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Refere 10/88/37 REFERE 10/88/37 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CANONICAL GROUP ACTIONS Jose F. Carinena INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION T IC/88/3T International Atomic Energy Agency and I. LOCALLY HAMILTONIAN DYNAMICAL SYSTEMS United nations Educational Scientific and Cultural Organization 1. Review of the theory of symplectic manifolds. In these notes (M,(*>) denotes a BYtnpleetic manifold: A/ is a finite-dimensional differen- tiate manifold and u is a nondegenerate 2-form which satiafiei du = 0 (that i&, u» e Zi(M)). INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS The dimension of M is necessarily even. For general resultB, reference textbooks are, for instance, [l] and [2]. The following well-known result characterises these manifolds completely, from the local [joint of view: THEOREM I (DARBOUX). Around each point u e M Acre is a local chart (U:ip) such that if 4 - (»\--.>4w;pi,---,p<v). th™ CANONICAL GHOUP ACTIONS ' A dpi. (1.1) Recall that the nondegeneracy of w is equivalent, by definition, to the property that the maps utu : TVM —*T^M, given by: {wu(v),v') :=r wu(v,v'} (1,2) Jose F. Carinena * have maximal rank. International Centre for Theoretical Physics, Trieste, Italy. Now, the resulting map w : I'M •—* T' M is a base-presfrving Jibered mrtp, \.t\, the following diagram commutes: TM I- 1- M —^- M Tims it itnlutes a niap]>ing beLwten sections which, with a slight abuse of notation , we will alsrj write w, Now let H £ Cno(Af), the set of infinitoty dilTercntiabl*1 functions on A/, and lot. us ronfli'ler the 1-form dli. We define the llamiltonian vector field C// assorifit«cl wilh the llamiltuiiiaii // to l>e the unique vector fteid satisfying MIRAMARE - TRIESTE March l°8fi Nol.f that with Die usual definition of contraction of a p-form u by ft vector field ,V: [i(-VH(V, l'p-,):="(-V, >',,.., VP-,) we get. from (1.2): HV,,)v = dll. PROPOSITION 1. C\Hu = 0. PHOOF: For any r» (; A1")^/) we iiave C^" = t'(-V)rfo + rf[i(.Y)n]. Applying I his lirimnU 2 idrntjt.v of (tartan 1.o iht> prcs^nt caat-, we iinrncdiately get CyHuj ^ d ff ~ 0 | Permanent address: uepartamento de Fisica Teorica, Universidad de Zaragoaa, Zaragozn, Spain. This work vaa nroDented during Professor Carinena1 r, visit to the Universidad rte Costa Rica ar> a Visiting Scholar of the ICTP Office of External Act i vi t,i an . His nssociation with Costa Rica is made po:; r, 11 -1 r> ly <*erif"roun funding from tlif Italian Government, The proof of this proposition suggests a more general definition than that of "Hamiltoni.in COROLLARY. XH is an rde-Ai in XLH, in the sense of Lie algebras. field": we Bay that a vector field X € X(M,ui) is locally Hamilioman tfi(X)u G Z'(M). This ! The POMSOTI Anaciei of two functions fTg e C°°(A/) is the function {/,g] j^iveii hy: in equivalent, by Proposition 1, to the vanishing of Cxw, If i(,Y)j E B (A/), then X is TIamiltonian. We have: By the proof of I,h« previous theorem, we have: Here u'1 is an isomorphism of real vector spaces. PROPOSITION 2. In Darhoux coordinates; and from (t-3) we get at once the expression in Darboux roordinates: _ dH a en s (1.3) opt Sq1 oq' dpi oq* dpi tJpi vq' Here and in other parts of these notes, we use the Einstein convention for sums whenever it is convenient, to simplify the notation. iicm&rk, Tlie more general concept of a Poisson structure on a manifold Ai is derived as Examples, a) The classic example is the cotangent bundle T'{Ti.N). If ?r,..., qN arc cartesian follows. Let (/V, •) be An associative algebra of functions over M and let {-.-} ho a bilinear, coordinates on It", we write the induced coordinates in T'(RN) &»((',..,,«";pi , PN). aiitisymmfct.fic operation on N X N satisfying: Then (T'Tf." ,w) with u given by (LI) is a symplectic manifold. Note that w = ~~dO, with: i) {/. {ff- i§, {h,f)) + {h, {f,g}} = 0 (Jacobi iuentity) ii) {/,!7i t/iffi) -ffz+ 01 • {/-flj} (derivation property) (M) Then we say {•,-} defines a Paisson structure on A/. It is readily verified that the Poisson bracket defined earlier is a Poisson structure, if N = CfX>(M) and • denotes the ordinary More generally, if Q is the configuration apace of a mechanical system, we define the (commutative) product of funrlions. In this case, the Jacobi identity is assured by the property 1 N Liouville 1-form on Q by (1.4); (q t ,P\, • -,PN) now denotes the coordinates of a local rfw = U. chart oST'Q induced by a local chart of Q. We then call u := —d9 the lanonual igmplcctir form on T'Q. It is immediate that iLj ~ 0 and that u is nondegenerate. Whenever u> is a 2. Symmetries. differential, we say we have an etacl symplectic manifold. In T'Q we always use this structure Diffeomorphisms between symplectk manifolds induce transformations of locally Tlamil- of exact symplectic manifold. tonifin fields with respect tow into locally Hamillonian fields with refipect to /-.w := (F~1)*LJ, lltanmk, The necessity of considering more general symplectic manifolds than T'KN or even PROPOSITION 3. Let F: M -+ A/ be a difTcomorphism. Then T'Q arises from the process of "reduction" associated with the presence of constants of motion in mechanical systems, which define hypersurfaces in phase space thai in general are not exact symplectic manifolds. where t\w JS a notation for (F~ b) The tangent bundle TQ with local coordinates {g'.f') receives a symplectic structure via the Lagrangian L(q',v')- PROOF-: Let Y e .V(Af). Then by definition of contraction wt = TT*"rS—rrftf1 A (iv1 -f -T—rr—rdtj* A tiq1. tJVx yi'J I7U' i)/}1 by definition of F+u The nondegeneracy condition may in tliis case be written : Tims it is clear that i(X)u> - 0 => t(F.A')(F*w) = 0. I We say that a difleomorphism F: jl/ —* A/ is a sytnplcctomorphism if £'*<*> — w. We have finished the preliminaries. We now introduce some of tin- main definitions of THEOREM 2. If X,Y <trr locally //airu'ftonian vector fir/i/s, then their rommurateir [A'.V] is these notes: Heuniltoni&n. i) The triple (Af,w,//) is called a Hamiltoniau dynamical system, PROor: We use the known result: i(.V)£,a- CyHX)a = i([X,Y])a, wliich is valid for any ii) The triple {M,w,T), with T € X{M) satisfying £rw = rf[i(T)w] = 0, is railed a locally form a. We then obtain : Hamiltoniau dynamical system. iii) A sympleclomorphisnn <f> which satisfies ^.// = // is called a symmetry of the llamil- tonian dynamical system (Af,jjp//); a symplectomorphism ^ which satisfies ^.F = F is calkd a symmetry of the locally Haniiltonian dynamical system (M.w.r), = -,(Y)d[i(X)u] - d[i(VW.\>] = • 2 Remark. A symmetry of (M,w,rH) is not necessarily asymmetry of (M,u, H). For example, the quotient. The image of g under X lies in Xa(M, w) if and only if X = 0. There exist conditions which assure this; one of them, obviously, is that $ be semisimple, because in this if Q = R, u = dq A dp on T*R, let // = p. Then T - *-. The symplectomorphism q (-. 5, H case [B,B)= 9 [3]. 6 p — p + r is a symmetry of (R, <fy A dp, T-), but not of (R, dq A dp, p), Let us see how this works for an exact symplectic manifold (A/, — d6). We say that the dq action of G on M is exact if *(t7 = B Vj € M. For instance, take M = T'Q. Then, Remark. Under the hypotheses of (iii), if 7 is an integral curve of I", then $ ° 7 is also an for a general symplectic action, —Cx.dS = —d(Cx.f>) — 0, which happens if and only if integral curve of V. f»o := Cx.9 e Zl{M). The action is Hamiltonian if and only if each <*„ lies in fl'(Af) because °a = HX )<^ + rf[i(A" )ff], so in general it suffices to show exactness of i(X )d$. For an exact Our interest is not centred on isolated symmetries, but rather on jroaps of symmetries. a d a action, we find that jCj.fl = 0 by a formula analogous to (1.6) or (1.7), and the action is Let G be a Lie group. A (differentiabie, left) action of G on » manifold M is a differentiate Hamiltonian. map *: G X M —. M such that: For example, suppose that a Lie group G acts on the bundle T*Q by the lifting of an i) *(e,u) = u Vu G M (e = identity of C); action of G on the base Q. Let A" = £' 7?-^ be the infinitesimal generator of a one-jiarameter 1 ) Vji, j, 6 G, Vu e Af. oq Q flfi 0 subgroup of transformations of Q\ its lifting XL to T'Q is given by A'1 = f— - JJ, — -— L We denote: t,{u) :- *(j,ti); *u(s) := *(»,")• The oritt of u under G, <?„, is the set in the induced coordinate system and satisfies £^ " = 0. G : *U(G) for any fixed u. The ijoiropj jiiijroBj) of u ie *«'(«) = {* £ *«(») = " ) II. THK COMOMENTIIM MAPPING AND LIE ALGEBRA COIIOMOLDHY Note that the <bs are diffeomorphisms and that (*,)""' = *,-'• We say that G, acting on a symplectic manifold (M,w), is a symmetry grouj/ of the 1.
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