k. Then for m := m(k)m(1) ∈ F1 we have exp(ad(m)) = exp(ad(m(k))) exp(ad(m(1))).

Examples 8.9. For K = R and C we give explicit descriptions of the automorphism exp(ad(m)) in each of Examples 8.2. (a) The subspace F1 T = T (n, K) consists of those upper triangular n n matrices with all zero diagonal elements; for P ∈ F1 and Q ∈T we claim that exp(ad(P ))Q = eP Qe P . Indeed, for f(t):=etP Qe tP one easily veriesP that f (n)(t)=etP (ad(P ))nQ tP tP tP ∞ 1 n n e , whereupon evaluation of e Qe = f(t)= n=0 n! (ad(P )) Q t at t =1 gives the result. (To make sense of dierentiation assume a norm has been chosen on T . Since dim(T ) < ∞ any two norms are equivalent, and the derivative is therefore norm independent.) (b) From (a) we see that the action of F1 on H(2n) induced by the representation is given by P Q = eP QeP , but a ner description proves useful. From the observation that the exponential of a Hamiltonian matrix is symplectic one sees that the exponential eB eBTˆ of P =(B : T ) must have the form eP = , where Tˆ is symmetric, and 0(eB) for Q =(A : S) we therefore have exp(ad(P ))Q =(eBAeB : eB(S (ATˆ +(ATˆ) ))(eB) ). In particular, exp(ad((B : 0))(A : S)=(eBAeB : eBS(eB) ), and, since T = Tˆ when B =0, exp(ad((0 : T ))(A : S)=(A : S (AT +(AT ) )).

(c) The F1-action on Hˆ(4n) is the straightforward analogue of that in (b). (d) If we replace the Poisson bracket { , } on L = Hamil(2n) by the negative { , } the result remains a graded Lie algebra which we again denote by L. This modication 1 guarantees that for K ∈ F1 and H ∈ Hamil(2n)wehaveexp(ad(K))(H)=H ϕ , A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 111

t where ϕ is the (formal) ow of the (formal) Hamiltonian vector eld XK . Indeed, for f(t)=K ϕt we have f (n)(t)=(ad(K))nH(ϕt), and the assertion follows as in Example (a). (e) Similarly, if we replace the Lie bracket by its negative on L = Vect (n) then for 1 t Y ∈ F1 and X ∈ Vect (n)wehaveexp(ad(Y ))X =(ϕ ) X, where ϕ is the (formal) ow of Y . A proof is easily adapted from that given in the previous example.

Proposition 8.10. (Van der Meer (1985, Corollary 2.19, p. 31)) Suppose m ∈ F1,`∈L, and k 1. Then [m, `] ∈ Fk i exp(ad(m))` ` ∈ Fk.

Proof. The forward implication is clear from the formula

exp(ad(m))` = ` +[m, `]k + {terms in Fk+1}. To prove the reverse we argue by induction on k, with the case k = 0 being immediate 0 from exp(ad(m))` = exp(ad(`0 + `1 + )) = `0 + `1 + . Now assume the result for 0 j k, k 0, and suppose exp(ad(m))` ` ∈ Fk+1, i.e. that

exp(ad(m))` = `0 + + `k1 + `k + , (i) where the nal (but not initial) omitted terms may dier from the corresponding homogeneous terms of `. From (i) we see that exp(ad(m))` ` ∈ Fk+1 Fk, and so by induction we have [m, `] ∈ Fk. Comparing

exp(ad(m))` = `0 + + `k1 +(`k +[m, `]k)+ with (i) we conclude that [m, `]k = 0, hence that [m, `] ∈ Fk+1. 2

For each k 0 dene the k-jet of ` = `0 + `1 + ∈Lto be Jk(`):=`0 + + `k. Then an equivalent statement of Proposition 8.10 is: [m, `] ∈ Fk+1 i exp(ad(m)) xes the k-jet of `.

9. Normal Forms and the Normal Form Algorithm

Again K is a eld of characteristic zero and L =j0Lj is a graded Lie algebra over K; for ease of reference we repeat the dening condition

[Li, Lj] Li+j (9.1) for a grading. Note for ` = `0 + `1 + ∈Land m = m1 + m2 + ∈F1 that 0 exp(ad(m))` = exp(ad(m))(`0 + `1 + )=`0 + `1 + , (9.2) and that k 1 and m ∈ F ⇒ k (9.3) exp(ad(m))` = Jk1(`)+(`k [`0,mk]k)+{terms in Fk+1}. For k 0 and ` ∈Llet

Ck(`):={m ∈ F1 : ad(`)(m)=[`, m] ∈ Fk+1} (9.4)

(the “kth-order centralizer” of `); from Proposition 8.10 we see that m ∈ Ck(`)i exp(ad(m)) xes Jk(`). Note from (9.1) that

Lk+1 Fk+1 Ck(`),k >0, (9.5) 112 R. C. Churchill and M. Kummer and from (9.2) that

C0(`)=F1. (9.6) Now set

Vk(`):=(ad(`)(Ck1(`))) ∩Lk,k 1, (9.7) and use ad(`)(m)=[`, m]=[`0 + `1 + ,m1 + m2 + ]= k1 i { } (9.8) i=1 (j=1[`ij,mj]i)+ terms in Fk to check that

Vk(`)=Vk(Jk1(`)),k 1, (9.9) and that

ad(`0)(Lk) Vk(`),k 1. (9.10)

Examples 9.11. (a) For     abc 0 xy     ` = 0 de ∈T(3,K) and m = 00z ∈ F1 00f 000 the bracket [`, m] of (9.8) takes the form   0(a d)x ex +(a f)y + bz [`, m]=`m m` =  00 (d f)z  . 00 0 Assume det(`)=adf =0.6 We have      0   0 0    T   C0(`)=F1 =  00  , 1 =  00  , 000 000 and from the displayed formula for [`, m] we conclude that

V1(`)=T1when a =6 d =6 f (f = a is not excluded),  000   6 V1(`)= 00  when a = d = f, 000    0 0    6 V1(`)= 000 when a = d = f, 000    000 { }   V1(`)= 0 , i.e.  000, when a = d = f. 000

Now observe that    00  T   F2 = 2 =  000 , 000 and as a consequence that m ∈ C1(`)i(d a)x =(f d)z =0.Thus A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 113

T 6 6 C1(`)=2when a =d = f(the case f = a is not excluded),  0    6 C1(`)= 000 when a = d = f, 000    00    6 C1(`)= 00  when a = d = f, and 000 C1(`)=F1 when a = d = f.

It follows that   a =6 d =6 f and f =6 a,or   a = d =6 f, or V2(`)=T2 if a =6 d = f, or ,   a = d = f and at least one of b and e is not zero V2(`)={0} otherwise.

(b) Suppose ` =(I + N : S) ∈H(2n), where ∈ K,   01 0 0  .   .   00 1 0 .   . . .  N =  . .. ..  ,   0 01 0 00 and S =(sij) ∈MK (n) is symmetric. Then ` = `0 + `1 + `n + `n+1 + , where    0 00   . .    .. .    .  `0 =(I :0),`1 =(N :0),`n = 0: .  , .00. 0 0 s    nn 0 0   . .    .. .       .  `n+1 = 0: .000 , etc.   .  .00sn1,n 0 0 sn1,n 0 Now recall that for any m =(B : T ) ∈H(2n)wehave ad(`)(m)=[`, m]=[(I + N : S), (B : T )] =([I + N,B]:(I + N)T +((I + N)T ) (BS +(BS) )), and therefore ad(`)(m)=([N,B]:2T + NT +(NT) (BS +(BS) )). (i) It follows easily that

Ck(`)={(B : T ) ∈ F1 :[N,B] ∈Tk+1},k=0, 1,...,n 1, 114 R. C. Churchill and M. Kummer

(where Tj is as in Example 8.2(a)), and that

Vk(`)={([N,B]:0)∈Hk :(B :0)∈ Ck1(`)},k=1, 2,...,n 1.

For k n it is evident that Ck(`) consists of those (B : T ) ∈ F1 satisfying [N,B]=0 and (ii) (0:2T + NT +(NT) (BS +(BS) ) ∈ Fk+1}.

To determine the spaces Vk(`) for k n it is useful to distinguish two cases. The Case =06 . In this instance

Vk(`)=Hk,k n. (iii)

Indeed, given (0 : T ) ∈Hk let m := (0 : (1/2)T ) ∈ Ck1(`) ∩Hk. Then from (i) we see that (ad(`)(m))k =(0:T ), and the result follows. The Case =0. First note that

Vn(`)={0}. (iv)

This is seen from the observation that for any m =(B : T ) ∈ F1 the matrix B ∈T(n, K) must have all zero diagonal terms, and as a consequence that the nn-entry of the matrix NT +(NT) (BS +(BS) ) in (i) must vanish. For k n a bit more work is required. For n k 2n 1 with k n even let

fk := (0 : Sk) ∈Hk, (v) where   0 0  . .   . .     1   .   .  Sk :=  . 10 row 2n k, (vi)      . .  . 1 . 0 10 0

(the number of entries in (1, 1, 1, 1,...,1, 1) is odd) and let hfkiHk denote the one-dimensional span of fk. For k>nwe claim that

Vk(`)=Hk if k 2n 1 and k n is odd, (vii) that Hk or Vk(`)= if k 2n 1 and k n is even, (viii) ad(`1)(Hk1) =6 Hk and that

Vk(`)=Hk if k>2n 1. (ix) These assertions are simple consequences of the following elementary observations:

dim(Hk)=dim(Hk1) when n3n 1 (in which case the dimension is zero); dim(Hk) and dim(Hk1) dier by one when n

One has   0 0  . .   . .   . .     a1   .  ad(`1)  .   . a2 0       . .  . a2 .  0 a1 0 0  0 0  . .   . .   . .     0 a1   .   .  =  . a1 + a2 0  .    0       . .  . a1 + a2 . 0 a1 00 0

the mapping ad(`1)|Hk1 : Hk1 →Hk is injective for n2n 1. fk, for all n

We note that both possibilities listed in (viii) can occur. For example, check that for         010 001 0 xy qrs ` =  001 : 0 10 ,m=  00z  :  rtu 000 10ε 000 suv one has     00z x 2(r y) t + x + s zu εy [`, m]= 00 0 :  t + s + x z 2uv εz  . 00 0 u εy v εz 0

Using the matrix on the right, one sees easily that f5 cannot be in V5(`)ifε = 0, and therefore V5(`)=ad(`1)(H4), whereas f5 ∈ V5(`)ifε = 1, in which case V5(`)=H5. The example also illustrates the next observation.   Suppose n r 2n 1,r n is even, and 6 ∈  ` = `1 + crfr + , where 0 = cr K. (x) Then Vj(`)=Hj for all j>r.

By (viii) it suces to prove that Vj(`)=ad(`1)(Hj1) is impossible when r

8.4(b)), thereby contradicting the fact that the decomposition Hj = ad(`1)(Hj1) hfji is orthogonal. For later examples it is useful to be more explicit about the proof of (x) in the case 1 r = n. To compress notation dene j := [[ 2 (j n) + 1]], where [[ ]] denotes the greatest integer function, and write an element    0 00   . .    . .. .    . . .       0       0 s1  `j = 0: .    .    . s2 0       0    . .  .0s2 0 . 0 0 s1 0 0 of Hj as [s1,s2,...,sj1,sj,sj1,...,s2,s1]ifj n is even, and as [s1,s2,...,sj,sj,..., s2,s1]ifj n is odd, where n0, and consider an element ` ∈ Hˆ(4n) b 0 of the form ` = `0 + `1 + `n + , where       0ˆ Iˆ 0ˆ 0ˆ ˆb 0ˆ 0ˆ  .    .    0ˆ Iˆ 0ˆ .    ˆ ˆ ˆ .       0 b 0 .    . .. ..    . . .    . . .   `0 =  ......  :0 ,`1 =   :0 .     .. ˆ    .    . 0   . 0ˆ ˆb 0ˆ .  . ˆ ˆ  0ˆ 0ˆ ˆb . 0 I 0ˆ 0ˆ 0ˆ Then from (i)–(v) of Example 8.4(c0) we see that     0ˆ 0ˆ    . .    . .. .    . . .  Vn(`)= 0: ˆ ˆ  , (i)   0 0     0ˆ 0ˆ  and that

Vj(`)=Hˆj,j>n. (ii) A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 117 ab (c00) Given a + ib ∈ C with ab =6 0 and 0 0, and ba consider an element ` ∈ Hˆ(4n) of the form ` = `0 + `1 + `n + , where       0ˆ Iˆ 0ˆ 0ˆ Eˆ 0ˆ 0ˆ .  ˆ ˆ ˆ .    .    0 I 0 .    0ˆ Eˆ 0ˆ .    . . .       . .. ..    ......    .   `0 =  . . .  :0 ,`1 =  .  :0 .  .    .. 0ˆ   . 0ˆ Eˆ 0ˆ  .    . ˆ ˆ  0ˆ 0ˆ Eˆ . 0 I 0ˆ 0ˆ 0ˆ Then from (9.10), (i) of Example 8.4(c00) and (v) of Example 8.4(c0) we see that

Vj(`)=Hˆj for all j n. (i)

∈L 0 0 Returning to generalities, suppose ` = `0+`1+ and that for each ` = `0+`1+ in the G-orbit of ` and each i 1 the vector space Li admits a direct sum decomposition 0 0 Li = Wi(Ji1(` )) Yi(Ji1(` )) (9.12) with 0 0 Wi(Ji1(` )) Vi(Ji1(` )),i 1. (9.13) Then we refer to the collection (9.12) as a splitting convention for `, or say, with (9.12) and (9.13) in mind, that a splitting convention for ` is understood. (The terminology is not standard.)

Examples 9.14.

(a) Suppose `0 ∈L0 “splits” L, i.e. suppose

Li = im(ad(`0)) ker(ad(`0)),i 1. Then from (9.10) we see that these decompositions comprise a splitting convention for any ` ∈Lof the form ` = `0 + `1 + . This particular convention is common in the context of Example 8.2(d), e.g. see Churchill et al. (1983, Section 1). (b) (From Baider (1989)) Suppose an inner product h, i is dened on i0Li such that hLi, Lji = 0 for i =6 j. (Assume h, i is Hermitian when K = C.) Then Li = 0 0 ⊥ Vi(` ) (Vi(` )) constitutes a splitting convention for any ` ∈L. (One could assume an inner product on L, but at the cost of added generality: not all inner products on i0Li extend to inner products on L.)

Now suppose ` = `0 + `1 + ∈L, that 0 k ∞, and that a splitting convention (as in (9.12) and (9.13)) is understood for `. Then ` is in normal form (w.r.t. the given splitting convention) to order k if k =0,orifk>0 and

`i ∈ Yi(Ji1(`)) for all 1 i k. (9.15) The element ` is in normal form if:

it is in normal form to order innity (i.e. to order k for all k 0); or if there is a t>0 such that Lj = {0} for all j t (as in our matrix examples) and ` is in normal form to order t. 118 R. C. Churchill and M. Kummer

In particular, when the splitting convention of Example 9.14(a) is assumed ` = `0 + `1 + is in normal form (to order k)i[`0,`j] = 0 for all j ( k). (This is the classical denition of “normal form” in the case of a nonlinear Hamiltonian function.)

Examples 9.16. Let   abc ` =  0 de ∈T(3,K). 00f (a) Assume a = d = f and the splitting convention of Example 9.14(a). Then the matrix ` is in normal form. In particular, the normal form of an element of T (n, K) need not coincide with the Jordan form. (b) Suppose a, d and f are nonzero and distinct, an inner product has been chosen, and the splitting convention of Example 9.14(b) is assumed. Then ` is in normal form i b = c = e = 0 (as can be seen from Example 9.11(a)).

Proposition 9.17. (The Normal Form Algorithm) Suppose ` ∈Ladmits a splitting convention, k 0, and ` = `0 + + `k + is in normal form to order k. W Y ∈ Write `k+1 = `k+1 + `k+1 in accordance with (9.12) and choose m Ck(`) such that W 0 [`, m]k = `k+1. Then for ` := exp(ad(m))` the following assertions hold:

0 (a) Jk(` )=Jk(`); and (b) `0 is in normal form to order k +1.

Proof. { } W Y exp(ad(m))` = ` +[m, `]k+1 + terms in Fk+2 = `0 + + `k +(`k+1 + `k+1) W Y Y ∈ `k+1 + = `0 + + `k + `k+1 + . This gives (a), and (b) then follows from `k+1 0 Yk+1(Jk(`)) = Yk+1(Jk(` )). 2

Example 9.18. Assume the inner product hM,Ni = tr(M N)onT (3, R), the corresponding splitting convention of Example 9.14(b), and let   35 7   ` = 023 , i.e. ` = `0 + `1 + `2, 00 1 where       300 05 0 007       `0 = 020 ,`1 = 003 ,`2 = 000 . 001 00 0 000

We will see that the normal form algorithm produces a matrix in T (3, R) which conjugates ` to `0. To convert ` to normal form to order 1 rst note from Example 9.11(a) that V1(`)=T1, W hence that `1 = `1 . Indeed, one can see directly from that example that `1 =[`, m(1)]1 = A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 119   05 0   (`m(1) m(1)`)1, where m(1) = 003 = `1. Now check that 00 0   0015 2   (m(1))1 = 00 0 , 00 0         100 05 0 0015 1515/2 m(1)     1     e = 010 + 00 3 + 2 00 0 = 01 3 , 001 00 0 00 0 00 1 and that   1 5 15/2 (em(1))1 = em(1) =  01 3 . 00 1 By Proposition 9.7 the element `ˆ := exp(ad(m(1))) must be in normal form to order 1, and from Example 9.11(a) we see that exp(ad(m(1)))` = em(1)èm(1). Direct computation now gives   307 `ˆ:= exp(ad(m(1)))` = em(1)èm(1) =  020 , 001 which is indeed in normal form to order 1. We next convert `ˆ to normal form, i.e. to normal form of order 2, rst noting from ˆ T ˆ ˆW Example 9.11(a) that V2(`)= 2, and hence that `2 = `2 . Here we see that for   007/2 m(2) :=  00 0 00 0 we have   007   [`, m(2)]2 = `ˆ2 = 000 , 000     107/2 107/2 em(2) =  01 0 ,em(2) =  01 0 , 00 1 00 1 and   300 exp(ad(m(2)))`ˆ= em(2)èˆ m(2) =  020 , 001 and the algorithm thus gives this last matrix as a normal form for ` (uniqueness has yet to be discussed). Moreover, the algorithm produces the conjugating matrix, i.e. we have shown that     35 7 300 P 023  P 1 =  020 , 00 1 001 120 R. C. Churchill and M. Kummer where       107/2 1515/2 154 P = em(2)em(1) =  01 0  01 3  =  013  . 00 1 00 1 00 1 Note that the explicit calculation of m(2)m(1) in Corollary 8.8 has not been necessary for obtaining these results. (This is typical.) On the other hand, if the logarithm of the conjugating matrix is desired one can simply check that   057/2 m(2) m(1) = m(2) + m(1) =  003  00 0 (see the displayed formula in Proposition 8.7(b)), and that em(2)m(1) = P .

For examples of explicit normal form calculations in (nonlinear) Hamiltonian and vector eld settings see, e.g. Baider (1989), Baider and Sanders (1991, 1992), and Churchill et al. (1983). 0 0 A splitting convention (9.12) is maximal if Wj(Jj1(` )) = Vk(Jj1(` )) for all j 1 (see (9.13)), e.g. the splitting convention of Example 9.14(b). (The terminology is not standard.)

Proposition 9.19. Suppose a splitting convention is understood for ` ∈Land 0 k< ∞. Then:

(a) the G-orbit of ` admits an element `N in normal form to order k; N (b) the lowest order nonzero term (if any) of Jk(` ) `0 is unique when im(ad(`0)|Lj) 0 0 Wj(Jj1(` )) for all 1 j k and all ` in the G-orbit of `; and N (c) Jk(` ) is unique when the splitting convention is maximal.

Assertion (a) is standard, and (b) is also well-known (e.g. see Baider and Churchill (1988b)). Assertion (c) is from Baider (1989). The hypothesis of (b) is satised by both splitting conventions of Example 9.14; that in (c) by Example 9.14(b).

Proof. (a) Since ` is automatically in normal form to order 0 we may assume k>0. Using the normal form algorithm (Proposition 9.17) we can then inductively choose elements m(1), ,m(k) ∈ F1 with the property that exp(ad(m(k)) exp(ad(m(1))` is in normal form to order k, whereupon the result becomes immediate from Corollary 8.8. N N ˆ ˆN N 6 (b) Suppose ` = `0 +ì + and ` = `0 +`j + are in the G-orbit of `, where ì = 6 ˆN ∈ N ˆN 0 = `j and w.l.o.g. i j. Then there must be an m F1 such that exp(ad(m))` = ` , whereas from (9.3) we have N N exp(ad(m))` = `0 +(ì +[m, `0]i)i + . N ˆN N ∈ N If ` and ` are in normal form to order k j i then ì Yi(Ji1(` )) while A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 121

N [m, `0]i = ad(`0)(m) ∈ im(ad(`0)|Li) Wi(Ji1(` )), the last inclusion by hypothesis. Since N N Wi(Ji1(` )) ∩ Yi(Ji1(` )) = {0} (i) N 6 N 6 N N ∈ and ì = 0 this gives ì +[m, `0]i = 0, forcing j = i as well as ì +[m, `0]i = ì ˆN N ˆN Yi(Ji1(` )). By viewing this last equality in the form ì ì =[`0,m]i and noting N N that Yi(Ji1(`ˆ )) = Yi(Ji1(` )), a second appeal to (i) then gives [`0,m]i = 0, and N ˆN ì = ì follows. (c) Suppose `N and `ˆN are in normal form to order k and in the same G-orbit, say N N N N exp(ad(m))` = `ˆ for some m ∈ F1. Then J0(` )=J0(`ˆ ) by (9.2), so we may N N N N assume k>0 and that for some 0 i

N ˆN From this point the argument of (b) is easily adapted to establish that `i+1 = `i+1, hence N N that Ji+1(` )=Ji+1(`ˆ ), and the proof is complete. 2

N Remark 9.20. Proposition 9.19 remains true when k = ∞, in which case Jk(` )in(c) is replaced by `N . The idea is to topologize L in such a way as to allow an extension of Corollary 8.8 to convergent innite products. For details see Baider (1989), particularly Proposition 1.8 and Theorem 3.7.

In some contexts successive steps involved in applying the normal form algorithm (Proposition 9.17) can be combined, as is illustrated by the following result.

Proposition 9.21. Suppose k>n>0 and ` = `0 +0++0+`n + `n+1 + ∈Lis in normal form to order (at least) n 1 w.r.t. a splitting convention satisfying

(a) im(ad(`0)|Lj)=Wj(Jj1(`)) for n j k.

Moreover, suppose

(b) [Li, Lj]={0} for n i, j k.

W Y For n j k write `j = `j +`j in accordance with the splitting convention, (use (a) to) ∈L W ∈ choose m(j) j such that ad(`0)(m(j)) = `j , and set m := m(k)+ + m(n) Fn. Then Y Y 0 exp(ad(m))` = `0 +0+ +0+`n + + `k + `k+1 + is in normal form to order (at least) k, and the same element can be obtained through k n +1 applications of the normal form algorithm. 122 R. C. Churchill and M. Kummer

Proof. For n j k hypothesis (b) gives 1 exp(ad(m(j))` = ` +[m(j),`]+ 2 [m(j), [m(j),`]] + = ` +[m(j),`]+{terms in Fk+1} = ` +[m(j),`0]j + {terms in Fk+1} W { } = ` `j + terms in Fk+1 , W which shows that successive applications of the normal form algorithm eliminates the `j terms of ` for n j k without altering any other term below k +1. Finally, observe from (b) that m(2n) m(n +1)=m(2n 1) + + m(n +1)=m, and therefore exp(ad(m)) = exp(ad(m(2n))) exp(ad(m(n + 1))). The result is now clear from the normal form algorithm. 2

Examples 9.22. kI S (a) Suppose 0 =6 k ∈ K and H = ∈H(2n, K). Then the normal form 0 kI kI 0 of H is , and the normal form algorithm, utilized as in Proposition 9.21, 0 kI I 1 S produces P := 2k as the conjugating symplectic matrix. 0 I DS (b) More generally, suppose H = ∈H(2n, K), where 0 D

D = diag([1,...,n]) ∈MK (n) 6 ∈M with i + j= 0 for all 1 i, j n and S =(sij) K (n) is symmetric. Set sij D 0 T := ∈MK (n). Then the normal form of H is and the normal i+j 0 D IT form algorithm produces P = as the conjugating symplectic matrix. 0 I Of course the fact that PHP1 has the stated form is obvious in both cases; of interest here is the observation that P results from the normal form algorithm. Note that (b) includes the well-known “nonresonance” case in which 1,...,n are assumed linearly independent over Q.

10. Applications to the Construction of Elementary Williamson Blocks In this section V =(V,ω,T) is an indecomposable 2n-dimensional Hamiltonian triple of height m over K = R or C. The minimal polynomial of V is denoted m(t). We assume the existence of T -invariant Lagrangian plane W V and a corresponding symplectic basis representation of V of the form AS H = , (10.1) 0 A where A ∈MK (n) is an elementary Jordan block and S =(sij) ∈MK (n) is symmetric. We regard H as an element of the appropriate graded Lie algebra H(2n) MK (2n)or A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 123

Hˆ(4n) MR(4n), assume the inner product hM,Ni = tr(M N) (where N = N when K = R), and use ⊥ to indicate orthocomplements w.r.t. this inner product. Assuming the splitting convention of Example 9.14(b) we will use the results of the previous two sections to compute the normal form of H. Note from Proposition 9.19(c) that this normal form is unique. Moreover, note that H = H0 + H1 + Hn + + H3n2 is already in normal form to order n 1.

Proposition 10.2. (The Case of a Non Nilpotent Lagrangian Splitting) Suppose K = R or C and m(t)=((t + )(t ))r, where 0 =6 ∈ K, in which case A 2 2 r has the form A(n) ∈MK (n), or that K = R and m(t)=((t + pt + q)(t pt + q)) , 2 where p, q ∈ R, 0

This result is of more theoretical than practical interest: it conrms that Baider’s methods will compute the Williamson form and conjugating symplectic matrix, but in this instance the techniques developed in Section 6 accomplish the same results with considerably less eort.

Proof. For the rst case the result is immediate from (iii) of Example 9.11(b); for the second use (i) of Example 9.11(c00). 2

Corollary 10.3. Under the hypotheses Proposition 10.2 there must be a symmetric matrix T ∈MK (n) with the following equivalent properties:

(a) S = AT +(AT ) ; and A 0 IT (b) PHP1 = , where P is the symplectic matrix . 0 A 0 I

In practice solving (a) for T is generally much easier than applying the normal form algorithm.

Proof. The equivalence of (a) and (b) is trivial, and the normal form algorithm produces a conjugating matrix of the form indicated in (b). Specically, from the statement immediately following (iii) of Example 9.11(b) we see that each m(j) chosen in the normal form algorithm is an element of Fn in the linear case, and the same is easily seen to hold in the case of complex eigenvalues. 2

Examples 10.4. We work out a simple example using the normal form algorithm, and then switch to the method suggested by the corollary. The ideas of Section 6 could also be applied. 124 R. C. Churchill and M. Kummer

(a) Write   4152 04 26  41 52 H =   = : ∈H(4) 0040 04 2 6 001 4 as ` + ` + ` + ` + ` , where 0 1 2 3 4 01 00 ` =(4I :0),`= :0 ,`= 0: , 0 1 00 2 0 6 etc.

From the comment immediately following (iii) of Example 9.11(b) we see that we can eliminate `2 by dividing that term by 8 and conjugating by the exponential of 00 the result. That is, for m(1) = 0: we have ` = ad(` )(m(1)), 0 6/8 2 0 and therefore 11 0 m(1) m(1) 41 5 4 H = e He = : 11 04 4 0 gives the normal form to order 3. To eliminate the (new) `3 term we again divide by 8 and conjugate by the exponential: 69 00 0 41 0 H = em(2)H e m(2) = : 16 04 00 11 0 32 gives the normal form to order 4, where m(2) = 0: 11 . 32 0 Finally, to obtain the unique normal form, i.e. the Williamson block representation, divide the (newest) `4 term by 8 and conjugate by the exponential: 000 00 41 00 H = em(3)H e m(3) = : 04 00 69 0 gives the normal form, where m(3) = 0: 128 . 00 Here one can check that m := m(3) m(2) m(1) = m(3) + m(2) + m(1), and so a symplectic matrix conjugating the original H to Williamson normal form is given by !! 69 11 em = I : 128 32 . 11 3 32 4

(b) We redo the previous example using Corollary 10.3. For H as in (a) we have 41 52 A = and S = 04 2 6 t t (in the notation of (10.1)). Writing T = 11 12 the equation S = AT +(AT ) t12 t22 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 125 reduces to a simple linear system with unique solutions t11 =69/128,t12 =11/32 and t22 = 3/4, and a symplectic matrix conjugating H to Williamson normal form is therefore given by   10 69 11  128 32   11 3  01  P =  32 4  . 00 1 0 00 0 1

For the remaining (unexceptional) cases minor normalizations in (10.1) are necessary if the goal in applying the normal form algorithm is to produce the Williamson normal form. These adjustments, already encountered in Section 7, are collected for ease of reference in the following remark.

Remark 10.5. For any 0 =6 k ∈ K conjugation of the matrix H in (10.1) by the sym- kI 0 Ak2S plectic matrix P := results in PHP1 = . When applying 0 k1I 0 A the normal form algorithm we can therefore assume one of the following normalizations, where S =(sij):

(a) snn ∈{1, 0, 1} if K = R; (b) snn ∈{0, 1} if K = C; s s (c) S˜ := n 1,n 1 n 1,n = εI + if K = R and n>1, with ε ∈ sn,n1 snn {1, 0, 1}. Specically, by expressing the initial S˜ in the form s s 10 n 1,n 1 n 1,n = 1 tr(S˜) s s 2 01 n,n 1 nn 1 2 (sn 1,n 1 sn,n) sn 1,n + 1 sn,n 1 2 ((sn 1,n 1 sn,n) we see that normalization is not required if tr(S˜q) = 0, in which case ε = 0, and | 1 ˜ |1 otherwise can be achieved by the choice k = 2 tr(S) , in which case ε = tr(S˜)/|tr(S˜)|.

Proposition 10.6. (The Nilpotent Cases) Suppose K = R or C and m(t)=tr,in which case A = A0(n). Moreover, assume S in (10.1) is normalized as in Remark 10.5(a) or (b). Then the normal form algorithm (Proposition 9.17) applied to H ∈H(2n) results in the following unique normal form in each of the two indicated cases.

(a) For m odd one obtains the elementary Williamson block A0(n) sgn(V)BK (n) , 0 (A0(n))

where sgn(V):=1if K = C. Moreover, m is odd ⇔ snn =06 ⇔ m =2n 1, and n1 when this is the case and K = R we have sgn(V)=(1) snn. 126 R. C. Churchill and M. Kummer

(b) For m even one obtains the elementary Williamson block A0(n)0 . 0 (A0(n))

Moreover, m is even ⇔ snn =0⇔ m = n 1 and n is odd.

Keep in mind that that we are assuming the indecomposability of V.

Proof. Formula references in this proof, e.g. (viii), refer to Example 9.11(b). The element fk ∈Hk is as in (v). ⊥ Note that Hn = snnfn, and by (iv) that (Vn(H)) = Hn = hfni. For this nilpotent case the given H is thus in normal form to order n. If snn =6 0 we see from (x) and the uniqueness of Williamson blocks that the normal form must be the matrix in (a). Otherwise by (x) it must have the form H˜ =(A0(n): 0) + cfr + , where c ∈ K and n

Example 10.7. We illustrate the result for the case of odd m; the other case in handled in a completely analogous way. Specically, we revisit Example 7.9(a), i.e. we consider the nilpotent Hamiltonian matrix   01 75 7  4 2  00 7 1  75 7  2  4 2 H =   = A0(2) : 7 . 0000 2 1 0010 (In that example H was written as H˜ .) Note that as an element of the graded Lie algebra H(4) we can write H = `1 + `2 + `3 + `4, where 01 00 ` = :0 ,`= 0: , 1 00 2 0 1 7 75 0 2 4 0 `3 = 0: 7 , and `r = 0: . 2 0 00

We know from Proposition 10.6 that the normal form is `1 +`2; here we quickly indicate how the remaining terms can be eliminated using the normal form algorithm.

To eliminate `3 simply conjugate H by the exponential of 0 7 m(1) = 2 :0 . 00 (The choice is evident from the proof of (x) in Example 9.11(b).) By direct calcu- A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 127

lation one then obtains 31 0 H = em(1)He m(1) = A (2) : . 2 0 0 1 31 0 2 To nish conjugate H2 by the exponential of m(2) = 31 :0 . 2 0

The symplectic matrix   7 217 31  2 4 2   31  00 0  P = em(2)em(1) =  2  00 1 0 7 00 2 0 is therefore such that PHP1 is in Williamson normal form.

0 Corollary 10.8. Assume the hypotheses of Proposition 10.6 and let S ∈MK (n) denote the symmetric matrix with entry snn of S in the nn position and 0’s elsewhere. (That is, S0 has0or1 in the lower right corner and 0 in every other position.) Then the following statements hold. For m odd (⇔ snn =0)6 there are matrices B,T ∈MK (n) satisfying the four properties listed in (a), or, equivalently, the property stated in (b).

(a) (i) B ∈T(n, K) (i.e. B is upper triangular) and all diagonal entries are 1; (ii) [A, B]=0; (iii) B1T is symmetric; and 0 1 (iv) BS S (B ) = AT + TA . 0 1 AS ∈ (b) PHP is the Williamson normal form of H, where P G is the 0 A BT symplectic matrix . 0(B )1.

Alternatively, for m even (⇔ snn =0)there is a symmetric matrix T ∈MK (n) as in the statement of Corollary 10.3.

Propositions 7.8 and 7.13 are immediate consequences, and examples of the use of this corollary follow those results.

Proof. From the discussion within Example 9.11(b) we have either Hj = ad(`1)(Hj1) or Hj = ad(`1)(Hj1) hfji for each j n. Moreover, from the comments leading to (xiii) of that example we see that the m(j) chosen in an application of the normal form algorithm will be from F1 (rather than Fn) if some term of the matrix being normalized involves fj. But there can only be such involvement if m is even. Indeed, for m odd we have snn = 0, hence fn plays no role, and if some fr for r>ncontributes to H then as in the proof of Proposition 10.6 we see that the Jordan form of H would be inconsistent with that given in Proposition 2.27(d). 2 128 R. C. Churchill and M. Kummer

Proposition 10.9. (The First Purely Imaginary Eigenvalue Case) Suppose K = R, that m(t)=(t2 + b2)r, where 0

0 ⊥ Proof. From equality (i) of Example 9.11(c ) we see that (Vn(H)) is the one-dimensional span of the matrix    0ˆ 0ˆ   . .. .    . . .  ∈ Hˆ 0:  n, 0ˆ 0ˆ 0ˆ 0ˆ Iˆ ⊥ and from (ii) of the same example we have (Vj(H)) = {0} for all j>n. The normal form algorithm therefore leads to the elementary Williamson block Aib(n) CR(n) , 0 (Aib(n)) where =(1)(n/2)1ε. The assertions now follow from the uniqueness of the Williamson normal form. 2

Example 10.10. We revisit Example 7.22(b), now computing the Williamson normal form using the normal form algorithm. We assume the Hamiltonian matrix is normalized as in (7.19), i.e. we begin with the real matrix    01 0 1   3       1000 H = H1 = A2i(4) :   .   1 4  00 3 3 1 4 5 3 0 3 3

(In Example 7.22(b) this matrix is labeled H˜ .) Then H1 = `0 + `1 + `2 + `3 + `4, where       02 00 0010 20 00   0001  ` =   :0 ,`=   :0 , 0  00 02  1 0000 0020 0000    0000        0000 `2 = 0:    1 4  00 3 3 4 5 00 3 3 A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems 129    00 0 0       00   00 0 0 = 0: + 0:  = `N + `R 0 I   2 4  2 2 00 3 3 4 2 00 3 3 (as in (7.19)),   1     0003 0100   0000   1000 ` = 0:  , and ` = 0:  . 3 0000 4 0000 1 3 000 0000 N By Proposition 10.9 the Williamson normal form is `0 + `1 + `2 ; we use the normal form algorithm to construct a symplectic matrix which conjugates away the remaining terms. In the discussion the matrices Eˆ(j) and Eˆ(j)(i; u, v) are as in Example 8.4(c0).

R To eliminate `2 rst note that ! 2 4 4 2 3 3 = Eˆ(2) + Eˆ(3). 4 2 3 3 3 3 0 R Using (iii) of Example 8.4(c ) we see that we can eliminate `2 by conjugating by the symplectic matrix em(2), where 3 1 m(2) = Eˆ(2)(2; 2, 2) + Eˆ(3)(2; 2, 2). 2 6 By direct calculation one then computes that    1 1 013 6   1 1  m(2) m(2)   106 3  H2 :=e H1e = A2i(4) :  1 1  3 6 10 1 1 6 3 0 1 1 1 =(A (4):0) Eˆ(1)(2; 2, 2) + Eˆ(3)(3; 1, 2) + Eˆ(4)(3; 1, 2) + Eˆ(2)(4; 1, 1). 2i 3 6 In the same manner we see that conjugating by the exponential of m(3a):= 1 ˆ(2) 1 ˆ(3) 0 12 E (3; 1, 2) eliminates 3 E (3; 1, 2), but from (v) of Example 8.4(c )wesee that to eliminate 1 Eˆ(3; 1, 2) we must conjugate by the exponential of 6    1 00 0 6 00 1 0   m(3b)= 6  :0 . 00 0 0  00 0 0 By direct calculation we then see that    1 5 00   36 6    5 1  m(3b) m(3a) m(3a) m(3b)   6 36 00 H3 := e e H2e e = A2i(4) :     0010 00 01 130 R. C. Churchill and M. Kummer 1 5 =(A (4):0) Eˆ(1)(2; 2, 2) + Eˆ(1)(4; 1, 1) + Eˆ(2)(4; 1, 1). 2i 36 6 1 ˆ(1) 5 ˆ(2) To eliminate 36 E (4; 1, 1) and 6 E (4; 1, 1) conjugate by the exponentials of 1 ˆ(1) 5 ˆ(3) m(4a):= 72 E (3; 1, 2) and m(4b):= 24 E (3; 1, 1). For the symplectic matrix P dened by P = em(4b)em(4a)em(3b)em(3a)em(2) one now sees by direct calculation 1 that PH1P is in Williamson normal form. (Readers who carry out the calculation will nd that P is not the same conjugating matrix encountered in Example 7.22(b).)

00 Corollary 10.11. Assume the hypotheses of Proposition 10.9 and let S ∈MR(n) denote the symmetric matrix with εI , in the lower right corner and 0’s elsewhere, i.e. 2  0 00  . . .   . .. .   . .  00   S =  000 .  .  . 0 ε 0 0 00ε

Then there must be matrices B,T ∈MR(n) satisfying the four properties listed in (a), or, equivalently, the property stated in (b).

(a) (i) B ∈T(n, R) and all diagonal entries are 1; (ii) [A, B]=0; (iii) B1T is symmetric; and (iv) BS S00(B )1 = AT + TA . (b) V 1 Aib(n) sgn( )CR(n) PHP = 0 (Aib(n)) is the Williamson normal form of V, where P ∈ G is the symplectic matrix BT . 0(B )1

Once again, solving for B and T is generally easier than applying the normal form algorithm. This result establishes Proposition 7.21, and examples follow that result.

Proof. From (ii) and (iii) of Example 8.4(c0) we see that when applying the normal form algorithm the m(j) are chosen from F1 (rather than Fn/2). 2

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Originally Received 12 February 1998 Accepted 09 August 1998