REFER hivU:- IC/89/61 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THEORY OF SINGULAR LAGRANGIANS

Jose F. Carinena

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION

IC/89/61

1. INTRODUCTION International Atomic Energy Agency There are different approaches in Classical Mechanics and one usually speaks of New- tonian, Lagrangian and Hamiltonian Mechanics. The main guide for Newtonian mechan- and ics is the determinism principle, according to which the knowledge of the positions and United Nations Educational Scientific and Cultural Organization velocities of the points of a mechanical system at a fixed time determines their future positions and velocities [1]. The idea is then to use a system of second order differential INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS equations in normal form i- = /U*J>), (l.i) because the theorem of existence and uniqueness of the solution of such systems fits well to the deterrninism principle. In the following, for the sake of simplicity, we will restrict ourselves to the study of cases in which the functions /' do not depend on the time t. Such a study may be reduced to that of an associated Erst order system THEORY OF SINGULAR LAGRANGIANS •

(1.2)

Jose F. Carinena ** International Centre for Theoretical Physics, Trieste, Italy. and these equations can be geometrically interpreted as a set of differential equations locally determining the integral curves of a vector field

(1.3)

which is a special kind of vector field in the tangent bundle of the configuration space, to be studied later on. The introduction of the Lagrangian mechanics is based on the observation that, at least for conservative systems, the equations of motion may also be Been as the Euler equations determining the solution curves of a variational problem [1-3]; the action J Ldi, with L = T ~ V, is extremal for the actual path when compared with other fixed-endpoint paths. More specifically, if the functions /* depend only on x} and there exists a function V{x') such that /' = — Jp-, the equations of motion (1.1) reduce to the MIRAMARE - TRIESTE Euler-Lagrange equations ar j t ar \ (1-4) April 1989 dtydx'

where £ = j £r=i (*') — V(x>), and it is well known that these equations can be seen as furnishing the curves with fixed ends i'(ti) = x\, I'((J) = x\ for which the action

5= L(x',i>)dt (1.5)

is an extremal.

Submitted for publication. Permanent address: Departamento de Fi'sica Te6rica, Universidad de Zaragoza, 50009 Zaragoza, Spain.

-1- There are very many interesting advantages in the use of this "Hamilton principle" Consequently, if the function V(i',wJ) is defined by because the extremal principle does not depend on a particular choice of coordinates and moreover it may incorporate the holonomic constraints with a particular choice, leading in this way to a considerable simplification. v-,(*-f A), In fact, if we have k independent holonomic constraints 4>"{x') = 0, a = 1,..., i, the freedom in the choice of coordinates allows one to choose the last k coordinates yi by then Fi can be written yTi-t+a _ ^"»(a-J) gjjd to complete this set to form a new set of (generalized) coordinates. SV d(8V The constraint forces can be simulated [1] by the introduction of a potential and a Lagrangian for the description of the motion of a charged particle is then given by

^) (1.7) and taking the limit of the real numbers TJ° going to infinity The last k Euler-Lagrange equations only mean that the motion is on the surface defined by the constraints " = 0; where T is the kinetic energy T = Jm(v • v). and n — k Euler-Lagrange equations, where the values of the last k y's are fixed to be Notice that the Lagrangian for a system (1,1) can be nonuniquely defined. In the zero, remain. particular case of (1.7), if we change from (A, ^) to a gauge equivalent description (A' = The use of Hamilton's principle has been shown to be very efficient for attacking A + Vx, 4>' — 4> - iff), we will get a different (but gauge equivalent) Lagrangian different problems in both Classical Mechanics and Classical Field Theory. The problem of what kind of second order differential equations can be obtained as the Euler-Lagrange equations of a Lagrangian function, is known as the "inverse problem", and it has received (1-8) a lot of attention [4,5] till the last few years in which geometric conditions, substituting for the well-known Hehnholtz conditions, have been given [6]. where x(x,t) is an arbitrary but differentiable function. The inverse problem can also be considered in a more generalized sense [7], namely, The Euler-Lagrange equations in generalized coordinates the question of whether there exist functions /ij (such that det jij ^ 0) and a Lagrangian function L such that the Euler-Lagrange equations defined by L are dtyd'J 8q' (1.9)

are a set of second order differential equations or in other words, what is physically relevant is not the system (1.1) but its set of solutions, the system (1.6) being then equivalent from this point of view. (1.10) A particulary interesting case of equations (1.1) admitting a Lagrangian description is that of the motion of a particle with an electric charge q in an electromagnetic field. We recall that such equations are given by the Lorentz force (summing over repeated indices is understood), which can be expressed in normal form (1.1) if the Hessian matrix: F = ,[E+lvAB].

The electric and magnetic fields can be expressed in terms of the vector and scalar potentials A,, by means ofE = -V<^-l^,B = rotA, and therefore, with some is regular. The function L is then said to be a regular Lagrangian. The Hamiltonian vectorial calculations we find that formulation is then introduced as follows [1-3]. Since det W / 0, the Bet of equations

Pi ~ 5-rr-1 = „0 (1.12) dq •

-2- -3- and the time-evolution of a dynamical variable f(q,p) can be written, making use of can be used, by means of the implicit function theorem, to define q* as a function Hamilton's equations, as

I -&••&• -«»>• (1.17) and the correspondence from (q, q) to (q, p) BO defined is called the Legendre transfor- 2. SINGULAR LAGRANGIANS: DIRAC'S THEORY mation. We can then define a function in the phase space T of the g' and pj by If the Hessian matrix W given by (1.11) is not everywhere regular, the Lagrangian function X will be called a singular Lagrangian and for technical reasons, to be explained (1.13) later, we will assume that the rank of W is constant. As far as the Euler-Lagrange equations (1.10) are concerned, we remark that they cannot be expressed in a normal •=] form because the matrix of the coefficients of the accelerations is singular. This implies called the Hamiltonian function. And then, that the theorem for existence and uniqueness of solutions of systems of differential equations does not apply; and in fact there can be some initial conditions for which there dH exists no solution at all, for only some initial positions and velocities are compatible with such equations. Indeed, if f (q, q) denotes an eigenvector function for W with zero Qqf Oof eigenvalue, then when left-multiplying (1.10) by the row £ we obtain as a necessary condition for compatibility: Therefore, if (q'(i),pj(t)) is the curve corresponding to a solution of (1.9), the Hamil- ton equations: (2.1)

hold, and conversely, if the curve (q>(.t),pj(t)) satisfies (1-14), the curve q\t) is a solution and therefore in order to have a solution with particular initial values of q' and ?*, a set of (1.9). The Lagrangian and Hamiltonian formulations for a regular Lagrangian are in of restrictions as given by (2.1) will appear. Of course, some of these restrictions may be this way put in a one-to-one correspondence. in some cases trivial identities. Moreover, particular initial values of g' and g' tor which a solution of (1.10) exists In the phase space T it is possible define an R-bilinear map associating with a couple do not uniquely determine only one solution, but a set of possible solutions. This is in of functions / and g on T, a new function, to be denoted {/,ff}, as follows [1-3]: conflict with the determinism principle assumed in the foundations of mechanics, and it is evidence for the existence of spurious degrees of freedom, namely gauge degrees (1.15) of freedom. One may think that a way of overcoming this problem is to go to the Hanultonian formalism which was proved to be, for regular L&grangians, equivalent to the Lagrangian description. However, we soon find that if L is singular, the Legendre Such a function {f,g} is called the Poitsen bracket of both functions, and it is easy transformation is not invertible because the implicit function theorem does not work, and to check that the following properties hold: then we cannot invert relation (1.12), and consequently a Hamiltonian function of q and p only is not well defined. Indeed, the fact of X being singular means that the image of the velocity phase space under tbe Legendre transformation is not all the phase space but only a subset of it, and the coordinates ?' and momenta p< are not independent but are related by some constraint functions a(g,p) = 0, where a = 1,...,M, called by Dirac "primary constraints". The Hamiltonian function may however be well defined on such The properties (i) and (ii) show that the space of functions on ? is endowed with on infinite-dimensional real Lie algebra structure, while property (iii) is for Poisson struc- tures. The Poisson brackets of the fundamental functions q' and pj are:

-5- -4- where now {T} runs over the set of all the constraints, both primary and secondary, obtained in the different steps of the algorithm. The explicit form of such a solution is a subset, but not its expression in terms of q and p, which are not independent.The value (2.5) of the Hamiltonian function at a point of the phase space T was defined for a regular L by the value of the energy function, Ej, — )'TT — L, at the corresponding point of where U is a particular solution of the system (2.4), and for any a. -v.x'} (2.7)

If H'a is an extension to T of Ho, all the Hamiltonian functions of the form a where x' = Va" and H' = BJ + Ua. The function HT = B' + vBx° is called the total Hamiltonian and the splitting of Hj as a sum of two terms corresponds to the fact that H' is well determined and the ambiguity is in the other term. are weakly equal, i.e. they have the same value on the Bubset Mi of the phase space that The ambiguity in the dynamics given by the arbitrary functions vff means that a is the image under the Legendre transformation of the velocity phase space V, whatever physical state is not actually described by a point (q',Pi) in R2n, but on the contrary, be the choice of the functions ua. This shows that there is an ambiguity in the description different 2rt-tuples describe the same physical state. The transformations generated by of the dynamics. This ambiguity may however be reduced by taking into account that each term in vrx' cannot change the physical state, or in other words, they generate the constraints 4" must be maintained in the time evolution [8-10]. This can restrict gauge transformations. the possible choice of the functions ua. In fact, from Another very relevant concept in Dirac's theory of constraints is that of "first class functions". Let us assume that {4>T • T = 1,...,J] is a complete Bet of independent constraint functions, both primary and secondary. They will define the set called the 1 we see that if we want ^ to be zero on the subset Mi defined by the constraints, the "final constraint submanifold ' C. A function F(q,p) is said to be a first clo4s function T following set of relations if {F, 4> ) »s 0 for any r = 1,...,/. Otherwise F is second class. Notice that the Jacobi identity for Poisson brackets suffices to show that the Poisson bracket {F, G) of two first class functions is a first class function too, because are to be satisfied. Some of these equations will be trivial identities, for instance when a a constraint function $" is such that f^",H'o) as 0 and { , ^*} ss 0 V/8 = 1,..., M. fi In other cases, when {", } w 0V£ = \,...,M, but {$",H'o) ijt 0, new constraint As another example, the gauge generators xr are first class, for functions, {4>°,H'a}, called by Dirac "secondary constraints", will arise. We must then proceed in similar ways with these new secondary constraints. The other equations will ix',*T} = {v,V,*'} = v/{#',*'} + *»{vy,#T} wo, be used to determine the values of some unknown arbitrary functions ua. Actually, we look for the general solution of the inhomogeneous linear system where the first term in the right hand side vanishes because of (2.6). J (2.4) The function H' is also a first class function, because

{#',*-} = {H'o

-6- -7- b) As a second example, we consider the case of a configuration space R3 \ {0} and a Lagrangian L given by [12] and therefore

There exists one primary constraint function which implies that i{r,9,z,pr,p,,p,) = p, = 0, {Hi, 0. and a possible Hamiltonian is Notice that the time evolution of first class functions is well defined, and in this sense first class functions will play a relevant role. It is not clear why one should consider only Ha = \pl + $ jjf- +p,z + V(r) + Xp,. primary first class constraint functions os being the generators of gauge transformations. It seems better to put all the first class constraint functions on the same footing and Consistency of the constraint ^j produces a secondary constraint therefore admit that the secondary first class constraint functions also generate gauge transformations and can be incorporated in the total Hamiltonian, which will be modified & = pt = 0 to become the extended Hamiltonian HE — H' + "MT*1! where {Tr*} is a complete set of first class constraint functions. and there is no other additional constraint. Of course, both constraints are first class and the extended Hamiltonian is Examples, a) Let us assume that the configuration space is R2 and consider a Lagrangian L given by [ll] L = |i2 + I*V There is no y appearing in L and therefore L is singular. In fact, the Legendre transfor- c) Another example is the one studied by Nesterenko and Chervyakov [13] that is left mation is given by 3 to the reader as an exercise, for a Lagrangian L defined in TR by L = \x\ - x2x3 . We p* - *, Pt = 0, will study here the case in which the configuration space is R* and L is given by which is not invertible. The Hamiltonian Ho is then given by

2 2 7 Ho = Plx + P|ry - \i - \x y « \p\ - Jx y, In this case there are three primary constraints and there is a primary constraint & = P2 = 0, i(*>V>P*,P*) = P> =0. 3 = P4 ~ Xi = 0. Consistency of this constraint leads to a secondary constraint A possible Hamiltonian Ho is

Ho = \p\ - x\ - x\ fu jpj - x\ - x\. and this one to a new secondary constraint: (j>i = pz = 0. The dynamical evolution is fully undetermined, the extended Hamiltonian being Since {^i,^j} = {^1,^3} = 0, we see that a consistency requirement for ^1 gives a secondary constraint because only 4>i is a first class constraint function.

-8- -9- while neither 4>i nor " is not a first class constraint function, trying here to give a small idea of the classical Dirac's theory before going to show its its square (°)2 is a first class constraint function. However, it is not an admissible one more general and beautiful geometric counterpart. because its differential vanishes on the final constraint submanifold. Notice that, given The geometric tools to be used in such a description are those of (pre-)symplectic geometry. We will first present the theory of Hamiltonian dynamical systems, showing a set of constraint functions {r?'}*Bl, none of the first class, it w aoroetiroes possible to replace this set by an equivalent one in which first class constraint functions with respect how both the Hamiltonian and Lagrangian approaches arise as particular cases when the to the set {r}"} J_J will arise. When it is no* possible to find such a replacement, the set Lagrangian L is regular, and we will carefully analyse the case of pre-symplectic systems of constraint functions is said to be of the tteoni class. This fact is equivalent to the corresponding to the case of L being singular. non-existence of a solution of the homogeneous system AQ {'?'*> T^} »* 0, $ — 1,..., JZ (in 3. SYMPLECTIC MANIFOLDS the weak sense), and this is equivalent to det{fj',?;0} ^ 0. Let w be a two-form on a differentiable manifold M. For any u € M, let wu denote Dirac showed that it is possible to choose an equivalent set of constraint functions in the map £„: T*M ~* T*M, given by: such a way that the new set is made up of a set of first class constraint functions together with a set of constraint functions which is a set of the second class . These last constraint {um(v),v') :=uu(v,v'). functions are inconvenient when trying to quantize the system, and furthermore, we Now, the resulting map Ci: TM -t T'M is a base-preserving fibcred map, i.e., the would not obtain the same results when a second class constraint is considered before following diagram commutes: taking the Poisson bracket as when we first take Poisson bracket and then we take account of the constraint. Dirac proposed a method to cope with these problems by TM i T'M eliminating such second class constraint functions. It is based on the regularity of the g fl1 matrix C = {TJ",^*}. In fact, if Caf is the inverse matrix: CafC = 61, let us define the "Dirac bracket" [10] by M Thus it induces a mapping between sections which, with a slight abuse of notation, we (2.8) = {F,G) - will also write w. We call rent >/UBI> point u e M the rank of uM. The case in which the rank of d>, does not depend on u € M will be relevant, and when such rank and it is easy to check that it has the same properties of the Poisson bracket {•, -}, namely, coincides with the dimension of M the form u will be said to be nondegenerate, which if F, G, K are functions: is only possible when the dimension of M is even. In this case Ci can be used to identify i) {Fi + XF ,G}* = {Fi,G}' + A{F2,G}* (R-bilinearity); vector fields with 1-forms on M: we will assume that w is closed. We say that a vector 3 1 1 ii) {F, G}* + {G, F}' = 0 (skew-symmetry); field X such that u(X) 6 B (M) is EamiUonian, and when w(X) £ Z (M), X ia locally iii) {F,{G,K)'}' + {G,{K,F}'}' + {K,{F,Gy)' = 0 (Jacobi identity); Ea.mHiov.ian. The set of Hamiltonian [respectively, locally Hamiltonian] vector fields is denoted XH{M,u) (resp. XLH(M,u)]. We will use the notation X, for the vector field together with u~l{df), where / £ C°{M). iv) {T,«,F}*=0; Notice that X is locally Hamiltonian if and only if Cx<*> — 0, because w is closed, and v) {F,G}' « {F,G} when G is first class. according to the Cartan homotopy identity, These last two properties are very important because (iv) means that when evaluating a Dirac bracket with a constraint of the set of second class, we can use the constraint £xu = i(X)tL> + condition either before or after the bracket computation. THEOREM 1. [14] Let w be a nondegenerate dosed two-form in M. UX,Y are locally The time evolution of a dynamical variable may be written, taking into account the HamiltonitM vector fields, X, Y £ Xm{M, w), then [X, Y) is a HamHtonian vector Geld, property (v), as: with Hamiltonian function w(V,X).

f = {f,HE}' (2-9) PROOF: The proof is based on the following identity: at

-10- -u- Tbis form is obviously closed, and u is regular because u A u A • • • A u (n times) is Taking into account that £ vw = 0 and the homotopy identity, proportional to the volume element di1 A • • • Adx2n. The coordinates are usually denoted 1 (j ,... ,fl";Pi,-. • ,pn), and then the expression of u) becomes:

(3.1) THEOREM 2. If w is nondegenerate, the R-biJiaear map {•,}: C^iM) X C°(M) C°°(.M) defined by {F, G] = ^ui'^dF^d-^dG)) satisfies tie properties: Another very important example is a cotangent bundle T*Q. It can always be en- dowed with a canonical structure, as we will show shortly. The important point is that the closedness of u) implies that it is possible to choose appropriate coordinates in the ii) {F, {G, A'}} + {G, {K, F)} + {K, {F, G}} = 0 (Jacobi identity), even dimensional manifold M in such a way that the local expression of w is like (3.1). This is the result known as the Darboux theorem. iii) {F,Gi • G2} = {F.d} - G2 4- Gj • {F,G2} (derivation property,) THEOREM 3 (DARBOUX). Around each point u e M there is a. local chart (U,) such characterizing the Poisson structures, if and oniy ifu is closed, u> £ 1 N that if = {q ,...,q ;p1,...,pN), then

PROOF: It suffices to note that {f,g} can be written {/,?} = Xtf = -X/j to show (i) and (iii). On the other hand, (3.2)

-u[[Xf,Xl],Xk) + »(\XttXk],Xl)-w([Xl1Xk],Xf). The explicit expression of the Hamiltonian vector field Xj in the coordinates of the previous theorem is Taking into account that Xfw(Xt,Xh) = {{ff,''},/} and the identity w{\Xj,Xa\,Xk) = d u>(X{s,f}>Xi') ~ {{ff,/},'»}, and also the corresponding equations obtained by reordering ?L±2L /, g and h, we get and the system of differential equations for the determination of its integral curves are Hamilton-like equations, the function / being the Hamiltonian. The expression in these MX/tXt,Xk) = {{g,h),f} - {{f,h},g} + {{/,*},*> coordinates of the Poisson bracket reduces to the well-known one:

(3.4) and therefore the Jacobi identity foUowe if and only if dm = 0. I dpi dq' 3q* dpi' This theorem shows why the condition of u> being closed plays such a relevant role. Another remarkable consequence will be given by the Darboux theorem, to be explained 4. POISSON MANIFOLDS later. As a corollary of Theorem 1, if f,g € C°°(M), then [Xf,X,] = -X{,,/}- DEFINITION 2. A Poisson structure on a manifold P is a skew-symmetric R-biiijiear map, denoted {;•}: C°°(P) x C°°(P) -> CX(P), such that DEFINITION 1. By a. tymplcctic manifold we mean a pair (M, w) in which w is a nondege- ntrsAe closed 2-Iorm cm the manifold M. a) {/i,{/i,/»}} + ViAhJi}} + {hAfuh}} = 0 (Jacobisidentity), b) {fif2,9}=fi{fi,9} + h{fu9} (Leibniz's identity). Examples, a) If M is a two-dimensional surface, any two-form ui that does not vanish at We remark that property (a) means that C°°(P) is endowed with a real Lie algebra any point p € M, endowes M with a symplectic structure because any two-form in M is structure while property (b) means that for every function /, there exists a vector field, to closed. be denoted Xf, such that X/g = {g, /}. In a Bet of local coordinates for F, {i',..., x'}, b) A prototype of symplectic manifold >B B2B with u> the 2-forai that in coordinates is the vector field Xj is written 9 • d

-13- -12-

T and with the cyclic sum of such expressions, we will find that the first set of terms where summation on repeated indices is understood. The system of differential equations vanishes because of (4.7), and the skew-symmetry of JIJ means that the remaining terms determining its integral curves looks like Hamilton equations: cancel. An intrinsic characterization of this property above can be given as follows. The ^ = {*%/}, (4.2) expression (4.5) for {/, g} shows that its value at a point p 6 P does not depend on g but on dg, and therefore the Poisson structure furnishes a skew-symmetric twice contravariant tensor A by where the function / plays the role of Hamiltonian. Therefore the vector fields Xj k(df,dg) = {f, ), (4.8) are called Eamiltonian vector fields and / is said to be the Eamiliontan of Xj. The g coordinate expression for the "Poisson bracket" {f,g} is because the correspondence / t-» Xj suffices to give a map *{p): T£P —* TrP. The rank of ?r(p) at the point p will be called the mnk of iht Poisson structure (also called "cosymplectic structure") at the point p. Lichnerowicz [15] showed that the relation (4.5) (4.3) is given in an intrinsic way by the vanishing of the so-called Schouten bracket [16], namely [A,A] = 0. and therefore the vector field Xt is given by (4.1). Finally, let us remark that the Jacobi identity for the Poisson bracket implies that In particular, the correspondence / i—> Xf is a Lie algebra anti-homomorphism. Examples, a) The simplest example of & Poisson structure on a differentiable manifold (4.4) is that of an Abelian Poisson structure for which {/, g] = 0 for any pair of functions /-J6C-(P). from which the expression for {/, g] becomes b) Given a symplectic manifold (M,UJ), the regularity of the map & may be used to define a skew symmetric contravariant tensor A by means of

(4.5) K(*tp)=u,{u-\a),w-\p)) (4.9)

1 This shows that in order to compute the Poisson bracket of any pair of functions in where a, 0 G A (M). In particular, when a = df and & = dg we find again the expression a particular set of local coordinates, it suffices to know the Poisson brackets (4.10) = {*',*'}, (4.6) for the Poisson bracket defined by the gymplectic structure, and likewise the expres-

J sion (4.3) for the Hamiltonian vector field X,. If the expression of the symplectic form corresponding to the coordinate functions themselves. The "structure functions" J' w in a set of arbitrary coordinates i1 is given by relative to these coordinates satisfy a skew-symmetry condition J'-'(x) + P'(x) = 0, together with a condition which is a consequence of the Jacobi identity: u> = %Cijdx' A dx' (4-11) where sum in all pairs of indices is understood, the matrix C is minus the inverse of the (4.7) matrix J. Its elements are the Lagrange brackets

Conversely, it is easy to check that if a set of skew-symmetric functions on P, Jjt, satisfying (4.7) is given, the expression (4.5) will define a Poisson structure on P, because and having in mind that i2i[u>~I(rfi')] = dx' and the expression (4.3) for the field associ- ated to the function dx', we see that

jtk J L df2 Wt T" dx>

-15- -14- and this property being true whatever x> is, the matrix C must be minus the inverse of J given by (4.6). The expression (4.2) for the Poisson bracket can be replaced in this particular case of Poisson structures defined in symplectic manifolds by the elements £,£, 6 7 are given by = a, (4.16) if,9) = {*', (4.13) and consequently, if a and b are elements of 7, The point to be remarked is that if w is symplectic, the associated Poisson structure has a constant maximal rank: if {/,?} = 0 for any function g, the function / must be (4.17) constant. On the contrary, in the case of a generic Poisson structure on P, there will B The Jacobi identity for {-, •} follows from that of [-, •]. exist functions C such that {C,g} = 0 for any g € C° (P). Such functions are called Let {} aFflF^'J*^*' ( ) associated. It suffices to translate the tensor A using the identification of TrP and T*P given by A, namely where Cif are the structure constants of 7 relative to the basis {a,-} of 7. ,( ,t) = A( r(pr1(«)^(p)"1(f))- The orbits in 7* when the coadjoint action of G on 7* is considered, turn out to W U 1 be symplectic manifolds, and they are the prototype of symplectic manifolds on which The condition [A, A] = 0, equivalent to the Jacobi identity, implies that u is closed. G acts by eymplectomorphisms (sometimes G has to be replaced by a central extension). Regularity of w comes from that of A. For a more detailed explanation, see e.g. [17] and references therein. Of course, the In the general case, the rank of A is not constant. There are very general results pullback of £, on every leaf is the Hamiltonian for the fundamental vector field. concerning the structure of Poisson manifolds (P, A). An equivalence relation may be introduced by considering that two points are equivalent if they are related by an integral In the case of a constant rank Poisson structure, there is a modified version of the curve of a Hamiltonian vector field. This gives a partition of P in symplectic manifolds, Darboux theorem that tells us the local expression for such a Poisson structure. c) An important example is that of the Poisson structure of the dual space 7* of a Lie THEOREM 4. Let (P, A) be up-dimensional Poisson manifold of constant rank 2n < p. At 1 n algebra 7, that is given as follows. If / € C°°(t'), the differential (Df)z at * e 7* is each point m € P, there exist a set of local coordinates (j ,. ..,q ;pi,.. • ,pn;*i.... ,zt) a linear map (Df)x : Tzf* -» R. There is a natural identification of 7* with Tti*, and (with In + I = p) such that the Poisson brackets of the coordinate functions are in this sense (Df)z can be thought of as an element of (7*)* and therefore of 7. Let us assume that Szf denotes such an element, i.e., = l*i,*i) = 0 and therefore

'S' dq'dpi dpid^ The Poisson structure on 7* is then defined by The leaves of the symplectic foliation intersect the coordinate chart in the slices { zi = Cj,..., zi = ci ) determined by the Casimir coordinates. The proof, based on the (4.14) Erobenius theorem, can be found in [18].

In particular, if for any a e 7, the function {, £ C°°(7*) is defined by 5. HAMILTONIAN DYNAMICAL SYSTEMS {.(x) = <*,«), (4.15) DEFINITION 3. By a. Samiltonian dynamical system we mean a triple (M,w,H) such that (M,u>) is a symplectic manifold and H is a function H € C°°(M)• The associated it is quite easy to see that since dynamical vector field is the Hanultcmian vector Geld X» = ^(dH). A triple (M,i*!,r) in which T is a locally Hamilionian vector fieW will be called a locally Hamiltonian dynamical system. In both cases the expressions in Darboux coordinates for the determination of the integral curves of Xtf or T respectively, are Hamilton-like equations. In the latter case,

-16- -17-

T the Hamiltonian is a local Hamiltonian for T, i.e., a function H such that t(F)uf = dH 1 n (locally). The coordinate expression for © in coordinates (g ,..., q ;pi, ...,pn) of T*Q induced from The H&miltonian formulation of the phase space corresponds to the case in which the coordinates (s1,...,}™) in the base Q is manifold M is the cotangent bundle T'Q of the configuration space, endowed with its natural symplectic form. The case considered here is more general and actually we will 6 = p.dg'. (6-3) see that in the cases of regular Lagrangians, it is possible to find a Hamiltonian dynamical i system in the velocity phase space describing the time-evolution by means of the integral Actually 0 can be written Ba = ai(a)dq +b>(Q)dpj where a,(a) and b>(a) are functions curves of the dynamical vector field T which is determined from the tymplectic fonn UIL to be determined by and the energy EL (both depending on the function L). We will study how they are related to the Euler-Lagrange equations determined by L.

6. THE GEOMETRY OF THE TANGENT AND COTANGENT BUNDLES. Opj can Let Q be a diiferentiable manifold. The set U.eQ TtQ ^ endowed with a vector bundle structure over Q and will be denoted TQ. When Q is the configuration space from which the local expression (6.3) follows. of a mechanical system, the tangent bundle TQ is usually called velocity phase space. As a consequence of this we see that the local expression for il = -dQ is like (3.1), The differentiable structure of TQ is given as follows. If (W,v) is a local chart of Q, (2 = dq A dp^, (6.4) V>(i) — (9l>"->9n)i then the set of vector fields yr, for t running from 1 to n, gives a J paralleUzation of the portion T~ (W) of TQ. Here r is the natural projection T : TQ—*Q what shows that il is a canonical symplectic form in T'Q. Therefore the cotangent that assigns to every vector u£T,Q the corresponding point g£Q. The parallelization n bundle of a manifold Q can be endowed with such a canonical symplectic structure. gives a product structure U x R on r"^!/) and we get in this way a chart forth e bundle. In a similar way, the tangent bundle TQ is also endowed with a canonical (1,1')-tensor Similarly, the dual bundle T'Q over Q is made up from U.eo T*tQ. The natural bundle field 5, [6], called vertical endomorphism, that for a long period of time was almost projection will be denoted it and local trivializations for the bundle are obtained from unnoticed but in the last years it is playing a relevant role in the study of Lagrangian charts (U, tensors, called Liouville one-form, 0 € f\* (T'Q), In order to introduce this concept, we first remark that any vector space V is a and vertical endomorphism, S€T*(TQ), respectively. We will start by introducing the differentiable manifold in which translations enable us to identify V with the tangent Liouville one-form, 9, which provides a good example of a symplectic manifold. space TVV for every i?£V, by associating with tueV the vector at ugV given by DEFINITION 4. Tie Liouville one-form is the one-form 6 € ft*(T'Q) given by X{v, w)f = ~f(v + iw)iM>, , V). e.(u) = (6-1) We will now be concerned with the tangent space TtQ as a vector space. It may be thought of as a submanifold of TQ, the fibre over the point ?€Q. The vectors tangent where *.„ is the differential of * : T'Q—Q at a € T'tQ. to this submanifold TfQ at vtTQ are the vertical vectors at that point, namely, those Let us remark that for any oei-^Q there is a map *£ : T'wia)Q-+T'o(T'Q) defined of Ker T,V = ^(TQ). Therefore, there exists an isomorphism £ * of the linear space T,Q as the dual map of sr,o or more specifically, Q onto Vu(TQ)cr.,(rQ), which is called vertical lift [6].

DEFINITION 5. Let v be a vector v€T,Q. Tie vertical lift f : TtQ^Tv(TQ) is defined

In the particular case in which $ = a, we see that (*;a)(u) = a(x.av). Therefore the (6.5) above definition for the Liouville one-form can also be rewritten Correspondingly, if JV€JE(Q) we shall call vertical lift of X the vector e. = *:*. (6.2) defined by (6.6)

-18- -19- 2 In this and other later formulae the notation (q, v) is used instead of v for indicating Notice that S = 0 and, furthermore, in 5(f,„) = Ker 5(|p,) = V(f$r)(TQ). the point Q of the base over which v lives. AB far as the coordinate expressions are Most geometric properties of tangent bundles are expressed in terms of the vertical concerned we see that if u>eX,Q has a local expression endomorphism (see e.g. [6]). However, for the time being, we will only use it in order to define the two-form wi arising from a Lagrangian function LeC'fTQ). Such a form WL is symplectic when L is regular. Another very relevant role in the description of the geometry of the tangent bundle is going to be played by the LiouviUe vector field A. A Bimilar concept may be defined and v£T,Q, then in any vector bundle, in particular in the cotangent bundle T'Q. DEFINITION 7. The LiouviUe vector £e!d in TQ is the vector Beld AeX(TQ) generating from which we get the local expression for £*, dilations along the Sbres, namely, it is defined by

(6.7) where f £ C°°(TQ). When X€X(Q) has the local coordinate expression In a natural coordinate system for TQ, its expression is Q (6.11)

its vertical lift is given by The tangent bundle T(TQ) has two different vector bundle structures over the base TQ. One is given by the natural projection TTQ : T(TQ)-*TQ, while the other is given by the bundle map r. : T{TQ)-*TQ, the tangent map of r : TQ-*Q. The vector fieldsi n TQ are sections with respect to the first projection and those vector fields that axe also We are now ready to introduce the concept of vertical endomorpbism. a section with respect to T» are a special type of vector fields which are going to play a. DEFINITION 6. The vertical endomorphism is the fibre bundle map S : T{TQ)^>T(TQ), relevant role in the geometric formulation of the mechanics. This is so because we will Ebred over the identity at TQ, given by show that they are the geometric version of second order differential equation systems (to be shortened as S.O.D.E.). S(q, v)U = C(.T.(,,,V)U) (6-9) DEFINITION 8. A second order differential equation is * vector £eld X€X(TQ) such that TmoX = id J-Q. where UeT^v){TQ). The image under S of a section for TTQ : T(TQ)-*TQ, a vector Geld in TQ, is a new vector £eld in TQ, again. This correspondence, with an abuse of Note that if the assignment X : u—»(u,Xu) is a vector field in TQ, the condition notation, will also be denoted S. for X to be a S.O.D.E. is TtHXu = «. That property means that its local coordinate expression is We remark that in local coordinates of the bundle x = j& + fu .,£ because the function g' such that

for which we obtain the following local expression for S is determined by (6.10) g\q,v) = JW^or) = iM^X^W) = v(g<) = „'.

-20- -21- r The coordinate expression for X shows that the 2n differential equations for the deter- Note that in the first term summation on both indices t and j is understood and therefore mination of its integral curves are it may be replaced by

The matrix representation in coordinates of wj,: X(TQ)-+ f^fTQ), defined by contrac- tion, will be A -W a system that is equivalent to one of n second order differential equations which justifies ' 0 the name of S.O.D.E. given to these vector fields X. The intrinsic characterization of where we have used a matrix notation, the elements of the matrices A and W being such a S.O.D.E. may be given in terms of the vertical endomorphism: a vector field XeX(TQ) is a S.O.D.E. if and only if S(X) = A. In fact it Buffices to recall that if

t} dq'dv* dv'dq> ,- .->

*J dv'dv* then respectively. Therefore the condition for regularity of L is

det W ^0. (7.5) Therefore S(X) = A if and only if g'(q, v) = v\ that is the property characterizing the S.O.D.E. vector fields. Another ingredient of the theory is the energy function determined by a Lagrangian 7. THE GEOMETRIC FORMULATION OF THE LAGRANGIAN DYNAMICS function L. It will be defined by

1 DEFINITION 9. tf L£C°°{TQ), we will denote 0t tie one-form S^eA ^*?) given by (7.6) 01 = dLoS. Jt will be called the Euler-Poinc&re one-form. The expression for $L in natural coordinates (?*,«') of the bundle induced by a chart its local coordinate expression being in Q, is

dv' because of DEFINITION 10. if LeC°°(TQ) is reguiar, tie Hamiitonian dynamical system defined

by L is the triple (TQ,U>L,EL)- If the Lagrangian L is regular, it is easy to see that the Lagrangian vector field Ti ~) = dL{0) = 0. uniquely defined by the dynamical equation We will say that a function LeC°°(TQ) is a Lagrangian if the rank of two-form ui given i(T )u = dE , (7.7) by ui = — dBi is constant. In particular, when wj, is symplectic, the function L is said L L L to be regular. The coordinate expression for wj, is obtained from that of 9^ and it turns out to be is a second order differential equation, S(TL) = A, i.e. TL is written in coordinates as

(7.2) r = „• JL + &•• Jl, (7.8) dv'dq'

-22- -23- 8. THE LEGENDRE TRANSFORMATION

•where the 6' satisfy DEFINITION 11. If I is a function LeC^TQ), the Legendre transformation 71 is the (7.9) fibre buncUe map TL : TQ-*T*Q over the identity in Q given by This is so because the Bystem of equations determining the components of TL are

where L denotes the map i : T,Q—R given by L,{v) - L(q,v). \W 0 )\h) ~ \ Wv ) ^710' q f We remark that dL,(v) : TV(T,Q)-»R, while f" : TtQ->Tv(TQ) with q = r(v). Both v which splits into two smaller subsystems, maps are linear and therefore dLt(v)o£ is an element of T'VQ. The Legendre map TL gives us a one-form TL'Q in TQ as a pull-back of the Liouville one-form in T'Q. The point is that TL'Q reduces to Si, namely,

(8.2) Now, the regularity of W means that a = v is the only solution of (7.11.a), and if M eL = denotes the inverse matrix of W, the values of b are determined by and the two-form ut defined by the Lagraagian L is the ^"i-pull-back of the canonical symplectic form fi in T*Q. In fact, the coordinates of the point TL{q,v) in T'Q are (7.12) given by dL where aj denotes P' ' QJ (8.3) (7.13) Indeed, if v,weT Q, then dg> v The integral curves of Tt, will be determined by

(7-14) from which we Bee that and therefore the Euler-Lagrange equations hold on the projection on Q of these inte- .») = (9. J£) gral curves. These equations arise then as associated to the determination of the integral This local expression shows, by taking into account the local expression for 0, that curves of the dynamical vector field and they are not coming from any variationat prin- the ^i-pull-back of the Liouville 1-form 6 in T'Q is the one-form 6 in TQ given by ciple. L Finally, we remark that in this case of regular Lagrangians the dynamical equation eL = dios. (7.7) is equivalent to THEOREM 5. A Lagrangian /unction L is regular if and only ilTL is a local diffeomor- phism. CT9L = dL. (7.15) This is because the expression for Crfl Pvcs PROOF: Let assume that 7L is a local diffeomorphism. If a vector field X€X(TQ) is such that wL(X, Y) = 0 VY€X(TQ), then tl(FL,X, TL.Y) = 0. In a neighborhood of i(T)d9L + d[i(T)9L] = dL a point in T'Q the values of vector fields TL,Y generate the tangent space. The form n is symplectic and therefore TL,X = 0. But the transformation TL was assumed to and T being a S.O.D.E., i(T)$i = A(Z), from which the equivalence of both equations be a local diffeomorphism, i.e. TL, is regular and consequently X = 0 . The two-form (for S.O.D.E. fields T) follows. t±>L is therefore symplectic. It is also noteworthy that there may exist different regular Lagrangian functions giving Conversely if UL is nondegenerate, since Ker^I.C Kerw/, we see that TL. is regular rise either to the same Hamiltonian system according to the Definition 10 or only to the and TL will be a local diffeomorphism. | same vector field TL- These Lagrangian functions are then said to be gauge equivalent in the first case and simply equivalent in the last one.

-25- -24-

T DEFINITION 13. A presymplectic manifold is a pair (M,w) where u is a closed two-form of constant r&nJc on tie differentiate manifold M. If a is a closed one-form in M, the triplet (M,i~\a) is said to be a locally-Eamiltoniim presymplectic dynamical system. We will present later more properties concerning the relation between Keruij, and KtrTL,. More specifically, it will be shown that KerTL. is the vertical part of Kerwi,. Particular cases of such systems are the triplets (XQ,u)£,dEi) for A singular La- The matrix representation of TL. in coordinates of TQ is grangian L. In case of an almoet-regulw Lagrangian function L, a concept introduced by Gotay and Neater [19] that means that the map TL is a submersion onto its image (8.4) and the fibres TIT* {TL(v)) are connected for each v€TQ, the energy function EL may ; be shown to be .Fl-proyectable, i.e. there exists a smooth function Ho on Mj such where that H0oTL = EL. A proof of this property was given in [19] and another one will be given later on (see expression (12.12)). It suffices to show that the energy function Ei is (8.5) t} dq'dv'' annihilated by the vector fields in the kernel of TL.. The function J?g allows us to define a presymplectic system (Afi,ii>i = ji*w,dH ) , which is TL -related to (TQ,wi,d£i). This expression also shows that TL. is regular if and only if L is regular. 0 Given a generic locally-Hamiltonian presymplectic dynamical system (M,tt>,a) we DEFINITION 12. A Lagrangian function L is hyper-regular if TL is a (global) diSeomoT* can consider the equation phism. *(I> = a (9.1) The relation between the Lagrangian and Hamiltonian formulations for hyper-regular corresponding to the dynamical equation (7.7) for locally-Hamiltonian dynamical sys- Lagrangians is given in the following theorem. tems. At each point m£M, this looks like a linear system Ax = y, where the matrix A is THEOREM 6. Let L be an hyper-regular Lagrangian. If Jf€C°°(r*Q) is defined by skew-symmetric and singular. Such an equation has a solution if and only if the vector 1 y is annihilated by the elements in the kernel of the dual operator A*. In this case, the H = ELOTL' , the Hamiltonian dynamical systems {TQ,uL,EL) and (T'Q,il,H) are isomorphic. solution is not uniquely defined but the general solution is X€XQ + KerA, where xa is a particular solution. Consequently, only at those points of M at which the solvability PROOF: The definition of H shows that BoTL = EL, and furthermore we know that conditions hold, the equation (9.1) has a solution. This gives rise to a subset TL'il = u>L as a consequence of (8.2). The vector fields XHeX{T'Q) and TLtX(TQ) will be consequently jFL-related. In fact, if Y€X(TQ), Mi — {m£Mi\i(Tm)u>m = dm has a solution}

where we have denoted the manifold M by M\. According to the previous remark, Mi while may be rewritten dELY = d{H oTL)Y = (dHoTL.)Y and taking into account that dH = i(Xit)^t w* fiod

= where TmMf is the subset OljlZ it(Jiif1J'li.I). These relations are true for every TL.Y€X(T'Q), which is arbitrary, and therefore »(Ft)wt = dEL if and only if TL.Ti - XH. Now, it is to be noticed that some points in Mi have been removed to obtain the 9. PRESYMPLECTIC SYSTEMS submanifold M2 in which (9.1) has a solution. The consistency of the theory means that only solutions of the dynamical equation that are tangent to Mj must be accepted and When the Legendre transformation TL : TQ->T*Q is not a local diffeomorphism, this leads to a new submanifold Ms and BO on. By iteration of the procedure we will its image Mi = TL(TQ) is a lowerdimenBional manifold, and if ji : M\-*T'Q denotes obtain a sequence of submanifolds ..—*Mi—»...—'A/j-*M\, which are defined by the natural imbedding of Mi in T'Q, the pull-back u, = ji*O is a closed two form but it may be degenerate, for instance if Mj is odd-dimensional. In a similar way, the M, = {m€M,-i|=0 pull-back wi = TL'Q may be a degenerate closed two form on TQ, and this suggests m the convenience of studying geometric structures generalizing the symplectic structure and (locally-) Hamiltonian dynamical systems.

-26- -27-

m an and therefore there will exist a dosed one-form an in MjT such that a — IJ'UR. The triplet {MjT,UR,aR\ is called the reduced locaUy-HBraUtonian dynamical system asso- •where ciated to (A/,w,a). = o

The submanifolds M3,Afj,... are called the secondary, tertiary,..., and /-ary constraint 10. THE GEOMETRIC VERSION OF DIRAC'S THEORY OF CONSTRAINTS. submanifolds, or simply secondary constraint submanifolds, and any function ^eC°°(Af) If the LfLgrangian L is singular, the Hamiltonian presymplectic dynamical system that is constant on Aft will be called Jfc-ary, or simply secondary, constraint function by to be considered is (Mi,u>i = ji*fi,dH ), as indicated above. The general theory of similarity with what happens for the classical Dirac's theory of constraints. The sequence 0 locally-Hamiltonian presymplectic dynamical systems applies but there is an additional {Mi} eventually terminates at some final constraint Bubmanifold C and when C / $ the particularity, since the primary constraint submanifold M\ = PL(TQ) is a submanifold equation of T'Q and the two form Wj is the pull-back of a symplectic form fi. We can then make use of such additional ingredients. In particular, ft allows us to associate, in a one-to-one posseses at least a consistent solution in C. For more details, tee the papers by Gotay correspondence, functions in M (up to addition of a constant) with vector fields. This and coworkers [19,20], Once the final constraint submanifold C has been found, we can will give us a geometric interpretation of Dirac's classification of constraints in first and define a new locally-Hamiltonian presymplectic system (C,wc,oc) where use = Jc'w second class respectively. and ac = jc'o, with jc • C-*M being the imbedding of C in M. The next step is to Let 4> be a constraint for the final-constraint submanifold C. That means that jc'4> — reduce this system, when possible, to a locally-Hamiltonian system on a reduced space. 0. The restriction onto C of the vector field X$ such that In order to avoid notational difficulties we will assume that M = C, otherwise we would have to change {C,uic,otc) f°r (M,u,a). The set Kern; = {veTM|»(v)w(r(tr)) = 0} is a subbundle of TM, because u is assumed to be of constant rank. The same notation Keru will also be used for the set takes values in of vector fields in M taking values in the subbundle Kerw. Since u> is closed, such a (TC)^ — {uercAf|n(t?,u) = o Vuerc}. (io-i) distribution is, moreover, involutive: let us assume that X and Y axe two such fields in In fact, if YeX(M) is such that the restriction of Y to C is tangent to C (we will simply Kerw. Then, for any other vector field say that Y is tangent to C and write YtX(C)), then, for any constraint funcion ^ 0 = du{X, Y, Z) - Yu>(X, Z) + Zu(X, Y) -U{\X,Y\,Z)+u{\XtZ],Y)-w{\Y,ZlX) and therefore X+ takes values in (TCJ^.The same property can be shown for any sub- and therefore, taking into account that X,Y€Kcru> we find that u([X,Y],Z) = 0, i.e. manifold of M with the corresponding definitions referred to the Bubmanifold because [,] we have not used at all the fact that C is the final constraint submanifold. The distribution defined by Kerw being involutive, it will be integrable (because of A function FeC°°(M) is called a first-class function (with respect to the submanifold the well-known Frobenius theorem, see e.g. the book by Crampin [21]) and this gives C) if {F,4>}\c — 0 f°r ""y constraint function ^ for C. In particular, when F is a a foliation T of M, where the leaves are the integral submanifolds of Kerw. In the best constraint function too, it will be called a first-class constraint funtion. The vector fields case the set M(T of the leaves can be endowed with & differentiable structure and then corresponding to first class functions are tangent to C because in this manifold we can define a symplectic structure w* such that i)*wj = u, with r) : M~*M/f being the natural projection, by means of [22]

and therefore the value of X F at a point of C is Q-orthogonal to the set of values of vector fields corresponding to constraint functions which was shown to be (TC)X. Consequently, where U and V are tangent vectors in M such that t}*(U) = u,T),(V) = v. The two the values of the vectors defined in C by Xp will be tangent to C. Moreover, if 7 is a form Wft is notidegenerate by construction, and closed because TJ is a submersion and w x first-class constraint function, then Xy takes values in (TC)n(TC) . This lost set is just is closed. The point to be remarked is that the closed one-form a is tj-projectable, since the kernel of fie and the vector fields associated to first class constraint functions are for any vector field Z€ Kerw, those in i(Z)a = i(Z)i{T)w = -i(T)i(Z)u = 0,

-29- -28-

T but the indexes only running from 1 to (2n — t). Therefore, we can write Returning to the equation (9.1), it is to be noticed that its solution is by no means unique but undetennined up to addition of a vector field in Kerne, which corresponds to a first class constraint function. When dealing with Dirac's constrained Hamiltonian systems, both first class and where a,fi = 1,... ,2Jt, and Caj denote the elements of the inverse matrix of C^ = second class constraints will appear. However, Sniatycki [23], proved that it is possible {$", 4>t}- The expression on the right hand ride is the one introduced by Dirac for the to imbed coisotropically C in a new symplectic manifold, and this way the second class definition of the modified bracket, now usually called Dirac bracket. Therefore, this is constraints are eliminated. but a Poisson structure such that its restriction to P coincides with that denned by £. In the general case of a generic presymplectic manifold (M,u), Gotay has Bhown [24] that there exists a minimal symplectic manifold (P, (1) which is essentially unique, 11. THE PRESYMPLECTIC SYSTEM DEFINED BY A SINGULAR LAGRANGIAN such that j ; M-*P is a coisotropic imbedding (see also [25]). That means that every constraint function for M in P is a first-class constraint function. This result was recently Let us now consider the case of the Hamiltonian presymplectic dynamical system used for dealing with the time scaling as an infinitesimal canonical transformation [26]. (TQ,ui,dEt) denned by a singular Lagrangian L. Then we will have a set of dynamical A more general result concerning a general locally-Hamiltonian presymplectic system constraint functions determining the points at which there exists a solution of the dy- {M,u,a) was given in [27]: namical equation. However, the ambiguity in the definition of the vector field giving the dynamics, essentially Keriiii,, indicates that not every solution of such equation will be THEOREM 7. XAere exists a svznpJectj'c manifoid (P,£) and a coisotropic imbedding a S.O.D.E. field, because it will not become a S.O.D.E. field under addition of a vector j : C—P such that j*E = tc*w, and: field that is another solution of the dynamical equation, in a general case, but only under i) For each vector Seid T on M, tangent to C, satisfying (9.1), there is a locally addition of vertical fields will preserve the S.O.D.E. character. Therefore we must inves- Eaxnihonian vector £eld T( on P tangent to C »ucc thai T\C = Ttic- tigate at which points of TQ it is possible to find a solution of (7.7) that is furthermore ii) The vector Gelds F( satisfying the above condition are given by the restriction of a S.O.D.E. field. This may give rise to new constraint functions which are not dynamical but extra S.O.D.E. conditions. It is worthwhile to remember that the S.O.D.E. field character of the vector field giving the dynamics is directly related to the fact that the Euler Lagrange equations describe the time evolution. wiiere ap is a closed one-form on P such that j'ocp = oc and (, is any closed first dass We first recall that if T is a (1,1) tensor field in a manifold M, the Nijenhuis tensor constraint one-form on P for C. NT of T is a (1,2) tensor field given by Hi) The coisotropic imbedding and the family {F^} are locally unique. N (X X ) J + T*([XltX,]) - (11.1) We will not insiBt on this point but we only remark that in Dirac'B theory of con- T U 2 strained systems P can be chosen as being a submanifold of T*Q and the two-form E is where Xi and Xj are arbitrary vector fields in M. Obviously, N j- is skew-symmetric and the pull-back of 0 onto P. This manifold P being symplectic, the form £ will give a new it is easy to check that JVT is a tensor field, i.e. NT(fX+Z,Y) = fNT(X,Y)+NT(Z,X) Poisson bracket. Actually this Poisson bracket is related to the so called Dirac bracket. for any feC°°(TM) and three vector fieldB X,Y, ZeX(M). In fact, let P be the symplectic manifold abovementioned, assumed to be defined It is a matter of computation to show that the Nijenhuis tensor of the vertical endo- by the vanishing of the set of second class constraint functions {$*,...,2k). We can morphism S vanishes [21], namely, then choose a set {f'}?=1 of coordinates in T*Q that contains this set, i. e. £' = y' if JVS = 0. (11.2) 1 < i < 2(n - k), while £' = ^-2<"-*) when 2n - Jt < i<2n. Now, if F and G are functions in T'Q and Fp and Gp denote their pull-backs, then This fact expresses that S is integrable. Since S2 — 0, we only have to prove that

{F,G}(p) = -{F, lS(Vi),S(Ut)] - 5([5(F,), tr,]) -S(\Ui,S(Ua)]) = 0, for any pair Vi, U2 of fields in TQ. Since vertical and complete lifts of the vector fields X where the indexes i,j run from 1 to n, while of a basis of X(Q), denoted X^ and Xc respectively, give a basis for X(TQ), and because of the skew-symmetry of JVS, it suffices to show this property when both vector fields {FP,GP)(p) = -{F

-30- -31- are vertical lifts, when both vector fields are complete lifts and when one is & vertical lift and the other is a complete lift. We also recall that if XsX{Q), then S{XC) ~ X\ THEOREM 8. If L is a function Lef\"{TQ), then /28J Therefore (1) C C T c T NS{X ,Y ) = [x\yi] -s([x ,y ]) - s([x%y ]) =o. (11.6) In a similar way, (2) (11.7) because S(jVfT) = S(Y^) = 0. Finally, PROOF: (1) is but a particular case of (11.3) when a = dL. On the other hand, for any U£X(TQ), = o. i, U) = -d6L(A, U) = -A6L{U) + 1 The vanishing of the Nijenhuis tensor Ns of S can also be used to show that for any one-form at/^iTQ) and any couple of vector fields Ui^Vj in X(TQ), and taking account of

a o S){U S(U )). (11.3) ), S(V2)) = d{a o U 2 and Actually, using the definition of d(a o 5), we will find d(a o wi(A,U) = — ,U])L = -(A, S(17)]L - S(U)AL + d(Q o S)(UUS(V3)) = - which implies that and by addition of both expressions, taking into account that Ns = 0, we will find UJI(A,U) = — ), S(U2)). where we have used (11.5). This relation can also be rewritten as Now we recall that if A is the Liouville vector field in TQ and S the vertical endbmor- phism, (i{A)wL)V = -S{V)(AL - L) = d(L - AL)S(U). £*S = -S (11.4) Since EL is given by EL = A(X) - L, the relation (11.7) follows. | because for any UeX(TQ), The relations (11.6) and (11.7) can be used to prove that if T is a solution of the dynamical equation, «(rVi=<*£t, (7.7) and, in particular, if U is a vertical lift, V = X\ with XeX(Q), then 5(A"T) = 0 and then the difference S(T) - A lies in Kerwj;. In particular, had L been regular, then T would be a S.O.D.E. as indicated in Section 7. In fact, from (11.6) and (7.7) it follows [A,*T] = -JfT (11.5) that i(£(r))u>£ = -i(T)wL o S = -dEL o 5, and using (11.7) we iee that (11.8) and both sides in the preceding relation give the aero vector. On the other hand, if U is a complete lift of a vector field in Q, U = Xc, it is homogeneous of degree zero, namely C C The coordinate expression of T is [A, V] = 0, and 5(17) = 5(X ) = Xl. Consequently, (£AS)(* ) = [A.^t] = -Jft = -S(XC). Since £ 5 is a (l,l)-tensor field, the relation (11.4) follows. A (11.9)

-32- -33-

T and Kerwi,(u)/V,(Keru>i), and the relation S(Kerwi)cV(KerWi,). A remarkable con- with sequence is that for Lagrangians for which the equality sign in (11.13) holds, called Type II Lagrangians in [31] when they admit a global dynamics, there will be no ad- ditional S.O.D.E. constraints. However, the solution may be nontangent to the final Now it ie clear that (11.6) shows that S{KeiutL)cV(KeiujL) = KeruLnV(TQ) where constraint submanifold. V(TQ) is the subbundle of T{TQ) made up by vertical vectors, that is, the kernel of We are also interested in the condition to be satisfied for a vertical vector of Keru>i TTQ*- In order to show the relevance of the property 5(Keru>i) = V(Keru>i) we first to be in the image under S of Keru>£. For the sake of simplicity we first give the analysis remark that given a solution V of the dynamical equation, another vector field I" will of the problem by using coordinates [30]. We recall that if H is a Hilbert space and also be solution of (7.7) if and only if the difference T - T lies in Keru>i. Similarly, if T a bounded operator in such space, the closure of the image of T coincides with the T is a S.O.D.E., the vector field T' in a S.O.D.E. if and only if the difference T' - T is orthogonal of the kernel of its adjoint operator Tf (Bee e.g. Ref. [32] p.357 or [33] p.214). a vertical field. Thus, the idea is to modify a given solution T of (7.7) by adding an If we consider the particular case where W is a finite dimensional Euclidean space and T element in Kerwi, in order to obtain a S.O.D.E. solution of (7.7) too. It will be possible a (skew-) symmetric operator, we can conclude that if T is a (skew-) symmetric matrix, if the difference S(T) — A is the image under 5 of an element in Keruij,. then the linear system Tx = y has a solution if and only if {z,y) = 0, V*e Ker T, where (,) Since the presymplectic form n>i is the /"L-pull-back of the canonical two-form i*i in denotes the Euclidean inner product. Such a solution is not uniquely determined except T'Q, it is clear that Ker TL,C Keru>i- Furthermore, KerJ\L, is made up of vertical up to addition of an element of Ker T. This fact may be used to prove the following vectors as follows from the matrix representation (8.4) of KetTL, and consequently result: Ker^i,CV(Kerwi). Equality of these two sets can also be shown (see e.g. [29] for an intrinsic proof). In fact, the expression in local coordinates for vectors in KerTL-, and THEOREM 9. Let X = £'-g~ be a vector Geld in Ker.FZ. = V(KSTU>L). Then, there in V(Kerwi) is [30] exists Ze Kerwj, such that S(Z) = X if and only if (?, A£) = 0 V{' such that W? = 0. X = C± (11.10) PROOF: The condition for X = (^ to be in Ker.FI, is W£ = 0. There will exist a Z€ Keru>t such that S(Z) = X, i.e. Z = f'^r +17'^r , if and only if the system where W£ = 0. This is because they are solutions of

has a solution. Then the remark preceding the statement of the preceding theorem shows a, that this is equivalent to {t',A$ = 0 V£' such that W? = 0. | and In particular, the necessary and sufficient condition for S(Keruji) — KeiFL. is (11.12) (11.14)

respectively. Actually, we know that T must be of the form (11.9) with W£ = 0, and for any couple {, £' such that therefore S(T) - A = £'•£? lies in Ker^I.. The point is that if such difference is the W? = W( = 0. image under 5 of an Xt KeruL, S(X) = S{T) - A, then r - X is a S.O.D.E. that is a The intrinsic property characterizing the preceding vector fields was given in [29]: a solution of the dynamics as well. As a consequence, if the restriction of S to Kerwt is vector field X lies in S(Kerwi) if and only if for every vector field Z such that S(Z) = X onto V(Kerwt), there will be no additional constraints for the possibility of choosing as and each vector field Y such that S(Y)^V(Keru>L), the identity &i(Z,Y) = 0 holds. a solution of the dynamical equation the restriction of a S.O.D.E. Actually the local expressions for X,Y and Z are respectively It is noteworthy that there is a relation between the dimensions of Kerwi, and that of its vertical part, V(KCTWL). Actually,

dim Ker w^ < 2 dim V(Keru>i,) (11.13) e"Kr and the equality symbol works if and only if SfKerui) = V(Keru)t). The proof of this fact is as follows [18]: Let 5' denote the restriction of 5 to Keri»L> Then Ker Si = Vv(KcrwL). The result follows from the isomorphism between Sv(KeruL)

-35- -34- with W$' = W£ = 0. Then,

In the particular case in which {(p,.A£v) = 0 for any pair of indices, we will again find (12.3) as the constraint functions ( £ remains absolutely undetermined in Ker TV) but when the rank r of the RxR matrix (£fi,jl£,,) is different from zero, there will be R — r linearly independent combinations and therefore the above mentioned condition is but the local version of this last one. (12.5) 12. THE LAGRANGIAN CONSTRAINTS From this we will find the constraint functions The dynamical constraint functions will be the functions defining the set in which the dynamical equation (7.7) has a solution, i.e. they correspond to the compatibility *r,.«,,«)=0, Vr-1 R-T, (12.6) conditions for the existence of solutions of the dynamical equations. The coordinate while r of the values of the parameters A are determined by (12.4) in terms of the expression for such systems is remaining R — r values. The freedom in the choice of the basis of Keru>£ allows us to redefine a new basis in which the R — r first elements are the combinations Aa. - Wb = V E tl l (12.1) W(a-v) = 0, TV = irM{,., Vr = l,...,R-r in such a way that the dynamical constraints are just the R — r functions where the local expression for T is T = a'g— + b'-^r. The general solution of the second system is a = v -f- $ with £ such that W£ = 0, and therefore the first subsystem becomes {7r,<*) = 0, VT=1,...,R-T; (12.7) the adjective dynamical indicates that these constraints have nothing to do with the Wb = a + (12.2) S.O.D.E. condition but only with the existence of a solution for (7.7). It is noteworthy that the new basis is such that (Tr,fp) = 0, for any index \i and vhere therefore the vector field 7J.37T is such that there exists a Ze KeiuL with S{Z) = i'T-^r- Then we have essentially proved the following result: THEOREM 10. Let L be a singular Lagrangian. Then, there exists a basis {•),,(,] (with We first analyse the existence of a S.O.D.E. solution of (7.7), that is, with £ = 0. The r = 1,... ,R~r and fi = R-r + l,.,,R) of KerW such tiat subsystem to be solved is (1) The dynamical Lagrangian constraints are given by Wb = a (7r,*»)=0, (12.7) the compatibility condition being {€,«) = o, (12.3) (2) The constraints for the existence of a. S.O.D.E. solution of the dynamics are (12.7) together with for any £ such that W£ = 0. We therefore find for each £ such that W£ = 0, a con-

straint function selecting the submanifold 5 of TQ where (7.7) has a solution that is the (£M,a}=0, V/i = ,R-r + l,..,.R. (12.8) restriction of a S.O.D.E. (but it may be non-tangent to S). On the other hand, had we looked for a solution of (7.7), not a S.O.D.E. but a general Note, however, that in some cases the constraints may be reduced to identities. vector field, we would have had to search for a £ such that W£ = 0 in such a way that In order to understand better the meaning of the constraint functions we remark that there exists a solution of (12.2). Let us choose a basis { (^ }(/* = 1,.., R = dimKer W) of if Z€ Kerwj, is written as Z = £'j§r + i?'^, then (£, a) = -Z(Ei). In fact, the kernel of W and then any £e Ker W can be written as a linear combination £ = A,, (0. The conditions for the existence of a solution are now ZE, = ?•£?&==,- = 0 = ($„, a) = 1 R. (12.4) (12.9)

-36- -37- and the condition

where we recaM th&t (' : T^,)Q->Tt{TQ) denotes the vertical hit. l k (12.10) dv>'dv* ' dq'dv This vector field is differentiable. In fact, its nonvanishing components are the pull- back by F of the first components of X. which when substituted in (12.9) leads to A particular case of such definition is when E = TQ and F is the identity map in which R{idfQ) reduces to the well-known vertical endomorphism S. We will deal with (12.11) the case E = T*Q and F the Legendre transformation TL : TQ~*T*Q associated to a given Lagrangian function L. The notation R(L) will be used instead of R(TL). We also remark that a map R{L) of T{T*Q) in T(TQ) inducing the corresponding map and consequently the conditions (12.9) are simply X(TmQ)-*X(TQ) does not exist. However, for any v€TQ we can define a map

ZEL=0, (12.12)

The above expression shows that the conditions obtained for vertical vector fields Z in the given by vertical part of Keru;£, reduce to identities [34]. Moreover, we can give a new expression R(LUU) = £>.«(.)?], WeTrLM(T'Q). (13.3) that holds not only for dynamical constraints but also for S.O.D.E. conditions. It is We next list a set of interesting properties of R(L) : Jt(r*Q)-.3t"(TQ), which is defined in a pointwise sense by [29]: (i(ro>jt - dEL, Z) = 0 (12.13) [R(L)X](v) = C\r.rn.\ (13.4) with Z being any vector field such that S(Z)€V(Kera>i) and To is an arbitrary S.O.D.E.. This expression becomes (12.12) when Z is chosen in Kerut as assumed in the derivation The expression in coordinates of R(L)X with X = a*^ F 1B of (12.12). These conditions also assure that the energy function is ^X-projectable because KerTL = V(Keru>i).

13. THE CONNECTION WITH THE HAMILTONIAN FORMULATION a In particular, if feC°°(T'Q) and the vector field Xf£C° (T'Q) is denned by When L is a singular Lagrangian, the Legendre transformation PL is not a local diffeomorphism. We will only consider here the case in which L is an "almost regular i{Xf)il = df (13.6) Lagrangian"according to the terminology used by Gotay and Nester [19]. Then KeiFL. then the vector field R(L)Xf is given by is an involutive distribution which generates a foliation T in TQ and the quotient space TQ(T is a differentiable manifold which is canonically equivalent to the submanifold Mi, the primary constraint submanifold in Dirac's terminology [8,9]. If C°(T'Q,Mi) denotes the set of constraint functions for Mi, then and the point to be stressed here is that when ^ is a constraint function for Mj, = Const(TofL) = 0, and when Y is taken to be one of the vertical fields Y = & where Const(TQ) denotes the set of constant functions on TQ. it becomes DEFINITION 14. Lei r\ : E-*Q be a vector bundle on Q and F : TQ-*E a bundie map preserving tie base Q. We will denote R(F) the map R(F) : X{E)-*X(TQ), given by and then (13.7) together with this expression shows that R{L)X^ KerTL*. Moreover, (13.2) not only

but even both sides coincide, as a simple counting of dimensions shows.

-38- -39- field solution of the equation i(X)j'u = dH, then j,X is a solution of j* {t(V)w} = dH THEOREM 11. If and ' are two constraint functions 4>,4>'£.C°{T'Q,Mi), then which is just tangent to Afj and consequently there are no secondary constraints in the Hamiltooian formulation. Besides the aforementioned relationship between the dynamical or S.O.D.E. La- grangian constraints and the first or second class primary constraints, respectively, there with r# a vector Held such that S(y^) = R(L)X+ and similarly for 4'. exists a well-known correspondence between the dynamical Lagrangian constraints and the secondary Harniltouian ones, obtained by making use of the pull-back TV. There- PROOF: It is just a matter of checking because fore, there is a local basis [35] of .^X-projectable Lagrangian constraints defining the sub- manifold Pi of TQ in which the dynamical equation has a solution, such that TL{P\) = M]. On the other hand, if Si is the submanifold defined by both the dynamical and the S.O.D.E. Lagrangian constraints, it can be Been that FL{Si) = /X(Pi) = M, (the proof is similar to that of Prop. 3 in the paper by Gotay and Nester [36]). and therefore It means that every S.O.D.E. Lagrangian constraint is not .FL-projectable. Finally, while Ker.TZ.. is tangent to Pi, we can only assert the 5i-tangency for the elements of S(KetuiL). Moreover, if there is no S.O.D.E. Lagrangian constraint trivial on Pi, then each ZeKerfL. \ S(KetwL) will be non-tangent to St. On the other hand, no matter what the choice for arbitrary IJ and tj' is, if Y4 is given by JV = fisff + itTw aa^ similarly for Yy we can check that 14. THE DEFINITION OF THE TIME EVOLUTION OPERATOR K. Another useful tool for the study of the relations between the constraints arising in the Lagrangian and Hamiltonian formalisms was given in recent papers [37,38] and its geometric formulation was given in [39]. It is a time evolution operator K with a because of = 0. coordinate expresion given by For any constraint function is of the first class (at the Afj level), or in other words, X+ is tangent to Afi, as a consequence of the The operator K has the following basic properties: Theorems 10 and 11. These hist constraint functions are just the dynamical constraints, i) Even if the time-evolution is not well-defined for singular systems, the operator K while the remaining constraint functions, the second class (at the M\ level) primary gives the time evolution in the Lagrangian formulation when applied to functions defined constraint functions will be associated to S.O.D.E. conditions. in the cotangent bundle. Other intereresting results that may be straightforwardly deduced from the relation ii) The action of K over the Hamiltonian constraint functions generates both the pro- (13.7) in the preceding theorem are summed up in the following Theorem [31]: jectable (dynamical) and non-projectable (S.O.D.E. conditions) Lagrangiaji constraints. These two basic properties of K justified the interest of a deeper analysis of the THEOREM 12. i) All the primary constraint functions in T'Q are of the first class subject, which was carried out in [39]. In order to get an intrinsic definition of K (Mi is then called coisotropic) if and only if there are no S.O.D.E. conditions, i.e. all the we will first review the dynamical formalism developed by Skinner and Rusk [40,41] in primary constraints in the Lagrangian formulation are dynamics] ones. the Whitney sum W9 = T'QeTQ. This space allows UB to treat both velocities and ii) If all the primary constraints are of tie second class (Mi is said to be symplec- momenta on the same footing and, a crucial fact, the geometric dynamical equation in tic), all the primary constraint functions in the Lagrangian formulation are S.O.D.E. Wo includes automatically the S.O.D.E. restrictions. We will only give some definitions, conditions and there are no dynamical constraints. asking the reader to consult the papers by Skinner and Rusk for additional concepts and We also remark that in this last situation there will exist no secondary constraints notation. in the phase space formulation. In fact, the comment after formula (12.12) shows that We will only consider Lagrangian functions L such that the associated two-form wi the energy function is ^"L-projectable, that is, there exists a function H€C°°{M\) such is of constant rank in every submanifold defined by the geometric constraint algorithm, that HoTL = EL. Let the Bubmanifold j : Mi-*T*Q be Bymplectic. If X is the vector

-41- -40-

T BO that its kernel defines a foliation; therefore, we exclude the posibility of singularities 15. PROPERTIES OF K. RELATION BETWEEN THE LAGRANGIAN AND HAMIITONIAN of the presymplectic structure on the final constraint Bubmanifold. The notation Wo will FORMULATION be used for the Whitney sum WQ = T*Q®TQ with the natural projections on each factor pr,, for i = 1,2. The operator K playB the role of a time-evolution operator and may be used for the The generalized Hamiltonian system (Wf,,w,D), where the presymplectic form fl is study of the equivalence between the Lagrangian and Hamiltonian formulations. Let C be the pull-back the final constraint submanifold of the generalized presymplectic system (TQ,wL,EL). It fl = pr;(u;g) (14.2) is the maximal submanifold of TQ in which a consistent solution of the equation i{F Vi = dEi does exist. Some results about the existence of (non maximal) submanifolds of C of the canonical symplectic form on T'Q, UIQ, and the dynamical function D^C (WO) in which a consistent S.O.D.E. solution exists were given in [30], but the corresponding is formulation when developed on WB shows that there will exist a maximal submanifold •^ = (Pri>Prj) ~ Prj(^)i (1^.3) B with a consistent Bolution, restriction of a S.O.D.E. in TQ, B = pr2(.4) = JX (A), was shown [40] to be equivalent to the Euler-Lagrange equations, i.e., to the presymplec- where A is here the final constraint submanifold of the system (Wo ,U,D). Time evolution tic Lagrangian system (TQ,wL,EL) plus the S.O.D.E. restriction. It is also equivalent is not well defined in B due to the ambiguity V(Kerwi,)nTB of the S.O.D.E. solution, to the Hamiltonian dynamical system (M\,Wi,H), where jj : Afi-»T"Q is the primary except when considering time evolution of weakly ^X-projectable functions [36] on 27, Hamiltonian constraint submanifold Mi = TL{TQ), UI=J*(L>Q) and H is the Hamilto- because of the property V(Kerui£,) = Ker FL.. The next proposition shows that K gives nian function defined on M\, by ?L'(H) = EL- then this well-defined time evolution. In order to find a subset of Wj> in which there exists a consistent solution of the dynamical equation PROPOSITION 1. Let h^C°°{TQ) be a weakly TL-projectable function on B and T a. i(Z)tt = dD, (14.4) vector field whose restriction to B is a consisteat S.O.D.E. solution of the dynamics. If x the systematic procedure developed by Gotay et «i [20] can be applied: let W be the feC (T'Q) is such tiiat FL'(f)\B = hiB andT' is any solution of the dynamics, then t A'(/) r'(A) primary constraint submanifold, i.e., the set of points in Wo where the equation (14.4) has a pointwise solution. This submanifold W\ is (Bee [41]) the image of the section PROOF: We can take Z = ?X.(F) as a particular solution of (14.4) on A and then the over pr given by o 2 restriction to B of the definition of K(f) according to (14.7) leads to

Therefore W\ is diffeomorphic to TQ and pra(Wj) = Mi, Let us consider a point and a vector Zw€T*,Wa solution of (14.4) in w. Every other such Bolution takes the and since pr i oTL — J-L we will obtain the result of the statement of the proposition. | form Z'v -Z + V*,, with U G Kerfi . The property Kerfl = Ker(pn). implies that w W K By an appropriate handling of the operator K (see e.g. [39]) we can prove that if 4> € C^iT'Q) is a first class function, then Ktfr is weakly ^"L-projectable [35] and that and it allows to define the following map: the image under K of the set of primary and secondary Hamiltonian constraints gener- ates that of all the Lagrangian constraints, dynamical and extra S.O.D.E. constraints. 0D no DEFINITION 15. The K operator is the map K : C (T'Q)^C {TQ) defined as follows: Moreover, if 4 is a first class constraint function, then K

a K = PL o2opri" (14-7) In the example a) of Section 2 in which L £ C°°(rR ) is given by

with Z being any vector Seld in Wo whose restriction to Wi is a aoJution of (14.4). L = \v\ + i

-43- -42- which is the Lagrangian constraint \fri =03. However, the operator K, when applied to we sec that Ker W is one-dimensional while Kerwt is generated by •£- and 3^. The fa, will give no new Lagrangian constraint function because of K4>i ~ 0. The constraint primary constraint function will be V'i = <*2 = *J = 0 and in the submanifold defined by %j)i is ^i-projectable and actually t/'i it is possible to find the following solution of the dynamical equation: ^1 = pi o TL. a a a In the case of the example c) of Section 2 Ker W is three-dimensional. Actually there will where o2 and b? are arbitrary functions. Of course only the solutions for which 02 = «2 exist a solution of the dynamical equation in the submanifold defined by the constrained are restrictions of S.O.D.E. vector fields. The consistency requires that ^ = vj = 0, function V>i = 12 = 0, the general form for such a solution being which gives rise to a new secondary constraint function in the Lagrangian formalism. Notice that the image under K of the primary constraint fa = pj arisen in the i J- Hamiltonian formalism is 0V2

that can be chosen as being the restriction of a S.O.D.E. onto the submanifold denned by the constraints which coincides with V»i- Similarly, the image under K of the secondary constraint i>2=vt, il>3=2xi-v3 $% = ii reduces to the secondary Lagrangian constraint ^2- Finally, the image of As a secondary condition, the consistency for the constraint ^1 leads to \pi =1)2. ^3 = px does not lead to any new constraint because according to the constraint V>i, 1 We remark that the image under K of the constraint functions arising in the K3 = X1X1 as 0. Notice that since 4>\ is firstclass , then V ! is ^t^projectable. Hamiltonian formulation are simply the constraints t/>, because As far as the example b) of Section 2 is concerned, proposed by Christ and Lee [12], with a Lagrangian in T(R3 - (0,0, z)) given in cylindrical coordinates by = -3— = 2l2 the kernel of W is one-dimensional and the primary constraint arising in this case,

ft»

is dynamical. The kernel of wL is generated by the vector fields 5; + — and £, and = v,. there are no secondary constraints in the Lagrangian formulation. In fact, the general solution of the dynamics is Finally other examples and applications can be found in [42] where the theory of La- grangians linear in the velocities, and therefore singular, is developed, as well, as its relation to the Hamiltonian formulation. The results were then applied for giving a new approach to the so-called Inverse Problem of the Mechanics and for studying symmetries of the Lagrangian system that are not point transformations and consequently they are and Y j{B - z) = 0, and thus there are no secondary constraints in this approach. We a not symmetries of the Lagrangian itself. recall that in the Hamiltonian counterpart the momenta are given by

Pr = r, p, - z), p, = 0, Acknowledgements. These notes formed a series of lectures given by the author as a Visiting Scholar at the University of Costa Rica in August 1988, under the auspices and therefore there exist a primary constraint p, = 0 and a secondary constraint pe = 0, of the International Centre of Theoretical Physics. The support of the ICTP and the but both of the first class. The K operator when applied to the primary Hamiltonian cooperation of the UCR is gratefully acknowledged. The final form of these notes profited constraint gives the constraint from conversations held with J. M. Gracia-Bondia and J. C. Vanity on that occasion. The author would also like thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

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