A Hamilton-Jacobi theory for implicit differential systems
aa r X i v : . [ m a t h - ph ] A ug A Hamilton–Jacobi theoryfor implicit differential systems
O˘gul Esen † , Manuel de Le´on ‡ , Cristina Sard´on ∗ Department of Mathematics † Gebze Technical University41400 Gebze, Kocaeli, Turkey.Consejo Superior de Investigaciones Cient´ıficas ‡ C/ Nicol´as Cabrera, 13–15, 28049, Madrid. SPAINInstituto de Ciencias Matem´aticas, Campus Cantoblanco ∗ Consejo Superior de Investigaciones Cient´ıficas
Abstract
In this paper, we propose a geometric Hamilton-Jacobi theory forsystems of implicit differential equations.In particular, we are interested in implicit Hamiltonian systems,described in terms of Lagrangian submanifolds of
T T ∗ Q generatedby Morse families. The implicit character implies the nonexistenceof a Hamiltonian function describing the dynamics. This fact is hereamended by a generating family of Morse functions which plays therole of a Hamiltonian. A Hamilton–Jacobi equation is obtained withthe aid of this generating family of functions. To conclude, we applyour results to singular Lagrangians by employing the construction ofspecial symplectic structures. Implicit differential equations (IDE) do not only arise in purely mathemat-ical frameworks, as in relation with minimizers of integrals in the calculusof variations, or as intermediate steps for the integration of differential alge-braic equations [26], they do appear in many various areas in science as well.Their applications are important in relativity, control theory, chemistry, etc.For example, they describe exchanges of matter, energy, or information thatvary in space and time.Unfortunately, differential equations and, particularly, IDE cannot al-ways be solved analytically. For this matter, different mathematical methods1ave been precisely designed. It is desirable that these differential equationsare reducible to quadratures, and many attempts have been tried throughFuchsian, Lie’s theory, etc. Nowadays, the methods of numerically integra-tion has been increasingly developed.The Hamilton–Jacobi theory (HJ theory) has been long known as a pow-erful problem solving tool [1, 2]. It is particulary useful for identifying con-served quantities for a mechanical system, which may be possible even whenthe mechanical problem itself cannot be solved completely. Therefore, itconstitutes an alternative way of finding solutions of Hamilton’s equations.It is equivalent to other classical formulations of mechanics and it roots invariational calculus. The action functions are solutions of Hamilton–Jacobiequation (HJE). It is important to remark that the classical HJ theory onlydeals with explicit Hamiltonian systems, but, in the literature there existtons of physical models governed by IDE. Hence, the necessity of construct-ing a Hamilton–Jacobi theory for implicit systems.For example, recall the number of theories described by singular La-grangians in the sense of Dirac-Bergmann [3, 9, 17], including systems ap-pearing in gauge theories [31]. The Euler–Lagrange equations (EL) give riseto differential equations that are implicit, and because of the degeneracy ofthe Lagrangian they cannot be put in a normal form. Some authors have in-troduced a geometric formalism for dealing with dynamical systems in theirimplicit form [43, 46] and a unified approach for the Lagrangian descriptionof (time-independent) constrained mechanical systems is provided througha technique that generates IDE on T ∗ Q from one-forms defined on the to-tal space of any fiber bundle over T Q [5]. Other authors have designedalgorithms following the Driac-Bergmann prescription, to be able to dealwith singular Hamiltonian and Lagrangian theories, see for example, the ge-ometric Gotay–Nester algorithm [22, 23, 24, 25] (see brief description in ourAppendix). In the local coordinate formalism, the classical EL displayingconservative or nonconservative force fields or subject to linear or nonlin-ear nonholonomic constraints, also arise in implicit form from d’Alambert’sprinciple of virtual work. In the geometric formalism, the correspondingequations should then be expected to arise in implicit form equivalentlyfrom a suitable expression of the above principle.Our main aim is to generalize the geometric Hamilton Jacobi explainedfor explicit systems to the realm of implicit systems on T ∗ Q . We inter-pret IDE in terms of arbitrary submanifolds of a higher-order tangent bun-dle, particularly, Lagrangian submanifolds of T T ∗ Q in the case of implicitHamiltonian systems (IHS), not necessarily in the horizontal form. As anapplication, we shall concentrate the problem of Hamilton Jacobi theory for2ingular Lagrangian theories.Let us summarize the problem in more technical terms. We consider afirst-order IDE as a submanifold E of T T ∗ Q . We project E to T Q by thetangent mapping
T π Q to a submanifold T π Q ( E ) of T Q , which is anotherIDE on Q . The philosophy of the geometric Hamilton–Jacobi theory is toretrieve solutions of E , provided the solutions of T π Q ( E ). In similar fashionas the classical Hamilton-Jacobi theorem, in order to lift the solutions in Q to T ∗ Q , we are still in need of a closed one-form γ on Q , but two ingredientsof the theory are missing. One is that the base manifold C = τ T ∗ Q ( E )is not necessarily the whole T ∗ Q , but possibly a proper submanifold of it.The second is the nonexistence of a Hamiltonian vector field due to theimplicit character of the equations. In the classical theory, the major roleof the Hamiltonian vector field is to connect the image space of γ and thesubmanifold E . To overcome these two difficulties, we need to introducea auxiliary section σ of the fibration τ T ∗ Q defined on C ∩ Im γ and takingvalues in E . If, particularly, the dynamics E is a Lagrangian submanifoldthen, according to generalized Poincar´e theorem [7, 27, 38, 52, 55], thereexists a Morse family (a family of generating functions) defined on the totalspace of a smooth bundle linked to T T ∗ Q by means of a special symplecticstructure. A Morse family also establishes a link from the base space T ∗ Q to E , so that for this particular case, there is no need for an auxiliary section.The plan of the manuscript is the following: in section 2 we review thefundamentals of Hamiltonian mechanics, Section 3 develops a geometric in-terpretation of dynamics as Lagrangian submanifolds and their generationgwith the aid of Morse families of functions. Section 4 illustrates the geo-metric HJ theory both for IDE and IHS. In section 5, we contemplate theconstruction of complete solutions. Section 6 concerns applications of ourconstructed theory to the case of degenerate Lagrangians.We assume that functional analytic issues related with the present dis-cussion are satisfied in order to highlight the main aspects of our theory.Accordingly, we assume that all manifolds are connected, all mathematicalobjects are real, smooth and globally defined. Let Q be the configuration space, T Q is the tangent bundle, and T ∗ Q is thecotangent bundle. Consider the tangent and cotangent bundles of T Q and T ∗ Q , these are the possibilities: T T Q, T ∗ T Q, T T ∗ Q and T ∗ T ∗ Q . Here wecan establish the canonical projections for the first order tangent and cotan-gent bundles, denoted as π Q : T ∗ Q → Q and τ Q : T Q → Q . Furthermore,3onsider the projections, π T ∗ Q : T ∗ T ∗ Q → T ∗ Q , τ T ∗ Q : T T ∗ Q → T ∗ Q , T π Q : T T ∗ Q → T Q and the two last projections π T Q : T ∗ T Q → T Q and τ T Q : T T Q → T Q . For the last case there is another possibility
T τ Q : T T Q → T Q , and both possibilities are related through a diffeo-morphism we shall devise in the following lines.
Table 1.
Canonical coordinates and symplectic forms on second-ordertangent and cotangent spaces.
Consider Q a mechanical configuration manifoldand note that we are assuming summation over repeated indices. Space Coordinates Symplectic forms
Q q i T Q ( q i , ˙ q i ) T ∗ Q ( q i , p i ) ω Q = dq i ∧ dp i T T ∗ Q ( q i , p i , ˙ q i , ˙ p i ) ω TQ = d ˙ q i ∧ dp i + dq i ∧ d ˙ p i T T Q ( q i , ˙ q i , δq i , δ ˙ q i ) T ∗ T ∗ Q ( q i , p i , α i , β i ) ω T ∗ Q = dq i ∧ dα i + dp i ∧ dβ i T ∗ T Q ( q i , ˙ q i , a i , b i ) ω TQ = dq i ∧ da i + d ˙ q i ∧ db i Let us recall the definition of the pullback bundle, as we refer to itin forthcoming sections. Let (
P, π, M ) be a fiber bundle and assume theexistence of a differential mapping ϕ from a manifold P to the base manifold M . Define the following product manifold ϕ ∗ P = { ( n, p ) ∈ N × P : ϕ ( n ) = π ( p ) } and the surjective submersion ϕ ∗ π which simply projects an element in ϕ ∗ P to its first factor in N . The triple ( ϕ ∗ P, ϕ ∗ π, N ) is called the pullback bundleof ( P, π, M ) via the mapping ϕ . This definition can be summarized within4he following commutative diagram. ϕ ∗ P ε / / ϕ ∗ π (cid:15) (cid:15) P π (cid:15) (cid:15) N ϕ / / M (1)Here, ε is the projection which maps an element in ϕ ∗ P to its second factor P . Let us point out two particular cases which are important for the presentwork. If ϕ is a diffeomorphism, ϕ ∗ P and P become diffeomorphic. If M isan embedded submanifold of P , and N is an embedded submanifold of M in (1), then it is evident that ϕ ∗ P is an embedded submanifold of P . In thiscase, the map ε plays the role of an embedding.Henceforth, we refer to a general, arbitrary manifolds by M or N , wewill denote fiber bundles by π : P → N , where P is the complete space and N is its projection by π (also M instead of N indistinctly). IDE will bedenoted by E generally, and any general Lagrangian submanifold is denotedby S . We will also refer by E to IHS that are Lagrangian submanifoldsgenerated by a Morse families. By F we denote Morse families. In general, P is the total space of a fiber bundle related with a Morse family F and R is the total space of a special symplectic structure. This section is dedicated for reviewing fundamentals of Hamiltonian me-chanics from a geometric viewpoint, and basics of the Tulczyjew’s triple.Here, we are setting the notation we shall be using along the paper.
Consider a manifold Q and a cotangent bundle T ∗ Q with canonical projec-tion π Q : T ∗ Q → Q . We denote by ( q i , p i ) the fibered coordinates in T ∗ Q such that π Q ( q i , p i ) = ( q i ). A cotangent bundle is equipped with a canonicalone-form θ Q defined as follows: h X α q , θ Q ( α q ) i = h T π Q ( X α q ) , α q i , (2)where X α q ∈ T α q ( T ∗ Q ) and α q ∈ T ∗ q Q . In fibered coordinates, the canonicalone-form reads θ Q = p i dq i , which is known as the Liouville one-form on5 ∗ Q . Consider now the two-form ω Q = − dθ Q , namely, ω Q = dq i ∧ dp i . Thistwo-form has the two properties1. It has maximal rank 2 n , where n is the dimension of Q .2. dω Q = 0This two-form is called a symplectic two-form. More generally, a symplecticmanifold is a pair ( M, ω ) such that ω has maximal rank and dω = 0. There-fore, ( T ∗ Q, ω Q ) is a symplectic manifold and ω Q is the canonical symplecticform on T ∗ Q .Given two symplectic manifolds ( M , ω ) and ( M , ω ) and a map F : M → M , we say that F is a symplectomorphism if F ∗ ω = ω . A Hamiltonian system on T ∗ Q is determined by the triple ( T ∗ Q, ω Q , H ),where H being the Hamiltonian function. Geometrically, Hamilton’s equa-tions are defined by ι X H ω Q = dH, (3)where X H is the Hamiltonian vector field associated with the Hamiltonianfunction H , and ι X H is the inner contraction operator.In Darboux’s coordinates ( q i , p i ) on T ∗ Q , with i = 1 , . . . , n for an n -dimensional configuration manifold Q . The canonical one-form reads θ Q = p i dq i and the symplectic two-form turns out to be ω Q = dq i ∧ dp i . In thislocal picture, the Hamiltonian vector field X H is written as X H = ∂H∂p i ∂∂q i − ∂H∂q i ∂∂p i (4)whereas the Hamilton’s equations (3) turn out to be˙ q i = ∂H∂p i , ˙ p i = − ∂H∂q i . (5)Note that, the Hamilton’s equations are explicit by definition. Hence,an IDE cannot be recast in the classical Hamiltonian formalism presentedin (5). To deal with IHS, we shall redefine Hamiltonian systems in a moreabstract framework, as we shall present in the forthcoming sections.6 .3 Geometry of the tangent bundle Consider the manifold Q and its tangent bundle T Q together with its tangentbundle canonical projection τ Q : T Q → Q . We consider fibered coordinates( q i , ˙ q i ) such that τ Q ( q i , ˙ q i ) = ( q i ). Given a function f : Q → R , we defineits complete lift f T to T Q as the function: f T ( v q ) = df ( q )( v q ) ∈ R . (6)In local fibered coordinates, f T ( q i , ˙ q i ) = ˙ q i ∂f∂q i . (7)Given a tangent vector v q ∈ T q Q with components ( q i , v i ), we define itsvertical lift v Vq for a point w q ∈ T q Q by v Vq = ddt | t =0 ( w q + tv q ) (8)If X is vector field on Q , then its vertical lift to T Q is the vector field X V ( w q ) = ( X ( q )) Vw q ∈ T w q ( T Q ) (9)for all w q in T Q . Now, consider the flow φ t : Q → Q of a vector field X . Wedefine the complete lift X T of X to T Q as the generator of the tangent flow
T φ t : T Q → T Q . We are assuming, for simplicity, that the flow generatedby X is complete, but the construction is still valid in general. If X = X i ∂∂q i ,a direct computation shows that X T = X i ∂∂q i + ˙ q i ∂X i ∂q i ∂∂ ˙ q i . (10)Consider now a k -form ω on Q . We define its complete lift ω T to T Q a k -form characterized by: ω Q ( X T , . . . , X Tn ) = ω Q ( X , . . . , X n ) T (11)The following identity follows from a direct computation d ( ω T ) = ( dω ) T .We are particulary interested in the case of lifts of symplectic forms on Q .Therefore, if ω Q is symplectic on Q , then ω TQ is a symplectic form on T Q .Indeed, rank( ω TQ ) = 2 rank( ω Q ) (12)and d ( ω T ) = ( dω ) T = 0. 7 .4 Submanifolds of symplectic manifolds Let (
M, ω ) be a symplectic manifold, and N be a submanifold of M . Wedefine the symplectic orthogonal complement of T N as the set of tangentvectors
T N ⊥ = { u ∈ T M | ω ( u, v ) = 0 , ∀ v ∈ T N } . (13)Note that, the dimension of the tangent bundle T M is the sum of the tangentbundle
T N and its symplectic orthogonal complement
T N ⊥ . We list someof the important cases. • N is called an isotropic submanifold of M if T N ⊂ T N ⊥ . In this case,the dimension of N is less or equal to the half of the dimension of M . • N is called a coisotropic submanifold of M if T N ⊥ ⊂ T N . In this case,the dimension of N is greater or equal to the half of the dimension of M . • N is called a Lagrangian submanifold of M if N is a maximal isotropicsubspace of ( T M, ω ). That is, if
T N = T N ⊥ . In this case, the dimen-sion of N is equal to the half of the dimension of M . • T N is symplectic if
T N ∩ T N ⊥ = 0. In this case, ( N, ω N ) is a sym-plectic manifold.A diffeomorphism between two symplectic manifolds is called a symplec-tomorphism if it preserves the symplectic structures. Under a symplecto-morphism, the image of a Lagrangian (isotropic, coisotropic, symplectic)submanifold is Lagrangian (resp. isotropic, coisotropic, symplectic) sub-manifold. ω Q Consider the canonical symplectic manifold ( T ∗ Q, ω Q ). The tangent bundle T T ∗ Q of T ∗ Q carries a symplectic two-form ω TQ that derives from two po-tential one-forms, denoted by θ = i T ω Q and θ = d T θ Q . These one-formsare defined by the canonical forms ω Q and θ Q , respectively. The definition ofthe derivation i T is the manifestation of the double vector bundle structureof T T Q on T Q , and explicitly given by i T ω Q ( X ) = ω Q ( τ T ∗ Q ( X ) , T τ Q ( X )) . The derivation d T is the commutator [ d, i T ].8ccordingly, in the local coordinate chart ( q i , p i ; ˙ q i , ˙ p i ), the potentialone-forms are computed to be θ = i T ω Q = ˙ p i dq i − ˙ q i dp i , θ = d T θ Q = ˙ p i dq i + p i d ˙ q i . (14)The exterior derivatives of these one-forms are the same and define thesymplectic two-form ω TQ = d ˙ p i ∧ dq i + dp i ∧ d ˙ q i (15)is known as the complete lift to the tangent space of the symplectic two-form[49, 53]. Note that, the difference θ − θ is an exact one-form. Actually, it isthe exterior derivative of coupling function of the Legendre transformationbetween the Lagrangian and Hamiltonian formalisms. Further, existence oftwo potential one-forms for ω TQ leads to the existence of a Tulczyjew’s triple,which is exhibited in the following subsection. We now consider a particular kind of symplectic manifolds introduced byTulczyjew in [48, 49, 50, 51, 52, 53].A special symplectic manifold is a quintuple (
R, N, τ, θ, A ) where τ : R → N is a fiber bundle, θ is a one-form on R and A : R → T ∗ N is a diffemor-phism such that π = π N ◦ A and θ = A ∗ θ N . Since ( T ∗ N, ω N = − dθ N ) is asymplectic manifold, then ( R, ω = − dθ ) is symplectic too and A ∗ ω N = ω ,therefore, ( R, ω ) and ( T ∗ N, ω N ) are symplectomorphic. Consider the fol-lowing diagram. R τ ❅❅❅❅❅❅❅❅ A / / T ∗ N π N | | ②②②②②②②②② N (16)Tulczyjew’s symplectic space ( T T ∗ Q, ω TQ ) admits two special symplecticstructures. Let us study them.The non-degeneracy of the canonical symplectic structure ω Q on T ∗ Q leads to the existence of the following diffeomorphism β Q : T T ∗ Q T ∗ T ∗ Q : X ι X ω Q . (17)The mapping β Q is actually a symplectomorphism if the iterated cotangentbundle T ∗ T ∗ Q is equipped with the canonical symplectic two-from ω T ∗ Q .In coordinates, we have that β Q ( q i , p i , ˙ q i , ˙ p i ) = ( q i , p i , − ˙ p i , ˙ q i ) . (18)9t is a matter of a direct calculation to prove that, the quintuple( T T ∗ Q, T ∗ Q, τ T ∗ Q , θ , β Q )is a special symplectic manifold. Here, θ is the differential one-form definedin (14).We define a canonical diffeomorphism α Q : T T ∗ Q → T ∗ T Q as follows.First of all, let us recall that there exists a canonical involution S Q : T T Q → T T Q locally given by S Q ( q i , ˙ q i , δq i , δ ˙ q i ) = ( q i , δq i , ˙ q i , δ ˙ q i ) (19)(see [19]). Now, given v ∈ T T ∗ Q we define α Q ( v ) ∈ T ∗ T Q as h w, α Q ( v ) i = ddt h γ, ξ i| t =0 (20)where γ : R → T Q and ξ : R → T ∗ Q are curves such that j ◦ γ = S Q ( w ), j ◦ ξ = v and τ Q ◦ γ = π Q ◦ ξ . In local coordinates, we obtain α Q ( q i , p i , ˙ q i , ˙ p i ) = ( q i , ˙ q i , ˙ p i , p i ) . (21)The mapping α Q is a symplectomorphism if the iterated cotangent bundle T ∗ T Q is equipped with the canonical symplectic two-from ω T Q . Then itbecomes immediate to prove that(
T T ∗ Q, T Q, T π Q , θ , α Q )is a special symplectic manifold. Here, θ is the differential one-form definedin (14).As a result, we have derived two special symplectic structures for thesymplectic manifold ( T T ∗ Q, ω TQ ). Tulczyjew’s triple is the combination ofthese two special symplectic structures in one commutative diagram as givenbelow. T ∗ T Q π TQ ●●●●●●●●● T T ∗ Q T π Q { { ✇✇✇✇✇✇✇✇✇ β Q / / τ T ∗ Q ❍❍❍❍❍❍❍❍❍ α Q o o T ∗ T ∗ Q π T ∗ Q z z ✉✉✉✉✉✉✉✉✉ T Q T ∗ Q (22)10 Lagrangian Submanifolds
In this and following subsections, we will summarize the theory of Morsefamilies and Lagrangian submanifolds generated by them. We refer an in-complete list [8, 38, 50, 51, 52, 55] for more detailed discussions.Let (
M, ω ) a symplectic manifold. A sufficient condition for a subman-ifold N ⊂ M to be Lagrangian is T N = T N ⊥ . If N is an isotropic sub-space of a symplectic manifold ( M, ω ), then N is Lagrangian if an onlyif dim N = dim M/
2. For different types of manifolds (Poisson, Nambu–Poisson, etc), the definition of a Lagrangian submanifold has been accom-modated to its background. See for example [35, 36, 38].Two principal examples of Lagrangian submanifolds of the symplecticphase space T ∗ Q are the fibers of the canonical cotangent bundle projectionand the image space of closed one-forms γ : Q → T ∗ Q . The latter caseincludes the zero section of the projection T ∗ Q Q as well. This statementis the well-known Weinstein tubular neighborhood theorem [55]. Consider a differentiable fibration (
P, π, N ). A real valued function F onthe total space P can intuitively be understood as a family of functionsparameterized by the coordinates of the fibers of π . The critical set of F isdefined by Cr ( F, π ) = { z ∈ P : h dF ( z ) , V i = 0 , ∀ V ∈ V z P } (23)and is a submanifold of P . The dimension of Cr ( F, π ) is equal to thedimension of N . Here, V P is the vertical bundle on P consisting of verticalvectors projecting to the zero section of T N under the mapping
T π . Wedefine a bilinear mapping W ( F, z ) : V z P × T z P → R : ( v, w ) → D (1 , ( F ◦ χ ) (0 , , (24)where χ : R → P is the mapping such that the vector v is obtained bytaking the derivative of χ with respect to its first entry at (0 ,
0) and thevector w is obtained by taking the derivative of χ with respect to its secondentry at (0 , F defined on the total space of the fibration( P, π, N ) is said to be regular if the rank of the matrix W ( F, z ) definedin (24) is the same at each z ∈ Cr ( F, π ) . A family of functions F is said to11e a Morse family (or an energy function) if the rank of W ( F, z ) is maximalat each z ∈ Cr ( F, π ).Let us write the requirement of being a Morse family in terms of localcoordinates. Assume that the dimension of the manifold N is n with coordi-nates ( q i ), the dimension of a fiber is k with coordinates ( λ a ). The function F is called a Morse family if the rank of the n × ( n + k )-matrix (cid:18) ∂ F∂q i ∂q j ∂ F∂q i ∂λ a (cid:19) (25)is maximal. A Morse family F on the smooth bundle ( P, π, N ) generates a Lagrangiansubmanifold S T ∗ N = { w ∈ T ∗ N : T ∗ π ( w ) = dF ( z ) } (26)of ( T ∗ N, ω N ). In this case, we say that S T ∗ N is generated by the Morsefamily F . Note that, in the definition of S T ∗ N , there is an intrinsic require-ment that π ( z ) = π T ∗ N ( w ). Here, we are presenting the following diagramin order to summarize the discussion. R P π (cid:15) (cid:15) F o o T ∗ N π N (cid:15) (cid:15) N N (27)In order to exhibit the structure of the submanifold S T ∗ N , define a fiberpreserving mapping κ from the critical set Cr ( F, π ) of the Morse family F to the cotangent bundle T ∗ N according to the requirement h κ ( z ) , Z N i = h dF, Z P i , (28)which is valid for all π -related vector fields Z N ∈ X ( N ) and Z P ∈ X ( P ).Note that, this mapping is an immersion and that dim ( S T ∗ N ) equals todim ( Cr ( F, π )) = n . A direct calculation shows that the image space of κ is the Lagrangian submanifold S T ∗ N defined in (26).Let N be an immersed submanifold of Q , and T ∗ N Q denote the inverseimage π − Q ( N ) in the cotangent bundle T ∗ Q . We define the mapping ξ : T ∗ N Q → T ∗ N (29)12y requiring that the equality h ξ ( p ) , Z N ( n ) i = h p, Z N ( n ) i holds for each Z N ∈ X ( N ), where n = π Q ( p ) and p ∈ T ∗ N Q . So, ξ isthe identity if p ∈ N, that is, if p ∈ T ∗ N. Consider the canonical injec-tion i : T ∗ N Q → T ∗ Q . If S T ∗ N is a Lagrangian submanifold of T ∗ N thenthe preimage i ◦ ξ − ( S T ∗ N ) is a Lagrangian submanifold S T ∗ Q of T ∗ Q . If,particularly, the Lagrangian submanifold S T ∗ N is a generated by a Morsefamily F on the fiber bundle ( P, π, N ) then, using the same terminology, wesay that S T ∗ Q is generated by the Morse family F . We are presenting thefollowing diagram. R P π (cid:15) (cid:15) F o o T ∗ Q π Q (cid:15) (cid:15) N (cid:31) (cid:127) / / Q (30)Let us try to see this more explicitly in the following way. For every point p ∈ T ∗ Q such that π N ( p ) ∈ N , we can find a point z ∈ P satisfying π ( z ) = π N ( p ). Then, for every vector Z P ( z ), we have that T π ◦ Z P is avector field on N , and hence a vector field on Q . Then, the elements of theLagrangian submanifold S T ∗ Q are defined by the requirement h p, T π ◦ Z P ( z ) i = h dF ( z ) , Z P ( z ) i , where F is a Morse family on P .If the local coordinates ( q i , λ a ) are considered, then the Lagrangian sub-manifold generated by the Morse family F is defined by S T ∗ Q = (cid:26)(cid:18) q i , ∂F∂q i ( q, λ ) (cid:19) ∈ T ∗ Q : ∂F∂λ a ( q, λ ) = 0 (cid:27) . (31)Note that, we are not distinguishing here the base manifold N from Q . Let (
R, N = Q, τ, ω = − dθ, A ) be a special symplectic structure for a sym-plectic manifold ( R, ω ) with symplectomorphism A . If S T ∗ Q is a Lagrangiansubmanifold of T ∗ Q , then its pre-image S = A − ( S T ∗ Q ) under the sym-plectomorphism A is a Lagrangian submanifold of the symplectic manifold( R, ω ). If the Lagrangian submanifold S T ∗ Q is generated by the Morse fam-ily F as presented in the diagram (30), then we say that S is generated13y the Morse family F as well. To illustrate this, we draw the followingdiagram. R P π (cid:15) (cid:15) F o o T ∗ Q π Q (cid:15) (cid:15) R A o o τ } } ④④④④④④④④④ N (cid:31) (cid:127) / / Q (32)Let us record here some special cases of the diagram for future reference. • The simplest case occurs if A is the identity mapping (no special sym-plectic manifold) on T ∗ Q , P = N (no Morse family), and the subman-ifold N = Q (no constraints on Q ). Then, we have a function F (nota family) on Q and its exterior derivative is a Lagrangian submanifoldof T ∗ Q . Then, S turns out to be the image space of a closed one-form dF : Q → T ∗ Q . This case includes the zero section of the projection T ∗ Q Q as well [55]. • If A is the identity mapping on T ∗ Q (no special symplectic manifold), P = N (no Morse family). Then we have a function F (not a family)on a submanifold of N of Q , and the Lagrangian submanifold S T ∗ Q = { p ∈ T ∗ Q : π Q ( p ) ∈ N, h Z, θ Q ( p ) i = h T π ( Z ) , dF i} (33)for any Z ∈ T p T ∗ N such that T π ( Z ) ∈ T N ⊂ T Q , [31]. • Let P = N (no Morse family) and the submanifold N = Q (no con-straints on Q ). Instead, consider the existence of a non-trivial specialsymplectic structure ( P, N, π, ω, A ). Then, a Lagrangian submanifold S of ( P, ω ) is defined by the pre-image of dF under the isomorphism A , that is S = A − ( dF ) = { y ∈ Y : h y, u i = h dE, T τ ( u ) i , ∀ u ∈ T y Y } (34)where E is defined on Q . • Let P = N (no Morse family), N be a proper submanifold of Q ,( R, N, τ, θ, A ) be a special symplectic manifold. Then the set S = { y ∈ R : τ ( y ) ∈ N, h θ, u i = h dF, T τ ( u ) i , ∀ u ∈ T y Y } (35)is a Lagrangian submanifold of ( R, − dθ ), and said to be generated bythe function F : N → R . We cite chapter 7 of [31] for a proof of thisstatement in a more general framework.14 .4 Lagrangian submanifolds of Tulczyjew’s symplectic space
Assume that a submanifold E of T T ∗ Q is defined in terms of the constraintfunctions Φ A : T T ∗ Q → R , Φ A ( q i , p i , ˙ q i , ˙ p i ) = 0 . If E is an IHS, then it must be a Lagrangian submanifold of the symplecticspace ( T T ∗ Q, ω TQ ) and the number of constraints must be 2 n assuming thatthe dimension of Q is n . Note that, in this case, the Poisson brackets of theconstraint functions must vanish [38], { Φ A , Φ B } = 0 . (36)Here, the Poisson bracket in (36) is the one induced by the Tulczyjew’ssymplectic two-form ω TQ in the subsection (2.5). This Poisson bracket canbe computed as { f, g } = ω TQ ( X f , X g ) = ∂f∂ ˙ p i ∂g∂q i − ∂g∂ ˙ p i ∂f∂q i + ∂f∂p i ∂g∂ ˙ q i − ∂g∂p i ∂f∂ ˙ q i . (37)The image of a Hamiltonian vector field is a Lagrangian submanifold of T T ∗ Q . Conversely, if a Lagrangian submanifold of T T ∗ Q is the image of avector field on T ∗ Q , then this vector field is a (at least locally) Hamiltonianvector field. To see this, we present the following calculations. Assume that E is a Lagrangian submanifold of T T ∗ Q and there exists a vector field X satisfying E = Im( X ). In Darboux’s coordinates, we write X as X = φ i ( q, p ) ∂∂q i + φ i ( q, p ) ∂∂p i , (38)where φ i and φ i are arbitrary functions on T ∗ Q . This local picture enablesus to write the fiber variables ( ˙ q, ˙ p ) in terms of the functions of the basevariables ( q, p ), and the following definition of the Lagrangian submanifoldof E given by E = (cid:8) ( q i , p j ; ˙ q i , ˙ p i ) ∈ T T ∗ Q : ˙ q i − φ i ( q, p ) = 0 , ˙ p i + φ i ( q, p ) = 0 (cid:9) . (39)Since E is a Lagrangian submanifold, the Poisson brackets of the definingequations in (39) must be identically zero. Here, the Poisson bracket is theone in (37). This requirement dictates n number of conditions ∂φ j ∂p i − ∂φ i ∂q j = 0 . φ = φ j dq j + φ i dp i . Locally, every closed one-form is exact. This impliesthat there exists a Hamiltonian function H depending on ( q, p ) satisfying dH = φ . As a result, the system (39) can be written as in form of theHamilton’s equations (4).A stronger result follows from the generalized Poincar´e lemma [7, 27,38, 52, 55]. The generelized Poincar´e guarantees that for a Lagrangiansubmanifold E of T T ∗ Q , whether it is explicit or implicit, there exists aMorse family of functions generating E . This theorem is also known asMaslov-H¨ormander theorem [6, 8, 11, 12]. We record here a Morse familygenerating the Lagrangian submanifold E as follows R P π (cid:15) (cid:15) F o o T ∗ T ∗ Q π T ∗ Q (cid:15) (cid:15) T T ∗ Q β Q o o τ T ∗ Q y y ttttttttt N (cid:31) (cid:127) / / T ∗ Q (40)where N is a submanifold of the cotangent bundle T ∗ Q , and F is a Morsefamily defined on the total space P of the smooth bundle ( P, π, N ). On thelocal chart, the Lagrangian submanifold E generated by F is computed tobe E = (cid:26)(cid:18) q i , p i ; ∂F∂p i , − ∂F∂q i (cid:19) ∈ T T ∗ Q : ∂F∂λ a ( q, p, λ ) = 0 (cid:27) . (41)Note that, if the Morse family F does not depend on the fiber variables ( λ ),then E becomes explicit.Let us comment on a particular case. Consider the following constrainedHamiltonian (Dirac) system˙ q i = ∂H∂p i ( q, p ) , ˙ p i = − ∂H∂q i ( q, p ) , Φ α ( q, p ) = 0 , (42)where Φ α , for α = 1 , ..., k , are real valued functions defining a constraintsubmanifold M ⊂ T ∗ Q [41]. Note that, we prefer to denote the generatorby H to highlight that it is a Hamiltonian function in the classical sense.Diagrammatically, we have the following picture generating the dynamics R P F o o (cid:15) (cid:15) T ∗ T ∗ Q π T ∗ Q (cid:15) (cid:15) T T ∗ Q β Q o o τ T ∗ Q y y ttttttttt T ∗ Q T ∗ Q (43)where P is the product manifold T ∗ Q × R k and the Morse family is givenby F ( q, p, λ ) = H ( q, p ) + λ α Φ α ( q, p ).16 Hamilton–Jacobi theory for implicit systems
A HJE is a partial differential equation for a generating function S ( q i , t ) on Q and the time t given by ∂S∂t + H (cid:18) q i , ∂S∂q i (cid:19) = 0 . (44)Note that, the generalized momenta do not appear in (44), except as deriva-tives of S . This equation is a necessary condition describing the externalgeometry in problems of calculus of variations. Hamilton’s principal func-tion S = S ( q i , t ), which is the solution of the HJE, and the classical function H are both closely related to the classical action S = Z Ldt.
The function S is a generating function for a family of symplectic flowsthat describes the dynamics of the Hamilton equations. If the generatingfunction is separable in time, then we can make an ansatz S ( q i , t ) = W ( q i ) − Et, where E is the total energy of the system. Then, HJE in (44) reducesto H (cid:18) q i , ∂W∂q i (cid:19) = E. (45)Physically, this constant is identified with the energy of the mechanicalsystem.Let us summarize the geometric Hamilton-Jacobi theory. For this, firstconsider a Hamiltonian vector field X H on T ∗ Q , and a one-form section γ on Q . We define a vector field X γH on Q by X γH = T π ◦ X H ◦ γ. (46)This definition implies the commutativity of the following diagram. T ∗ Q π Q (cid:15) (cid:15) X H / / T T ∗ Q T π Q (cid:15) (cid:15) Q γ > > X γH / / T Q (47)We enunciate the following theorem.17 heorem 1.
The closed one-form γ = dW on Q is a solution of theHamilton–Jacobi equation (45) if the following conditions are satisfied:1. The vector fields X H and X γH are γ -related, that is T γ ( X γ ) = X ◦ γ. (48)
2. Or equivalently, if the following equation is fulfilled d ( H ◦ γ ) = 0 . Proof.
We refer [13, 32, 33] for proof of this theorem.The first item in the theorem says that if (cid:0) q i ( t ) (cid:1) is an integral curve of X γH ,then (cid:0) q i ( t ) , γ j ( q ( t )) (cid:1) is an integral curve of the Hamiltonian vector field X H , hence a solution of the Hamilton’s equations (4). Such a solution ofthe Hamiltonian equations is called horizontal since it is on the image ofa one-form on Q . In the local picture, the second condition implies thatexterior derivative of the Hamiltonian function on the image of γ is closed,that is, H ◦ γ is constant H (cid:0) q i , γ j ( q ) (cid:1) = cst . (49)Under the assumption that γ is closed, we can find (at least locally) a func-tion W on Q satisfying dW = γ . After the substitution of this, equation(49) retrieves the HJE (45). A first-order differential equation on a manifold M is a submanifold of itstangent bundle. The submanifold is said to be an explicit differential equa-tion (EDE) if it is image of a vector field defined on M , otherwise, it is calledimplicit (IDE) [27, 39, 41, 45, 47]. Consider a local coordinate system ( q i )on M , and the induced coordinates ( q i , ˙ q i ) on its tangent bundle T M . AnIDE can be written in form˙ q i = g i ( q, λ ) , f a ( q, λ ) = 0 (50)for i with certain values running from 1 , ..., n and a = 1 , ..., k . Here, g i and f a are real valued differentiable functions on ( q, λ ) ∈ M × R k . Note that f a ’s form a matrix of maximal rankrank (cid:18) ∂f a ∂q i , ∂f a ∂λ b (cid:19) = k. (51)18his implies that only certain ˙ q i ’s are expressible as in (50) depending oncertain ( q j , λ a ) for some a ’s equal to k ≤ n .A solution of an IDE is a curve φ on M satisfying that the tangentvectors ˙ φ ( t ) belong to E for all t . The submanifold E is called integrable iffor all vectors v ∈ E , there exists a solution φ satisfying v = ˙ φ ( t ) for some t .Equivalently, we may define “integrability” of an IDE without refering to thesolution curves as follows. We say that the IDE is integrable if the restrictionof the tangent bundle projection τ M to E is surjective summersion and if E ⊂ T ( τ M ( E )).In principle, an IDE is not necessarily integrable. For example, considerthe following case [47]. Example 1.
Let E ⊂ T T ∗ Q be a system of IDE, where Q is coordinated by q and T ∗ Q by ( q, p ) , defined by E = { ( q, p, ˙ q, ˙ p ) ∈ T T ∗ Q, q + p + ( ˙ q + 1) + ˙ p = k } (52) is not integrable in points q + p = k , ˙ q = − and ˙ p = 0 with q = 0 . Nonetheless, we can develop an algorithm to extract its integrable part[47]. This algorithm works as follows. We denote the projection of thesubmanifold E onto M by C . Each step of the algorithm, there is a fiberbundle (cid:0) E k , τ k , C k (cid:1) consisting of two submanifolds E k ⊂ E and C k ⊂ C ,and a (surjective submersion) projection τ k : E k → C k . The first step isinitiated by choosing (cid:0) E = E, C = C (cid:1) and, iteratively, the further stepsare defined by E := E ∩ T C C := τ Q (cid:0) E (cid:1) τ : E → C → ... → E k := E k − ∩ T C k − C k := τ Q (cid:0) E k (cid:1) τ k : E k → C k → ... . In finite dimensions, there is an end for the algorithm, that is, there exists athree tuple (cid:0) E f , τ f , C f (cid:1) satisfying that E f +1 = E f and C f +1 = C f . Notethat, the final manifold E f is integrable. We call E f the integrable part of E . Constrained Example 2.
Consider the following set ( x, p, r, s, ˙ x, ˙ p, ˙ r, ˙ s ) on T R , with thefollowing equations [47] E = (cid:26) r = p, s = 0 , ˙ r = − ∂H∂x ( x, p ) , ˙ s = ˙ x − ∂H∂p ( x, p ) (cid:27) (53)19 ere, E = E and C = { ( x, p, r = p, s = 0) } , such that T C = { ˙ x, ˙ p, ˙ r =˙ p, ˙ s = 0 } . So, E ∩ T C = (cid:26) ˙ x = ∂H∂p ( x, p ) , ˙ r = ˙ p = − ∂H∂x ( x, p ) , r = p, s = 0 (cid:27) (54) From here, any other iteration E k = E and T C k = T C . Hence, (54) isthe integrable part of (53) . In this section we develop a geometric Hamilton-Jacobi theory for IDE. Ourproblem is that given a set of IDE, we are not necessarily provided with aHamiltonian vector field as explained in former sections. Here, we proposetwo methods to construct our theory. The first method consists of a theorywhich does refer to vector fields. The second is based on the constructionof a local vector field defined on the image of a section, but not globally onthe phase space.Let us start with the first method. Consider a submanifold E of T T ∗ Q .By projecting E by the tangent mapping T π Q onto the tangent bundle T Q , we arrive a submanifold
T π Q ( E ) of T Q . Note that, E refers to anIDE on T ∗ Q , whereas T π Q ( E ) refers to an IDE on Q . If E is integrable,then T π Q ( E ) is integrable too. We see this by considering the projection T π Q ( V ) = v ∈ T π Q ( E ) of an element V ∈ E . Note that, if ϕ is a curve on T ∗ Q and it is tangent to V ∈ E , then π Q ◦ ϕ is curve on Q which is tangentto v ∈ T π Q ( E ). This shows that the projections of the solutions of E aresolutions of T π Q ( E ). Our aim is to discuss the inverse question, startingfrom the solutions of T π Q ( E ) construct solutions of E , that is to lift thesolutions on Q to the cotangent bundle T ∗ Q . This is the philosophy of ageometric Hamilton–Jacobi theory. (Recall the geometric Hamilton–Jacobitheory exposed in subsection 4.1.) Furthermore, if E were not integrable, wewould have to perform the integrability algorithm explained in subsection4.2.To answer the question we have to introduce a section γ : Q → T ∗ Q . Ifany solution ψ : R → Q of T π Q ( E ) is a solution γ ◦ ψ : R → T ∗ Q of E ,then we denote the submanifold T π Q ( E ) by E γ , and say that E and E γ are γ − related. We illustrate this in a diagram.20 T T ∗ QC T ∗ Q T Q E γ C ∩ Im( γ ) Q R i τ T ∗ Q T πQ i π Q τ Q ii γ ψ In coordinates, a submanifold E of T T ∗ Q can be given by the set offunctions Φ A : T T ∗ Q → R , Φ A ( q i , p i , ˙ q i , ˙ p i ) = 0The projection of E onto T ∗ Q by means of τ T ∗ Q results with a submanifold C of T ∗ Q given by the set of functionsΨ α ( q i , p i ) = 0 . Consider the intersection of the projected submanifold C and the imagespace of the one-form γ . We denote this globally by C ∩ Im( γ ) and locallyby Ψ α ( q i , γ j ( q )) = 0. If a solution curve is represented by ψ i ( t ) ⊂ Q , thecomposition γ ◦ ψ ( t ) = ( ψ i ( t ) , γ i ( ψ ( t ))) is a curve on T ∗ Q and the timederivative of the curve is ddt ( γ ◦ ψ )(0) = T γ ( ψ (0)) · ˙ ψ (0)= (cid:18) ψ i (0) , γ i ( ψ (0)) , ˙ ψ i (0) , ∂γ j ∂q i ˙ ψ i (0) (cid:19) . Then, the equations of the submanifold E along γ take the formΦ A (cid:18) q i , γ i ( q ) , ˙ q i , ∂γ j ∂q i ˙ q i (cid:19) = 0 , (55)provided that Ψ α ( q i , γ i ( q )) = 0 along γ ( Q ) ⊂ C . Here, we assumed that ψ i (0) = q i .In the second case, we consider an additional section σ : T ∗ Q → T T ∗ Q such that σ ( C ) ⊂ E . 21 T T ∗ QC ∩ Im( γ ) T ∗ Q T Q E γ Q R i τ T ∗ Q T πQ σ i π Q τ Q iγ ψ Since E is implicit, there may exist several vectors in E projecting to thesame point, say c , in C . The role of the section σ is to reduce this unknownnumber to one. We are additionally require that the domain of the section σ be the intersection of Im( γ ) and C since, for implicit systems, C may notbe the whole of T ∗ Q . As a result, we arrive at a vector field X σ . Note that X σ satisfies ι X σ ω Q = Θ( γ ( q )) . (56)for an arbitrary one-form Θ defined on γ ( q ).We define a vector field X γσ on the tangent bundle T Q by the commu-tation of the following diagram. T ∗ Q π (cid:15) (cid:15) X σ / / T ( T ∗ Q ) T π (cid:15) (cid:15) Q γ > > X γσ / / T Q
Explicitly, X γσ = T π ◦ X σ ◦ γ. (57)In local coordinates, the vector field X σ and its projection X γσ can bewritten as X σ = σ i ( q, γ ( q )) ∂∂q i + σ i ( q, γ ( q )) ∂∂p i , X γσ = σ i ( q, γ ( q )) ∂∂q i , (58)respectively. Using a one-form section γ on Q , the tangent lift of the pro-jected vector field X γσ is T γ ( X γσ ) = σ i (cid:18) ∂∂q i + ∂γ j ∂q i ∂∂p j (cid:19) (59)22sing (57), we find an expression relating the section σ and the vector fieldsas follows. σ i ( q, γ ( q )) ∂γ j ∂q i ( q ) = σ j ( q, γ ( q )) . (60)We are ready now to state the following lemma. Lemma 2.
Given the conditions above, we say that: the two vector fields X σ and X γσ are γ -related if and only if (60) is fulfilled. Again, this construction can be mimicked for nonintegrable IDE that aresubmanifolds E of a higher-order bundle, after performing the integrabilityalgorithm given in 4.2. As we have summarized in subsection (3.4), for every Lagrangian submani-fold E of T T ∗ Q , there exists a Morse family F : T T ∗ Q → R generating E .This enables us to write E locally as E = (cid:26)(cid:18) q i , p i ; ∂F∂p i , − ∂F∂q i (cid:19) ∈ T T ∗ Q : ∂F∂λ a = 0 (cid:27) (61)where F = F ( q, p, λ ). We cite two important studies [7, 42] related with theproblem addressed in this subsection.We introduce a differential one-form γ on the base manifold Q . Seethat, Im( γ ) is a Lagrangian submanifold of T ∗ Q so that there is an inclusion ı : Im( γ ) T ∗ Q . Use the inclusion ı in order to pull the bundle ( P, π, T ∗ Q )back over Im( γ ). By this, one arrives at a fiber bundle ( ı ∗ P, ı ∗ π, Im( γ )). Forthe present case, the commutative diagram for a generic pullback bundleexhibited in (1) takes the following particular form. ı ∗ P ε / / ı ∗ π (cid:15) (cid:15) P π (cid:15) (cid:15) Im( γ ) ı / / T ∗ Q (62)Here, the total space ı ∗ P = { ( γ ( q ) , z ) ∈ Im( γ ) × P : π ( z ) ∈ Im( γ ) }
23f the pull-back bundle is a submanifold of P with ε is the correspondinginclusion. A local coordinate system on ı ∗ P can be taken as ( q, γ ( q ) , λ ).Although restriction of the Morse family on ı ∗ P should formally be writtenas F ◦ ǫ , we will abuse notation by still denoting it by F ,but to highlight thedifference we shall write the arguments of the function as F = F ( q, γ ( q ) , λ ).The submanifold generated by F = F ( q, γ ( q ) , λ ) is given by E | Im( γ ) = (cid:26)(cid:18) q i , γ i ( q ); ∂F∂p i , − ∂F∂q i (cid:19) ∈ T T ∗ Q : ∂F∂λ a = 0 (cid:27) . (63)Note that, if the Lagrangian submanifold E is the image of a Hamiltonianvector field X H , then E | Im( γ ) reduces to the image space of the composition X H ◦ γ .The submanifold E | Im( γ ) exhibited in (63) does not depend on the mo-mentum variables. This enables us to project it to a submanifold E γ of T Q by the tangent mapping
T π Q as follows E γ = T π Q ◦ E | Im( γ ) = (cid:26)(cid:18) q i , ∂F∂p i ( q, γ ( q ) , λ ) (cid:19) ∈ T Q : ∂F∂λ a = 0 (cid:27) . (64)Note that the submanifold E γ defines an implicit differential equation on Q .We state the generalization of the Hamilton-Jacobi theorem (1) as follows. Lemma 3.
The following conditions are equivalent for a closed one-form γ :1. The Lagrangian submanifold E in (61) and the submanifold E γ in (64)are γ -related, that is T γ ( E γ ) = E | Im( γ )
2. And it is fulfilled that dF ( q, γ ( q ) , λ ) = 0 , where F is the Morse familygenerating E .Proof. The one-form γ = γ i dq i is closed, that is, ∂γ i ∂q j = ∂γ j ∂q i . The firstassertion in lemma 3 can be written locally as ∂γ j ∂q i ∂F∂p j ( q, γ ( q ) , λ ) + ∂F∂q i ( q, γ ( q ) , λ ) = 0 , (65)with the conditions that ∂F/∂λ a = 0. We make a simple calculation tocompute the exterior derivative of the Morse family as follows dF ( q, γ ( q ) , λ ) = ∂F∂q j dq j + ∂F∂p i γ i,j dq j + ∂F∂λ a dλ a . (66)24ote that, after the substitution of (65) into (66) and by employing thesymmetry ∂γ i ∂q j = ∂γ j ∂q i , we arrive at that the exterior derivative of F vanisheswhen p = γ ( q ). To prove the reverse direction, it is enough to repeat thesesteps in reverse order.Assume now that the one-form γ is exact so that γ = dW ( q ) for some realvalued function W called the characteristic function on the base manifold Q .Then the second condition in lemma 3 gives the implicit Hamilton-Jacobiequation (IHJ equation) F (cid:18) q, ∂W∂q , λ (cid:19) = cst , ∂F∂λ a (cid:18) q, ∂W∂q , λ (cid:19) = 0 . (67)In the lemma (3), if the Lagrangian submanifold E is the image of a Hamil-tonian vector field X H , then E γ becomes the image space of the vector field X γH in (46) and we retrieve the classical HJ theory given in (47).It is possible to generalize Lemma (3) by replacing the image space Im γ by an arbitrary Lagrangian submanifold S of T ∗ Q . Note that, according tothe generalized Poincar´e lemma, there exists a Morse family W on the totalspace of a smooth bundle ( R, τ, Q ) generating the Lagrangian submanifold S . We equip the total space R with the coordinates ( q i , µ α ). Then we havethat the Lagrangian submanifold S can be written as S = (cid:26)(cid:18) q i , ∂W∂q i ( q, µ ) (cid:19) ∈ T ∗ Q : ∂W∂µ β ( q, µ ) = 0 (cid:27) . (68)Now, the inclusion in the diagram (62) becomes ı : S T ∗ Q . In this case,the restriction of the Morse family F generating the submanifold E of T T ∗ Q to the inclusion ǫ defines the following submanifold E | S = (cid:26)(cid:18) q i , ∂W∂q i ; ∂F∂p i , − ∂F∂q i (cid:19) ∈ T T ∗ Q : ∂F∂λ a = 0 , ∂W∂µ β ( q, µ ) = 0 (cid:27) , (69)where W = W ( q, µ ) and F = F ( q, ∂W∂q i ( q, µ ) , λ ). The submanifold E | S doesnot depend on the momentum variable p explicitly. So that, its projection E S to the tangent bundle T Q by means of
T π Q is well-defined and given by E S = (cid:26)(cid:18) q i ; ∂F∂p i ( q, ∂W∂q ( q, µ ) , λ ) (cid:19) ∈ T Q : ∂F∂λ a = 0 , ∂W∂µ β = 0 (cid:27) . (70)We are now ready to state a generalization of the lemma (3) as follows.25 emma 4. Let S be a Lagrangian submanifold of T ∗ Q generated by Morsefamily W = W ( q, µ ) defined on the total space of a bundle ( R, τ, Q ) . Thefollowing conditions are equivalent1. The Lagrangian submanifold E in (61) and the submanifold E S in (70)are S -related, that is T ( dW | µ )( E S ) = E | S for every µ , where E | S is in (69).2. dF (cid:16) q, ∂W∂q ( q, µ ) , λ (cid:17) = 0 for all µ . Here, F is the Morse family gener-ating E . We only give some clues instead of writing the whole proof of this lemmasince it is very similar to the proof of lemma 3. The closedness of the one-form γ is replaced by the commutativity of the second partial derivatives ofthe Morse family W with respect to q .Let us comment on the notation T ( dW | µ ) as well. If the fiber variable µ is frozen, then the exterior derivative dW | µ of the Morse family W becomesa differentiable mapping from Q to T ∗ Q and its tangent mapping T ( dW | µ )goes from T Q to T T ∗ Q . We remark that this last comment is a generaliza-tion of the Hamilton-Jacobi theory for the Lagrangian submanifolds studiedin [4] as well. Example 3.
Let V be a nonholonomic k -dimensional distribution on Q spanned by the vector fields X a . We define the following Morse family onthe total space of the fiber bundle T ∗ Q × R k F ( q, p, λ ) = p i λ a X ia ( q ) . (71) Here F is a Morse family and determines a Lagrangian submanifold of T T ∗ Q given by ˙ q i = λ a X ia ( q ) , ˙ p j = − p i λ a ∂X ia ∂q j , p i X ia ( q ) = 0 . (72) This system is integrable according to [27]. The corresponding Hamilton-Jacobi equation is computed to be ∂W∂q i λ a X ia ( q ) = cst . (73)26 Complete solutions of the HJ equation for IHS
Before writing the complete solution of a HJ equation of IHS, we first inves-tigate the complete solutions of HJ equations for explicit systems in termsof Morse families and Lagrangian submanifolds.
In the classical sense, a solution W of the HJ equation (45) is called completeif it depends on additional variables that equal in number to the dimension ofthe base manifold Q [1]. To illustrate this, we start by considering two copiesof the configuration manifold and denote them by Q and ¯ Q . Endow thesemanifolds with local coordinates ( q i ) and (¯ q j ), respectively. The numberof arbitrary parameters for the general solution is given by j , which doesnot necessarily equal i . A complete solution is a real valued function W = W (¯ q, q ) on the product space ¯ Q × Q that resolves the HJ equation (45).This function generates three different Lagrangian submanifolds, let us showthem.Construct the fiber bundle structure ( ¯ Q × Q, ρ, Q ). Here, the bundleprojection ρ is assumed to be a projection to the second factor. As discussedpreviously, a real valued function W = (¯ q, q ) on the total space ¯ Q × Q iscalled a Morse family if the matrix [ ∂ W/∂ ¯ q i ∂q j ] is non-degenerate. Wedraw the following diagram R ¯ Q × Q ρ (cid:15) (cid:15) W o o T ∗ Q π Q (cid:15) (cid:15) Q Q (74)In this case, the Morse family W defines a Lagrangian submanifold of T ∗ Q given by S = (cid:26)(cid:18) ( q i , ∂W∂q i (¯ q, q ) (cid:19) ∈ T ∗ Q : ∂W∂ ¯ q i (¯ q, q ) = 0 (cid:27) . (75)Note that, by changing the roles of Q and ¯ Q , we may define a bundlestructure ( ¯ Q × Q, ¯ ρ, ¯ Q ) over the manifold ¯ Q and obtain a diagram similar to(74). In this case, W defines a Lagrangian submanifold ¯ S of T ∗ ¯ Q as follows¯ S = (cid:26)(cid:18) ¯ q i , ∂W∂ ¯ q i (¯ q, q ) (cid:19) ∈ T ∗ ¯ Q : ∂W∂q j (¯ q, q ) = 0 (cid:27) . (76)27nother Lagrangian submanifold generated by W is the result of fol-lowing observation. The cotangent bundle T ∗ ( ¯ Q × Q ) = T ∗ ¯ Q × T Q of theproduct space is a symplectic manifold equipped with the symplectic two-form ω ¯ Q ⊖ ω Q [55]. It is evident that image of the exterior derivative dW ofa complete solution W is a Lagrangian submanifoldˆ S = (cid:26)(cid:18) ¯ q, q ; ∂W∂ ¯ q , ∂W∂q (cid:19) ∈ T ∗ ( ¯ Q × Q ) (cid:27) . (77)It is known that a Lagrangian submanifold of T ∗ ( ¯ Q × Q ) defines a sym-plectomorphism between T ∗ ¯ Q and T ∗ Q . The Morse family W = W (¯ q, q )generates a symplectomorphism according to the following identity θ ¯ Q ⊖ θ Q = ¯ p i d ¯ q i − p i dq i = dW ( q, ¯ q ) , (78)where we assume the Darboux’ coordinates on the cotangent bundles. Inthe local picture, the induced symplectomorphism is given by ϕ : T ∗ ¯ Q → T ∗ Q : (cid:18) ¯ q i , ∂W∂ ¯ q j (cid:19) → (cid:18) q i , − ∂W∂q j (cid:19) . (79) Let us first try to geometrize the complete solutions of the HJ equation forexplicit systems. A function W = W (¯ q, q ) is a complete solution of the HJequation (45) if the Hamiltonian function H is constant when it is restrictedto S exhibited in (75). That is, a complete solution W is the one satisfying H (cid:18) q, ∂W∂q (¯ q, q ) (cid:19) = cst , ∂W∂ ¯ q = 0 . (80)Using the symplectic diffeomorphism ϕ in (79) generated by the function W , we pull the function H back to T ∗ ¯ Q and see that ϕ ∗ H is a constant. Sothat the dynamics generated by ϕ ∗ H is trivial.Now, assume that, we have a Lagrangian submanifold E of T T ∗ Q . Thenthere exists a Morse family F generating E . A complete solution of theimplicit Hamilton-Jacobi equation (67) is a smooth function W satisfying F (cid:18) q, ∂W∂q , λ (cid:19) = cst, ∂W∂ ¯ q (¯ q, q ) = 0 , ∂F∂λ (cid:18) q, ∂W∂q , λ (cid:19) = 0 . (81)We aim to pull the dynamics E or the Morse family F back to T ∗ ¯ Q . Toachieve this goal, we recall the definition of the pullback bundle in (1) and28pply it to the diffeomorphism (79). This way we obtain a fiber bundlestructure ( ϕ ∗ P, ϕ ∗ π, T ∗ ¯ Q ) where the total space is defined to be ϕ ∗ P = { (¯ z, r ) ∈ T ∗ ¯ Q × P : ϕ ( z ) = π ( r ) } equipped with the induced coordinates (¯ q i , ¯ p i , λ ), and ϕ ∗ π is the projectionto the second factor. We draw the following commutative diagram in orderto summarize the discussion. ϕ ∗ P ˆ ϕ / / ϕ ∗ π (cid:15) (cid:15) P π (cid:15) (cid:15) T ∗ ¯ Q ϕ / / T ∗ Q (82)Here, ˆ ϕ is a diffeomorphism and in the local coordinates readsˆ ϕ : ϕ ∗ P ↔ P : (cid:18) ¯ q i , ∂W∂ ¯ q i , λ (cid:19) ↔ (cid:18) q i , − ∂W∂q i , λ (cid:19) . (83)The pullback ¯ F = F ◦ ˆ ϕ of the Morse function F by ˆ ϕ is a Morse familyon the total space of the pullback bundle ( ϕ ∗ P, ϕ ∗ π, T ∗ ¯ Q ). Note that ¯ F isa constant function and the implicit differential equation generated by ¯ F renders trivial dynamics.Generalizing, the most general form of a Lagrangian submanifold of T ∗ ¯ Q × T ∗ Q is generated by a Morse family W defined on the total space ofthe fiber bundle ( R, τ, M ) where the base manifold M is a submanifold of¯ Q × Q . Let us depict it in a diagram. R R τ (cid:15) (cid:15) W o o T ∗ ¯ Q × T ∗ Q π ( ¯ Q × Q ) (cid:15) (cid:15) M ı / / ¯ Q × Q (84)A complete solution to the implicit Hamilton Jacobi equation (67) is a Morsefamily W defined on the total space R . Note that, the Morse family F gen-erating the dynamics on E reduces to a constant function on the Lagrangiansubmanifold generated by W . 29et us now depict the situation in coordinates. Assume that a subman-ifold M of ¯ Q × Q is defined by a number “ l ” of equations U a (¯ q, q ) = 0 , a = 1 , ..., l. (85)Consider a real function W ′ on the submanifold M and define an arbitrarycontinuation W = W ′ + ν a U a of W ′ to the product space P = ¯ Q × Q × Λ where ( ν a )’s are the Lagrangemultipliers defining a local coordinate system for Λ. This W is a completesolution of the implicit Hamiltonian dynamics (3) generated by F if F (cid:18) q i , ∂W ′ ∂q i , λ (cid:19) = cst, U a (¯ q, q ) = 0 , a = 1 , .., l. An implicit description of a Lagrangian submanifold of T ∗ ¯ Q × T ∗ Q gener-ated by W or equivalently of the corresponding diffeomorphism ϕ can becomputed by θ Q ⊖ θ ¯ Q = p i dq i − ¯ p i d ¯ q i = d (cid:0) W ′ ( q, ¯ q ) + ν a U a ( q, ¯ q ) (cid:1) , (86)where the θ ¯ Q and θ Q are the canonical one-forms on ¯ Q and Q , respectively.In this case, the momenta p ∈ T ∗ q Q and ¯ p ∈ T ∗ ¯ q ¯ Q can be explicitly stated as¯ p i = − ∂W ′ ∂ ¯ q i − ν a ∂U a ∂ ¯ q i p i = ∂W ′ ∂q i + ν a ∂U a ∂q i U a (¯ q, q ) = 0 , a = 1 , .., l. (87) A Lagrangian function L is a real valued function on T Q . Consider thecoordinates ( q i , ˙ q i ) on T Q induced those from Q . We define a vertical endo-morphism S given by S = ∂/∂ ˙ q i ⊗ dq i . Note that, S is a (1 , Q . In terms of S the Cartan one-form θ L is defined to be θ L = S ∗ ( dL ) = (cid:0) ∂L/∂ ˙ q i (cid:1) dq i dL is the exterior derivative of a Lagrangian density. The Cartantwo-form derivates from the Cartan one-form ω L = − dθ L . See that ω L issymplectic if the Hessian matrix( W ij ) = (cid:18) ∂ L∂ ˙ q i ∂ ˙ q j (cid:19) (88)is not singular. In this case, the fiber derivative (or the Legendre transfor-mation) F L : T Q T ∗ Q : ( q i , ˙ q j ) (cid:18) q i , ∂L∂ ˙ q j (cid:19) (89)becomes a symplectomorphism relating the Cartan two-form ω L on T Q andthe canonical symplectic two-form ω Q on T ∗ Q . The Lagrangian is said tobe hyperregular if the fiber derivative F L is a global diffeomorphism.The energy is defined as E L = ∆( L ) − L , a real valued function on T Q where the Liouville vector field is ∆ = ˙ q i ∂/∂ ˙ q i . The Hamiltonian is retrievedthrough H ( q, p ) = E L ◦ F L − . (90)If the Lagrangian is regular, or equivalently, if ω L is symplectic, then theLagrange equations can be expressed geometrically as ι ξ L ω L = dE L , (91)whose solution ξ L is called a Euler–Lagrange vector field explicitly given by ξ L = ˙ q i ∂∂q i + ξ i ( q, ˙ q ) ∂∂ ˙ q i . (92)The integral curves ( q i ( t ) , ˙ q i ( t )) are lifts of their projections ( q i ( t )) on Q andare solutions of the system of differential equations dq i ( t ) dt = ˙ q i , d ˙ q i ( t ) dt = ξ i , (93)which is equivalent to a second-order differential equation d q i ( t ) dt = ξ i . (94)The curves ( q i ( t )) in Q are called the solutions of ξ L that correspond withthe solutions of the Euler–Lagrange equation ddt (cid:18) ∂L∂ ˙ q i (cid:19) = ∂L∂q i . (95)If the Lagrangian is regular, then the fiber derivative (89) has the fol-lowing geometry. 31 T T Q, ω L ) ( T T ∗ Q, ω TQ ) T ∗ T QT Q R T F Lτ TQ α Q dLξ L L In this case, the Hamiltonian vector field X H associated with the Hamilto-nian function H in (90) and ξ L presented in (91) are related as X H ◦ F L = T F L ◦ ξ L . The diffeomorphisms α Q and β Q maps Lagrangian submanifolds into La-grangian submanifolds, α Q (Im( X H )) = Im( dL ) , β Q ◦ α − Q (Im( dL )) = Im( dH ) , whereas the Hamiltonian and the Lagrangian vector fields are related totheir corresponding Lagrangian submanifolds as β Q ◦ X H = dH, α Q ◦ T F L ◦ ξ L = dL, respectively. In this section we depict the geometric interpretation of a HJ theory forLagrangian dynamics. For it, we present the EL equations in terms of Morsefamilies and special symplectic structures.Recall the special symplectic structure on the left side of Tulczyjew’striple. T ∗ T Q π TQ (cid:15) (cid:15) T T ∗ Q α Q o o τ TQ z z ✉✉✉✉✉✉✉✉✉ R T ∗ Q L o o . (96)In the induced local picture on T T ∗ Q , by following the procedure presentedin subsection (3.2), we compute the Lagrangian submanifold E generatedby the Lagrangian L as E = (cid:26)(cid:18) q i , ∂L∂ ˙ q i ; ˙ q i , ∂L∂q i (cid:19) ∈ T T ∗ Q (cid:27) (97)which is equivalent to the EL equations (95). We can generate this La-grangian submanifold from the right wing (the Hamiltonian side) of the32riple (22) by defining a proper Morse family F L → H on the Pontryagin bun-dle P Q = T Q × Q T ∗ Q over T ∗ Q . On a local chart, the energy function F ( q, p, ˙ q ) = p i ˙ q i − L ( q, ˙ q ) (98)satisfies the requirements of being a Morse family. Hence, F generates a La-grangian submanifold of T ∗ T ∗ Q as defined in equation (26). In coordinates( q i , p i , α i , β i ) of T ∗ T ∗ Q , this Lagrangian submanifold is given by α i = ∂F∂q i = − ∂L∂q i , β i = ∂F∂p i = ˙ q i , ∂F∂ ˙ q i = p i − ∂L∂ ˙ q i = 0 (99)The inverse of the isomorphism β Q maps this Lagrangian submanifold to theLagrangian submanifold E presented in (97). Here, we record the followingdiagram for this. R P Q π (cid:15) (cid:15) F o o T ∗ T ∗ Q π T ∗ Q (cid:15) (cid:15) T T ∗ Q β Q o o τ T ∗ Q y y ttttttttt T ∗ Q T ∗ Q (100)For regular cases, the Morse family F on P Q can be reduced to a Hamil-tonian function H on T ∗ Q . For degenerate cases, a reduction of the totalspace P Q to a subbundle larger than T ∗ Q is possible depending on degen-eracy level of Lagrangian function [8]. There exists an intrisecally geometricprocedure for dealing with constraints in Hamiltonian and Lagrangian me-chanics. It has been available since 1979, with advantages over the Dirac-Bergman algorithm, it is the Gotay-Bergman algorithm [22, 23, 24, 25] (readAppendix for brief description of the method). To write the associated HJ equation of the EL equations generated by (pos-sibly) degenerate Lagrangian densities, we apply lemma 3 to the Lagrangiansubmanifold presented in (97). Accordingly, we arrive at that the implicitHJ equation F ( q, γ ( q ) , ˙ q ) = γ i ( q ) ˙ q i − L ( q, ˙ q ) = cstfor a closed one-form γ = γ i ( q ) dq i . Taking the exterior derivative of thisequation, we arrive at the following local picture of the Hamilton-Jacobiequation for a Lagrangian L ˙ q i ∂γ i ∂q j ( q ) − ∂L∂q j ( q, ˙ q ) = 0 , γ i ( q ) − ∂L∂ ˙ q i ( q, ˙ q ) = 0 . (101)33o illustrate this, we propose two particular problems [34]. Example 4.
Consider the degenerate Lagrangian L on T R given by L ( q, ˙ q ) = L ( q , q , q , ˙ q , ˙ q , ˙ q ) = 12 ( ˙ q + ˙ q ) , and the Whitney bundle T R ⊕ T ∗ R fibered on T ∗ R and parameterizedby ( q , q , q , ˙ q , ˙ q , ˙ q , p , p , p ). The Lagrange multipliers correspond with( λ i ) = ( ˙ q , ˙ q , ˙ q ). Following (98), we define the Morse family F ( q, ˙ q, p ) = ˙ q p + ˙ q p + ˙ q p −
12 ( ˙ q + ˙ q ) . that generates a Lagrangian submanifold E of T T ∗ R . Explicitly, E = { ( q , q , q , p , p , p ; ˙ q , ˙ q , ˙ q , , , ∈ T T ∗ R : p = ˙ q + ˙ q , p = ˙ q + ˙ q , p = 0 } . (102)It is evident that the projection of E to T ∗ R results with the following 4dimensional submanifold C = { ( q , q , q , p , p , p ) ∈ T ∗ R : p = p , p = 0 } . Consider now the closed one-form γ : R T ∗ R . According to theLagrangian HJ theorem (3), the system (101) in this particular case takesthe form: [1] ˙ q ∂γ ∂q + ˙ q ∂γ ∂q + ˙ q ∂γ ∂q = 0 , [2] ˙ q ∂γ ∂q + ˙ q ∂γ ∂q + ˙ q ∂γ ∂q = 0 , [3] ˙ q ∂γ ∂q + ˙ q ∂γ ∂q + ˙ q ∂γ ∂q = 0 , [4] γ − ˙ q − ˙ q = 0[5] γ − ˙ q − ˙ q = 0[6] γ = 0 . (103)It is immediate to see from equations [4] and [5] that γ = γ . If γ is closedand γ = 0, one obtains that γ and γ are independent of q . Then [3]in (103) is automatically satisfied. The substitution of [4] into [1] and [2]results in γ ∂γ ∂q = 0 , γ ∂γ ∂q = 0 . A nontrivial solution is possible if γ and γ are constants. Hence, we recordthe one-form γ ( q ) = ( q , q , q , c, c, .
34n terms of submanifolds, that is, according to the first condition intheorem 3, the picture is the following. The constant character of the one-form defines a constraint ˙ q + ˙ q = c , on the velocity variables. We firstrestrict the submanifold E to the image space of the γ , we arrive at E | Im( γ ) = (cid:8) ( q , q , q , c, c,
0; ˙ q , ˙ q , ˙ q , , , ∈ T T ∗ R : c = ˙ q + ˙ q (cid:9) whose generic version is given in (63). The projection of E | Im( γ ) to thetangent bundle T R by T π R results in a five dimensional submanifold E γ = (cid:8) ( q , q , q , ˙ q , ˙ q , ˙ q ) ∈ T R with ˙ q + ˙ q = c (cid:9) , of the tangent bundle T R . The system of equations (103) is equivalent tosaying that the tangent lift of E γ by the tangent mapping T γ equals E | Im( γ ) .It is indeed immediate to see that T γ ( E γ ) = E | Im( γ ) .In terms of vector fields, the situation is as follows. Consider a section σ of the tangent bundle τ T ∗ Q given by σ ( q , q , q , p , p , p ) = ( q , q , q , p , p , p ; c − ˙ q , ˙ q , ˙ q , , , C ∩ Im( γ ). Note that, Im( σ ) ⊂ E . Accordingly, we writethe following vector field X σ = c ∂∂q + ˙ q ◦ γ ( q ) ∂∂q + ˙ q ◦ γ ( q ) (cid:18) ∂∂q − ∂∂q (cid:19) . We project this vector field by
T π Q and arrive at the vector field X γσ , whichis locally the same as X σ . Composing with the section γ , T γ ( q , q , q , ˙ q , ˙ q , ˙ q ) = ( q , q , q ; c, c, , ˙ q , ˙ q , ˙ q , , , T γ ( X γσ ) = X σ (104)is obviously fulfilled. Example 5.
Consider the Lagrangian L on T R given by L ( q, ˙ q ) = L ( q , q , ˙ q , ˙ q ) = 12 ( ˙ q ) + q ( q ) . To recast the EL system generated by this Lagrangian density, we simplydefine the following Morse family on the Whitney sum T R ⊕ T ∗ R F ( q, p, ˙ q ) = p ˙ q + p ˙ q −
12 ( ˙ q ) − q ( q ) . (105)35his family generates the following Lagrangian submanifold E = (cid:8) ( q , q , p , p ; ˙ q , ˙ q , q q , ( q ) ) ∈ T T ∗ R : p = ˙ q , p = 0 (cid:9) . (106)The projection of E onto the cotangent bundle T ∗ R by the tangent bundleprojection τ T ∗ Q results with the following submanifold C = { ( q , q , p , p ) ∈ T ∗ R : p = 0 } . (107)According to theorem 3, we now introduce a closed one-form γ on R and require that the Morse family F in (105) is constant on the image space,that is F ( q, γ ( q ) , ˙ q ) = γ ( q ) ˙ q + γ ( q ) ˙ q −
12 ( ˙ q ) − q ( q ) = cst. For this case, the Hamilton Jacobi equations (101) turn out to be the fol-lowing set [1] ˙ q ∂γ ( q ) ∂q + ˙ q ∂γ ( q ) ∂q − q q = 0[2] ˙ q ∂γ ( q ) ∂q + ˙ q ∂γ ( q ) ∂q − ( q ) = 0[3] γ ( q ) − ˙ q = 0[4] γ ( q ) = 0 . (108)The closedness of γ , together with equation [4], imply that γ depends onlyon q . If we substitute this and equations [3] and [4] in [1] and [2], then wearrive at a partial differential equation and a constraint γ ( q ) ∂γ ( q ) ∂q − q q = 0 , q = 0 . Note that the constraint q = 0 implies that the system is automaticallysatisfied for any function γ = γ ( q ). Then, the one-form is described by γ ( q ) = ( q , q ; q , . (109)Let us assume that γ ( q ) = q as it is done in [34], and rewrite system(108) in terms of submanifolds. If the Lagrangian submanifold E in (106)is restricted to the image space of γ in (109), then the result becomes E | Im( γ ) = (cid:8) ( q , q , q , q , ˙ q , q q , ( q ) ) ∈ T T ∗ R : q = 0 (cid:9) . (110)This is projected to the tangent bundle T R by T π R in order to get thereduced dynamics E γ = { ( q , q ; ˙ q , ˙ q ) : q = 0 , ˙ q = q = 0 } . γ -relatedness of E and E γ can be checked with the following lift T γ ◦ E γ = ( q = 0 , q , q ,
0; ˙ q = 0 , ˙ q , ˙ q = 0 , . As a result, we have three constraints q = 0 , p = 0 and p = 0. So, theprojected submanifold C in (107) must be rectified as C = { ( q , q , p , p ) : q = 0 , p = 0 , p = 0 } ⊂ T ∗ R . Here, p = 0 is called the primary constraint, since it roots in the functionalstructure of the Lagrangian function, and the other two constraints q = 0and p = 0 are called the secondary constraints.Consider a section σ of the tangent bundle fibration τ T ∗ R given by σ ( q , q ; ˙ q , ˙ q ) = ( q , q , ˙ q , ˙ q ; ˙ q , ˙ q , q q , ( q ) ) . By restricting this section to the intersection Im( γ ) ∩ C , we arrive at thefollowing vector field X σ = ˙ q ◦ γ ( q ) ∂∂q . The projection of this vector field to T R is the vector field X γσ , and it looksexactly as X σ , at least locally. X γσ and X σ are γ related since the tangentlift by γ is given by T γ ( q, ˙ q ) = ( q , q , q ,
0; ˙ q , ˙ q , ˙ q , X γσ into X σ . Conclusions and future work
In this work we have presented a geometric Hamilton-Jacobi theory for sys-tems of implicit differential equations. In the general context, due to theimplicit character of the equations, the lack of a vector field has been solvedby the introduction of a local section σ . In the particular case of the im-plicit Hamiltonian dynamics, Morse families and special symplectic struc-tures have been employed to derive a Hamilton–Jacobi theory in which theMorse function plays the role of the Hamiltonian. This result has been par-ticulary applied to singular Lagrangians. We expect further applications ofthe theory in constraint Hamiltonian systems, Dirac systems, and etc.The obtainance of a Hamilton–Jacobi theorem through reduction is heresketched in terms of coisotropic reduction [37]. As future work, we aim atstating the problem of reduction of the implicit Hamilton-Jacobi theory un-der the Lie group symmetry of the implicit system of differential equations.37 cknowledgements This work has been partially supported by MINECO MTM 2013-42-870-Pand the ICMAT Severo Ochoa project SEV-2011-0087. One of us (OE) isgrateful for ICMAT for warm hospitality where some parts of the workshas been done. (OE) is also grateful Prof. Hasan G¨umral for valuablediscussions in the theory of Tulczyjew triples.38 ppendix
The Gotay–Nester algorithm
The Gotay-Nester algorithm is a suitable tool for reducing the dynamics ofsingular Hamiltonian or Lagrangian systems to a reduced manifold wherethe motion is well-defined. This algorithm was created as a generalization ofthe well-known Dirac-Bergman alrorithm which has local nature and doesnot cope with all the singularities appearing in dynamics. Let us brieflydescribe the Gotay-Nester algorithm [22, 23, 24, 25].Let us recall that given a singular Lagrangian L : T Q → R , the Legendretransform F L : T Q → T ∗ Q and the energy E L : T Q → R , one can define apresymplectic system ( M = F L ( T Q ) , ω ), where ω is the restriction of thecanonical symplectic form on T ∗ Q to M . We will assume that L is almostregular (the Legendre transformation is a submersion and surjective), then M = F L ( T Q ) is a submanifold of T ∗ Q The restriction of the Legendre mapping
F L : T Q −→ M to thissubmanifold is a submersion with connected fibers. In this case, M is calledthe submanifold of primary constraints. If L is almost regular, ker( T F L ) =ker( ω L ) ∩ V ( T Q ), where V ( T Q ) denotes the vertical bundle, and the fibersare connected, a direct computation shows that E L projects onto a function h : M → R . The inclusion of this submanifold is denoted by j : M −→ T ∗ Q and define ω = j ∗ ( ω Q ). The dynamics in the primary constraintmanifold is i X ω = dh , where h ∈ C ∞ ( M ) is the projection of the energy E L ∈ C ∞ ( T Q ). Now,there are two possibilities: the solution X defined at all the points of M issuch that X defines global dynamics and it is a solution (modulo ker ω ),in other words, there are only primary constraints. Or, we are in need ofa second submanifold M where ι X ω = dh and X ∈ T M . But sucha solution X is not necessarily tangent to M , so we have to impose anadditional tangency condition to M and obtain a new submanifold M along which there exists a solution. Continuing this process, we obtain asequence of submanifolds · · · M k ֒ → · · · ֒ → M ֒ → M ֒ → T ∗ Q where the general description of M l +1 is M l +1 := { p ∈ M l such that there exists X p ∈ T p M l satisfying i X ω = dh } .
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