An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications
aa r X i v : . [ m a t h - ph ] J a n An overview of the Hamilton–Jacobi theory: theclassical and geometrical approaches and someextensions and applications
Narciso Román-Roy ∗ Dept. of Mathematics. Universitat Politècnica de Catalunya.Ed. C3, Campus Nord. 08034 Barcelona, Spain.
January 12, 2021
Abstract
This work is devoted to review the modern geometric description of the Lagrangian andHamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical”Hamiltonian approach using canonical transformations is also analyzed. Furthermore, amore general framework for the theory is also briefly explained. It is also shown how, fromthis generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamicalsystems are recovered, and how the model can be extended to other types of physical sys-tems, such as higher-order dynamical systems and (first-order) classical field theories in theirmultisymplectic formulation.
Key words : Hamilton–Jacobi equations, Lagrangian and Hamiltonian formalisms, higher–ordersystems, classical field theories, symplectic and multisymplectic manifolds, fiber bundles.
AMS s. c. (2020):
Primary : 35F21, 70H03, 70H05, 70H20.
Secondary : 53C15, 53C80, 70G45, 70H50, 70S05.
Contents ∗ [email protected] ( ORCID : 0000-0003-3663-9861). . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. The Hamilton–Jacobi theory is a topic of interest in mathematical physics since it is a wayto integrate systems of first-order ordinary differential equations (Hamilton equations in thestandard case). The classical method in Hamiltonian mechanics consists in obtaining a suitablecanonical transformation which leads the system to equilibrium [2, 39, 41, 51], and is given byits generating function. This function is the solution to the so-called
Hamilton–Jacobi equation ,which is a partial differential equation. The “classical” Hamilton–Jacobi problem consists infinding this canonical transformation. Because of its interest, the method was generalized in otherkinds of physical systems; such as, for instance, singular Lagrangian systems [30] or higher-orderdynamics [17], and different types of solutions have been proposed and studied [19, 20].Nevertheless, in recent times, a lot of research has been done to understand the Hamilton–Jacobi equation from a more general geometric approach, and some geometric descriptions tothe theory were done in [1, 8, 31, 44, 45]. From a geometric way, the above mentioned canonicaltransformation is associated with a foliation in the the phase space of the system which isrepresented by the cotangent bundle T ∗ Q of a manifold (the configuration manifold Q ). Thisfoliation has some characteristic geometric properties: it is invariant by the dynamics, transversalto the fibers of the cotangent bundle, and Lagrangian with respect to the canonical symplecticstructure of T ∗ Q (although this last property could be ignored in some particular situations).The restriction of the dynamical vector field in T ∗ Q to each leaf S λ of this foliation projects ontoanother vector field X λ on Q , and the integral curves of these vector fields are one-to-one related.Hence, the complete set of dynamical trajectories are recovered from the integral curves of thecomplete family { X λ } of all these vector fields in the base. These geometric considerations can bedone in an analogous way in the Lagrangian formalism and hence this geometrical picture of theHamilton–Jacobi theory can be also stated for this formalism. The geometric Hamilton–Jacobiproblem consists in finding this foliation and these vector fields { X λ } .Following these ideas, the Lagrangian and Hamiltonian versions of the Hamilton–Jacobi the-ory, for autonomous and non-autonomous mechanical systems, was formulated in another geo-metrical way in [11]. The foundations of this geometric generalization are similar to those givenin [1, 40]. Later, this framework has been used to develop the Hamilton–Jacobi theory for manyother kinds of systems in physics. For instance, other applications of the theory are to the caseof singular Lagrangian and Hamiltonian systems [12, 23, 24, 43], higher-order dynamical sys-tems [15, 16], holonomic and non-holonomic mechanics [7, 13, 26, 38, 43, 46, 47], and controltheory [4, 56]. The theory is also been extended for dynamical systems described using othergeometric structures, such as Poisson manifolds [24, 36], Lie algebroids [3, 42], contact manifolds(which model dissipative systems) [21, 37], and other geometric applications and generalizations:[5, 14, 57]. Furthermore, in [6, 48] the geometric discretization of the Hamilton–Jacobi equationwas analyzed. Finally, the Hamilton–Jacobi theory is developed for the usual covariant formula-tions of first-order classical field theories, the k -symplectic and k -cosymplectic [22, 29] and themultisymplectic ones [25, 27], for higher-order field theories [52, 54], for the formulation in the Cauchy data space [9], and for partial differential equations in general [53, 55].This review paper is devoted, first of all, to present, in Section 2, the foundations of thismodern geometric formulation of the Hamilton–Jacobi theory, starting from the most generalproblem and explaining how to derive the standard Hamilton–Jacobi equation for the Hamilto-nian and the Lagrangian formalisms of autonomous mechanics. After this, the notion of completesolution allows us to establish the relation with the “classical” Hamilton–Jacobi theory based oncanonical transformations, which is summarized in Section 3, where this relation is also analyzed(this topic had been already discussed in [55]). We also present briefly a more general geometricframework for the Hamilton–Jacobi theory which was stated in [14], from which we can derive themajority of the applications of the theory to other kinds of physical systems, including the case. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches.
We summarize the main features of the geometric Hamilton–Jacobi theory for the Hamiltonianand Lagrangian formalisms of autonomous dynamical systems as it is stated in [11] (see also[1, 40]).
Typically, a (regular autonomous) Hamiltonian system is a triad (T ∗ Q, ω, H ) , where the bun-dle π Q : T ∗ Q / / Q represents the phase space of a dynamical system ( Q is the configurationspace ), ω = − d θ ∈ Ω (T ∗ Q ) is the natural symplectic form in T ∗ Q , and H ∈ C ∞ (T ∗ Q ) is the Hamiltonian function . The dynamical trajectories are the integral curves σ : I ⊆ R / / T ∗ Q of the Hamiltonian vector field Z H ∈ X (T ∗ Q ) associated with H , which is the solution to the Hamiltonian equation i ( Z H ) ω = d H . (1)(Here, Ω k (T ∗ Q ) and X (T ∗ Q ) are the sets of differentiable k -forms and vector fields in T ∗ Q , and i ( Z H ) ω denotes the inner contraction of Z H and ω ). In a chart of natural coordinates ( q i , p i ) in T ∗ Q we have that ω = d q i ∧ d p i , and the curves σ ( t ) = ( q i ( t ) , p i ( t )) are the solution to the Hamilton equations d q i d t = ∂H∂p i ( q ( t ) , p ( t )) , d p i d t = − ∂H∂q i ( q ( t ) , p ( t )) . Definition 1
The generalized Hamiltonian Hamilton–Jacobi problem for a Hamiltoniansystem (T ∗ Q, ω, H ) is to find a vector field X ∈ X ( Q ) and a -form α ∈ Ω ( Q ) such that, if γ : R / / Q is an integral curve of X , then α ◦ γ : R / / T ∗ Q is an integral curve of Z H ; that is, if X ◦ γ = ˙ γ , then ˙ α ◦ γ = Z H ◦ ( α ◦ γ ) . Then, the couple ( X, α ) is a solution to the generalizedHamiltonian Hamilton–Jacobi problem . Theorem 1
The following statements are equivalent:1. The couple ( X, α ) is a solution to the generalized Hamiltonian Hamilton–Jacobi problem.2. The vector fields X and Z H are α -related; that is, Z H ◦ α = T α ◦ X . As a consequence X = T π Q ◦ Z H ◦ α and it is called the vector field associated with the form α .3. The submanifold Im α of T ∗ Q is invariant by the Hamiltonian vector field Z H (or, whatmeans the same thing, Z H is tangent to Im α ).4. The integral curves of Z H which have their initial conditions in Im α project onto theintegral curves of X . . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches.
5. The equation i ( X )d α = − d( α ∗ H ) holds for the -form α . T Q T α T(T ∗ Q ) T π Q o o R γ / / Q X O O α T ∗ Q π Q o o Z H O O ( Proof ) (Guidelines for the proof):The equivalence between 1 and 2 is a consequence of the Definition 1 and the definition ofintegral curves. Then, the expression X = T π Q ◦ Z H ◦ α is obtained by composing both membersof the equality Z H ◦ α = T α ◦ X with T π Q and taking into account that π Q ◦ α = Id Q .Items 3 and 4 follow from 2.Item 5 is obtained from Definition 1 and using the dynamical equation (1).In order to solve the generalized Hamilton–Jacobi problem, it is usual to state a less generalversion of it, which constitutes the standard Hamilton–Jacobi problem. Definition 2
The
Hamiltonian Hamilton–Jacobi problem for a Hamiltonian system (T ∗ Q, ω, H ) is to find a -form α ∈ Ω (T ∗ Q ) such that it is a solution to the generalized HamiltonianHamilton–Jacobi problem and is closed. Then, the form α is a solution to the HamiltonianHamilton–Jacobi problem . As α is closed, for every point in Q , there is a function S in a neighbourhood U ⊂ Q suchthat α = d S . It is called a local generating function of the solution α . Theorem 2
The following statements are equivalent:1. The form α ∈ Ω ( Q ) is a solution to the Hamiltonian Hamilton–Jacobi problem.2. Im α is a Lagrangian submanifold of T ∗ Q which is invariant by Z H , and S is a localgenerating function of this Lagrangian submanifold.3. The equation d( α ∗ H ) = 0 holds for α or, what is equivalent, the function H ◦ d S : Q / / R is locally constant. ( Proof ) (Guidelines for the proof): They are consequences of Theorem 1 and Definition 2.The last condition, written in natural coordinates, gives the classical form of the HamiltonianHamilton–Jacobi equation, which is H (cid:18) q i , ∂S∂q i (cid:19) = E ( ctn. ) . (2)These forms α are particular solutions to the (generalized) Hamilton–Jacobi problem, but weare also interested in finding complete solutions to the problem. Then:. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Definition 3
Let Λ ⊆ R n . A family of solutions { α λ ; λ ∈ Λ } , depending on n parameters λ ≡ ( λ , . . . , λ n ) ∈ Λ , is a complete solution to the Hamiltonian Hamilton–Jacobi problem ifthe map Φ : Q × Λ −→ T ∗ Q ( q, λ ) α λ ( q ) is a local diffeomorphism. Remark 1
Given a complete solution { α λ ; λ ∈ Λ } , as d α λ = 0 , ∀ λ ∈ Λ , there is a family offunctions { S λ } defined in neighbourhoods U λ ⊂ Q of every point such that α λ = d S λ . Thereforewe have a locally defined function S : T U λ × Λ ⊂ Q × Λ −→ R ( q, λ ) S λ ( q ) which is called a local generating function of the complete solution { α λ ; λ ∈ Λ } .A complete solution defines a Lagrangian foliation in T ∗ Q which is transverse to the fibers,and such that Z H is tangent to the leaves. The functions that define locally this foliation arethe components of a map F : T ∗ Q Φ − −→ Q × Λ pr −→ Λ ⊂ R n , and give a family of constants ofmotion of Z H . Conversely, if we have n first integrals f , . . . , f n of Z H in involution, such that d f ∧ . . . ∧ d f n = 0 ; then f i = λ i , with λ i ∈ R , define this transversal Lagrangian foliation andhence a local complete solution { α λ , λ ∈ Λ } . Thus we can locally isolate p i = p i ( q, λ ) , replacethem in Z H and project to the basis, then obtaining the family of vector fields { X λ } associatedwith the local complete solution. If { α λ ; λ ∈ Λ } is a complete solution, then all the integralcurves of Z H are obtained from the integral curves of the associated vector fields { X λ } . The above framework for the Hamilton–Jacobi theory can be easily translated to the Lagrangianformalism of mechanics. Now, the phase space is the tangent bundle τ Q : T Q / / Q of theconfiguration bundle Q and the dynamics is given by the Lagrangian function of the system, L ∈ C ∞ (T Q ) . Using the canonical structures in T Q ; that is, the vertical endomorphism S ∈ T (T Q ) , and the Liouville vector field ∆ ∈ X (T Q ) , the Lagrangian forms θ L := d L◦ S ∈ Ω (T Q ) , ω L = − dθ L ∈ Ω (T Q ) , and the Lagrangian energy E L := ∆( L ) − L ∈ C ∞ (T Q ) are constructed.Then the Lagrangian equation is i (Γ L ) ω L = d E L , (3)and (T Q, ω L , E L ) is a Lagrangian dynamical system . Furthermore, the
Legendre transforma-tion associated with L , denoted by F L : T Q / / T ∗ Q , is defined as the fiber derivative of theLagrangian function. We assume that L is regular; that is, F L is a local diffeomorphism or,equivalently, ω L is a symplectic form (the Lagrangian is hyper-regular if F L is a global diffeo-morphism). In that case, the Lagrangian equation (3) has a unique solution Γ L ∈ X (T Q ) , whichis called the Lagrangian vector field , whose integral curves are holonomic , and are the solutionsto the Euler-Lagrange equations. (See [18] for details).
Definition 4
The generalized Lagrangian Hamilton–Jacobi problem for a Lagrangiansystem (T Q, ω L , E L ) is to find a vector field X ∈ X ( Q ) such that, if γ : R / / Q is an integralcurve of X , then X ◦ γ = ˙ γ : R / / T Q is an integral curve of Γ L ; that is, if X ◦ γ = ˙ γ , then Γ L ◦ ˙ γ = ˙ X ◦ γ . Then, the vector field X is a solution to the generalized LagrangianHamilton–Jacobi problem . Theorem 3
The following statements are equivalent: . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches.
1. The vector field X is a solution to the generalized Lagrangian Hamilton–Jacobi problem.2. The vector fields X and Γ L are X -related; that is, Γ L ◦ X = T X ◦ X .3. The submanifold Im X of T Q is invariant by the Lagrangian vector field Γ L (or, what meansthe same thing, Γ L is tangent to Im X ).4. The integral curves of Γ L which have their initial conditions in Im X project onto theintegral curves of X .5. The equation i ( X )( X ∗ ω L ) = d( X ∗ E L ) holds for the vector field X . T Q T X T(T Q ) T τ Q o o R γ / / Q X O O X T Q τ Q o o Γ L O O ( Proof ) (Guidelines for the proof): The proof follows the same patterns as Theorem 1.As in the Hamiltonian formalism, we consider the following simpler case:
Definition 5
The
Lagrangian Hamilton–Jacobi problem for a Lagrangian system (T Q, ω L , E L ) is to find a vector field X such that it is a solution to the generalized Lagrangian Hamilton–Jacobiproblem and satisfies that X ∗ ω L = 0 . Then, this vector field X is a solution to the LagrangianHamilton–Jacobi problem . Since X ∗ ω L = − X ∗ d θ L = − d ( X ∗ θ L ) then, for every point of Q , there is a neighbourhood U ⊂ Q and a function S such that X ∗ θ L = d S (in U ). Theorem 4
The following statements are equivalent:1. The vector field X is a solution to the Lagrangian Hamilton–Jacobi problem.2. Im X is a Lagrangian submanifold of T Q which is invariant by the Lagrangian vector field Γ L (and S is a local generating function of this Lagrangian submanifold).3. The equation d( X ∗ E L ) = 0 holds for X or, what is equivalent, the function E L ◦ d S : Q / / R is locally constant. ( Proof ) (Guidelines for the proof): They are consequences of Theorem 3 and Definition 5.The last condition leads to the following expression which is the form of the LagrangianHamilton–Jacobi equation in natural coordinates, ∂S∂q i = ∂ L ∂v i ( q i , X i ) . (4)As in the Hamiltonian Hamilton–Jacobi theory, we are interested in the complete solutionsto the problem, which are defined as:. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Definition 6
Let Λ ⊆ R n . A family of solutions { X λ ; λ ∈ Λ } depending on n parameters λ ≡ ( λ , . . . , λ n ) ∈ Λ , is a complete solution to the Lagrangian Hamilton–Jacobi problem if themap Ψ : Q × Λ −→ T Q ( q, λ ) X λ ( q ) is a local diffeomorphism. If we have a complete solution to the Lagrangian Hamilton–Jacobi problem, all the integralcurves of the Lagrangian vector field Γ L are obtained from the integral curves of all the vectorfields X λ .The equivalence between the Lagrangian and the Hamiltonian Hamilton–Jacobi problems isstated as follows: Theorem 5
Let (T Q, ω L , E L ) be a (hyper)regular Lagrangian system, and (T ∗ Q, ω, H ) itsassociated Hamiltonian system. If α ∈ Ω ( Q ) is a solution to the (generalized) HamiltonianHamilton–Jacobi problem, then X = F L − ◦ α is a solution to the (generalized) LagrangianHamilton–Jacobi problem and conversely, If X ∈ X ( Q ) is a solution to the (generalized) La-grangian Hamilton–Jacobi problem, then α = F L ◦ X is a solution to the (generalized) Hamilto-nian Hamilton–Jacobi problem. ( Proof ) (Guidelines for the proof): It can be proven that α = F L ◦ X ; then, bearing in mindthat T F L ◦ Γ L = Z H ◦ F L , the proof follows using items 2 and 5 of Theorems 1 and 3 (or item3 of Theorems 2 and 4). In this section we review the geometric description of the classical Hamiltonian Hamilton–Jacobitheory (for autonomous systems), based on using canonical transformations [1, 2, 39, 44, 45]. Itis stated in the Hamiltonian formalism .
First we remind the following well-known results [1]:
Proposition 1
Let ( M , ω ) , ( M , ω ) be symplectic manifolds and π j : M × M / / M j , j = 1 , .Then ( M × M , π ∗ ω − π ∗ ω ) is a symplectic manifold. Proposition 2
Let
Φ : M / / M be a diffeomorphism and : graph Φ ֒ → M × M . Φ is a symplectomorphism (i.e., Φ ∗ ω = ω ) if, and only if, graph Φ is a Lagrangian submanifoldof ( M × M , π ∗ ω − π ∗ ω ) . If ω j = − d θ j , j = 1 , ; being graph Φ a Lagrangian submanifold we have ∗ ( π ∗ ω − π ∗ ω ) = d ∗ ( π ∗ θ − π ∗ θ ) ⇐⇒ ∗ ( π ∗ θ − π ∗ θ ) | W = − d S . (5) S is a function defined in an open neighbourhood W ⊂ graph Φ of every point, which dependson the choice of θ and θ .. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Definition 7 S is called a generating function of the Lagrangian submanifold graph Φ andhence of the symplectomorphism Φ . If ( U ; q i , p i ) , ( U ; ˜ q i , ˜ p i ) are Darboux charts such that W ⊂ U × U , local coordinates in W can be chosen in several ways. This leads to different possible choices for S . Thus, for instance,if ( W ; q i ˜ q i ) is a chart, then (5) gives the symplectomorphism explicitly as ˜ p i d˜ q i − p i d q i = − d S ( q, ˜ q ) ⇐⇒ ˜ p i = − ∂ S ∂ ˜ q i ( q, ˜ q ) , p i = ∂ S ∂q i ( q, ˜ q ) . Now, let (T ∗ Q, ω, H ) be a Hamiltonian system. Definition 8 A canonical transformation for a Hamiltonian system (T ∗ Q, ω, H ) is a sym-plectomorphism Φ : T ∗ Q / / T ∗ Q . As a consequence, Φ transforms Hamiltonian vector fields intoHamiltonian vector fields. Definition 9
The
Hamilton–Jacobi problem for a Hamiltonian system (T ∗ Q, ω, H ) consistsin finding a canonical transformation Φ : T ∗ Q / / T ∗ Q leading the system to equilibrium; that is,such that H ◦ Φ = E ( ctn. ) . The canonical transformation Ψ is given by a generating function S : ∂ S ∂q i ( q, ˜ q ) = p i , − ∂ S ∂ ˜ q i ( q, ˜ q ) = ˜ p i , (6)where S the general solution to the Hamilton–Jacobi equation H (cid:18) q i , ∂ S ∂q i (cid:19) = E ( ctn. ) . (7)Then, the Hamilton equations for the transformed Hamiltonian function H ◦ Φ ≡ ˜ H are d˜ q i d t = ∂ ˜ H∂ ˜ p i (˜ q ( t ) , ˜ p ( t )) = 0 , d ˜ p i d t = − ∂ ˜ H∂ ˜ q i (˜ q ( t ) , ˜ p ( t )) = 0 ; (8)and solving (7), from (8) and (6), the dynamical curves ( q i ( t ) , p i ( t )) of the original Hamiltoniansystem (T ∗ Q, ω, H ) are obtained. The relation between the “classical” and the geometric Hamilton–Jacobi theories is establishedthrough the equivalence of complete solutions and canonical transformations (see also [55]).
Theorem 6
Let (T ∗ Q, ω, H ) be a Hamiltonian system. A complete solution { α λ ; λ ∈ Λ } tothe Hamilton–Jacobi problem provides a canonical transformation Φ : T ∗ Q / / T ∗ Q leading thesystem to equilibrium, and conversely. ( Proof ) In a neighbourhood of every point, consider a complete solution { α λ ; λ ∈ Λ } , and let S be a generating function of it. As S = S ( q i , λ i ) , we can identify λ i with a subset of coordinates λ i ≡ ˜ q i in T ∗ Q × T ∗ Q , and then S = S ( q i , ˜ q i ) can be thought as a generating function of alocal canonical transformation Φ and hence of an open set W of the Lagrangian submanifold. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. graph Φ ֒ → T ∗ Q × T ∗ Q . Making this construction in every chart, we have the transformation Φ and the submanifold graph Ψ . Now, as (2) holds for every particular solution S λ , we have that E = H (cid:18) q i (˜ q, ˜ p ) , ∂ S ∂q i ( q (˜ q, ˜ p ) , ˜ q ) (cid:19) = ˜ H (˜ q i , ˜ p i ) . Conversely, if we have the canonical transformation Ψ , from a generating function S = S ( q i , ˜ q i ) ,taking ˜ q ≡ (˜ q i ) = ( λ i ) ≡ λ , we obtain a family of functions { S λ } and, hence a local complete solu-tion { α λ = d S λ ; λ ∈ Λ } to the Hamiltonian Hamilton–Jacobi problem. Making this constructionin every chart, we have the complete solution.Geometrically, this means that, on each local chart of T ∗ Q , fixing the coordinates ˜ q i = λ i ofa point, we obtain a local submanifold whose image by Φ − gives the image of a local section α λ : Q / / T ∗ Q which is a particular solution to the Hamiltonian Hamilton–Jacobi problem. The geometric Hamilton–Jacobi theory can be stated in a more general framework which allowsus to extend the theory to a wide variety of systems and situations. Next we present a summaryof this general framework as it is stated in [14] (see also [3] for another similar approach).
In general, a dynamical system is just a couple ( P, Z ) , where P is a manifold and Z ∈ X ( P ) is avector field which defines the dynamical equation on P . Then, in order to state the analogous tothe Hamilton–Jacobi problem for this system in a more general context, consider a manifold M ,a vector field X ∈ X ( M ) , and a map α : M / / P , as it is showed in the following diagram: T M T α / / (cid:15) (cid:15) T P (cid:15) (cid:15) M α / / X H H P Z V V Proposition 3
The following statements are equivalent:1. If γ is an integral curve of X , then ζ = α ◦ γ is an integral curve of Z .2. The vector fields X and Z are α -related: T α ◦ X = Z ◦ α , (9) Furthermore, if α is an injective immersion, (inducing a diffeomorphism α o : M / / α ( M ) ), thenthese properties are equivalent to:3. The vector field Z is tangent to α ( M ) , and, if Z o = Z | α ( M ) , then X = α ∗ o ( Z o ) .Then, the map γ α ◦ ξ is a bijection between the integral curves of X and the integralcurves of Z in α ( M ) . ( Proof ) They are immediate, bearing in mind the commutativity of the above diagram.. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Definition 10 A slicing of a dynamical system ( P, Z ) is a triple ( M , α, X ) which is a solutionto the slicing equation (9) . If ( x i ) and ( z j ) are coordinates in M and P , respectively; and α ( x ) = ( a j ( x )) , X = X i ∂∂x i ,and Z = Z j ∂∂z j , then (T α ◦ X − Z ◦ α )( x i ) = (cid:16) a j ( x ) , ∂a j ∂x i X i − Z j ( α ( x )) (cid:17) , and ( M , α, X ) is asolution to the slicing equation if, and only if, ∂a j ∂x i X i ( x ) = Z j ( α ( x )) . We say that the vector field X gives a “partial dynamics” or a “slice” of the “whole dynamics”which is given by Z , and the whole dynamics can be recovered from these slices. In fact, theintegral curves of Z contained in α ( M ) ⊂ P can be described by a solution ( α, X ) to the slicingequation; but we need a complete solution to describe all the integral curves of Z and it can bedefined as a family of solutions depending on the parameters of a space Λ ⊆ R n . Definition 11 A complete slicing of a dynamical system ( P, Z ) is a map α : M × Λ / / P anda vector field X : M × Λ / / T M along the projection M × Λ / / M such that:1. The map α is surjective,2. for every λ ∈ Λ , the map α λ : M / / P and X λ : M / / T M are a slicing of Z . T M × Λ T α / / (cid:15) (cid:15) T P (cid:15) (cid:15) M × Λ α / / X H H P Z V V Thus, a complete slicing is a family of maps α λ ≡ α ( · , λ ) : M / / P and vector fields X λ ≡ X ( · , λ ) : M / / T M satisfying the above conditions.As for every point p ∈ P there exits ( x, λ ) ∈ M × Λ such that α ( x, λ ) = p ; the integral curveof Z through p is described by the integral curve of X λ through x by means of the map α λ .In addition, if each α λ is an immersion (for instance,when it is a diffeomorphism) then X λ aredetermined by the α λ .The hypothesis of α being an embedding holds in many situations; for instance, for thesections of a fiber bundle π : P / / M . Then we can consider the slicing problem for sections α : M / / P of π , as before. In this case, as α is an embedding, the equation (9) determines X ,and X is given from α by the equation X = T π ◦ Z ◦ α . In this case, Proposition 3 states that a section α of π : P / / M is a solution to the slicingequation for ( P, Z ) if, and only if, T α ◦ T π ◦ Z ◦ α = Z ◦ α . . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Consider the case of a Hamiltonian system ( P, ω, H ) , where ( P, ω ) is a symplectic manifold, H ∈ C ∞ ( P ) is a Hamiltonian function, and Z = Z H is its Hamiltonian vector field; that is, thesolution to (1). Then: Theorem 7 If ( M , α, X ) is a solution to the slicing equation (9) for ( P, Z H ) , then i ( X ) α ∗ ω − d α ∗ H = 0 . In addition, if α : M / / P is an embedding satisfying the condition α ∗ ω = 0 , then d ( α ∗ H ) = 0 ; and conversely, if dim P = 2 dim M and α satisfies this equation and α ∗ ω = 0 , then α is asolution to the slicing equation (9) . In the particular case where π : P / / M is a fiber bundle (for instance, M = Q and P = T ∗ Q ),we can consider the slicing problem as before, but only for sections of π , T M T α / / (cid:15) (cid:15) T P (cid:15) (cid:15) M α / / X H H P π j j Z V V Being α an embedding, the equation (9) determines X , and composing this equation with T π ,we obtain that X = T π ◦ Z H ◦ α . Therefore the slicing equation (9) reads T α ◦ T π ◦ Z ◦ α = Z ◦ α . In this way, the equation (9) can be considered as a generalization of the Hamilton–Jacobiequation in the Hamiltonian formalism, which is just the slicing equation for a closed -form α in Q . Therefore, α = d S locally, and the slicing equation looks in the ordinary form H ◦ d S = const .The same applies to the Lagrangian formalism. In this case P = T Q and, if L ∈ C ∞ (T Q ) isa regular Lagrangian function, Z = Γ L is the Lagrangian vector field solution to the Lagrangianequation (3). Then all proceeds as in the Hamiltonian case.The definitions 3 and 6 of complete solutions to the Hamiltonian and Lagrangian Hamilton–Jacobi problems respectively are particular cases of the definition 11 of complete slicings. Using the general framework presented in the above section, the Hamilton–Jacobi problem canbe stated for a wide kind of physical systems. Next we review two of them. (Other applicationsof the theory are listed in detail in the Introduction).
Let Q be a n -dimensional manifold) and let T k Q the k th-order tangent bundle of Q , whichis endowed with natural coordinates (cid:0) q A , q A , . . . , q Ak (cid:1) = (cid:0) q Ai (cid:1) , i k , A n . If. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. L ∈ C ∞ (T k Q ) is the Lagrangian function of an autonomous k th-order Lagrangian system, usingthe canonical structures of the higher-order tangent bundles, we can construct the Poincaré-Cartan forms and the Lagrangian energy whose coordinate expressions are ω L = − d θ L = k X r =1 k − r X i =0 ( − i +1 d iT d ∂ L ∂q Ar + i ! ∧ d q Ar − ∈ Ω (T k − Q ) ,E L = k X r =1 q Ar k − r X i =0 ( − i d iT ∂ L ∂q Ar + i ! − L ∈ C ∞ (T k − Q ) , where d T f (cid:0) q A , . . . , q Ak +1 (cid:1) = k X i =0 q Ai +1 ∂f∂q Ai ( q A , . . . , q Ak ) . Thus we have the higher-order Lagrangiansystem (T k − Q, ω L , E L ) . Assuming that the Lagrangian function is regular; that is, ω L is a sym-plectic form; the Lagrangian equation i ( X L ) ω L = d E L has a unique solution X L ∈ X (T k − Q ) (the Lagrangian vector field) whose integral curves are holonomic (that is, they are canonicalliftings j k − φ : R / / T k − Q of curves φ : R / / Q ) and are the solutions to the Otrogradskii or higher-order Euler-Lagrange equations (see [28, 35, 49] for details).The Hamilton–Jacobi problem for higher-order Lagrangian dynamical systems is just the slicing problem for the particular situation represented in the diagram T(T k − Q ) T s T(T k − Q ) T ρ k − k − o o T k − Q X O O s T k − Q ρ k − k − o o X L O O that is, for sections of the natural projection ρ k − k − : T k − Q / / T k − Q , s ∈ Γ( ρ k − k − ) ; and thuswe have the following settings (see [15, 16] for the details and proofs): Definition 12
The generalized k th-order Lagrangian Hamilton–Jacobi problem for thehigher-order Lagrangian system (T k − Q, ω L , E L ) is to find a section s ∈ Γ( ρ k − k − ) and a vectorfield X ∈ X (T k − Q ) such that, if γ : R / / T k − Q is an integral curve of X , then s ◦ γ : R / / T k − Q is an integral curve of X L ; that is, if X ◦ γ = ˙ γ , then X L ◦ ( s ◦ γ ) = ˙ s ◦ γ . Then,the couple ( s, X ) is a solution to the generalized k th-order Lagrangian Hamilton–Jacobiproblem . Theorem 8
The following statements are equivalent:1. The couple ( s, X ) is a solution to the generalized k th-order Lagrangian Hamilton–Jacobiproblem.2. The vector fields X and X L are s -related; that is, X L ◦ s = T s ◦ X . As a consequence, X = T ρ k − k − ◦ X L ◦ s , and X is said to be the vector field associated with the section s .3. The submanifold Im( s ) of T k − Q is invariant by the Lagrangian vector field X L (or, whatmeans the same thing, X L is tangent to s (T k − Q ) ).4. The integral curves of X L which have initial conditions in Im ( s ) project onto the integralcurves of X . . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches.
5. The equation i ( X )( s ∗ ω L ) = d( s ∗ E L ) holds for α . ( Proof ) (Guidelines for the proof): The proof follows a pattern similar to that of Theorem 1,but now using Definition 12.
Definition 13
The k th-order Lagrangian Hamilton–Jacobi problem for the higher-orderLagrangian system (T k − Q, ω L , E L ) is to find a section s ∈ Γ( ρ k − k − ) such that it is a solutionto the generalized k th-order Lagrangian Hamilton–Jacobi problem and satisfies that s ∗ ω L = 0 .Then, this section s is a solution to the k th-order Lagrangian Hamilton–Jacobi problem . Observe that that s ∗ ω L = − s ∗ (d θ L ) = − d( s ∗ θ L ) = 0 ; that is, s ∗ θ L is a closed -formand then there exists S ∈ C ∞ ( U ) , U ⊂ T k − Q , such that s ∗ θ L | U = d S . Theorem 9
The following statements are equivalent:1. The section s is a solution to the generalized k th-order Lagrangian Hamilton–Jacobi prob-lem.2. Im( s ) is a Lagrangian submanifold of T k − Q , which is invariant by the Lagrangian vectorfield X L (and S is a local generating function of this Lagrangian submanifold).3. The equation d( s ∗ E L ) = 0 holds for s or, what is equivalent, the function E L ◦ d S : Q / / R is locally constant. ( Proof ) (Guidelines for the proof): They are consequences of Theorem 8 and Definition 13.In natural coordinates, from this last condition we obtain that ∂S∂q Ai = k − i − X l =0 ( − l d lT ∂ L ∂q Ai +1+ l ! (cid:12)(cid:12)(cid:12) Im( s ) . This system of kn partial differential equations for S generalizes the equation (4) to higher-ordersystems. Definition 14
Let Λ ⊆ R n . A family of solutions { s λ ; λ ∈ Λ } , depending on n parameters λ ≡ ( λ , . . . , λ n ) ∈ Λ , is a complete solution to the k th-order Lagrangian Hamilton–Jacobi problem if the map Φ : T k − Q × Λ −→ T k − Q ( q, λ ) s λ ( q ) is a local diffeomorphism. For the Hamiltonian formalism, let h ∈ C ∞ (T ∗ (T k − Q )) be the Hamiltonian function of a(regular) higher-order dynamical system. Using the canonical Liouville forms of the cotangentbundle, θ k − = p iA d q Ai ∈ Ω (T ∗ (T k − Q )) and ω k − = d q Ai ∧ d p iA ∈ Ω (T ∗ (T k − Q )) , where ( q Ai , p Ai ) ( ≤ A ≤ n, ≤ i ≤ k − ) are canonical coordinates in T ∗ (T k − Q ) ; the dynamicalequation for the Hamiltonian system (T ∗ (T k − Q ) , ω k , h )) is i ( X h ) ω k − = d h , and it has aunique solution X h ∈ X (T ∗ (T k − Q )) . As we are working in the cotangent bundle T ∗ (T k − Q ) ,the Hamiltonian Hamilton–Jacobi problems for higher-order systems is stated in the same way as. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. slicing problem for the particular situation representedin the diagram T(T k − Q ) T α T(T ∗ (T k − Q )) T π T k − Q o o T k − Q X O O α T ∗ (T k − Q ) π T k − Q o o X h O O Therefore, all the definitions and results are like in the first-order case, and the relation betweenboth the Lagrangian and the Hamiltonian Hamilton–Jacobi problems is stated as in Theorem 5.
The Hamilton–Jacobi theory for multisymplectic field theories has been studied in [25, 27, 54].Next we state the Lagrangian and the Hamiltonian problems for these systems. (For details onmultisymplectic field theories see, for instance, [10, 32, 50] and the references therein).
Let π : E / / M a bundle, where M is an oriented manifold with dim M = m and dim E = n + m .The Lagrangian description of multisymplectic classical field theories is stated in the first-order jetbundle π : J π / / E , which is also a bundle ¯ π : J π −→ M . Natural coordinates in J π adaptedto the bundle structure are ( x i , y α , y αi ) ( i = 1 , . . . , m ; α = 1 , . . . , n ). Giving a Lagrangian densityassociated to a Lagrangian function L and using the canonical structures of J π we can definethe Poincaré–Cartan forms associated with L , Θ L ∈ Ω m ( J π ) and Ω L := − dΘ L ∈ Ω m +1 ( J π ) ,whose local expression is Ω L = − dΘ L = − d (cid:18) ∂L∂y αi d y α ∧ d m − x i − (cid:16) ∂L∂y αi y αi − L (cid:17) d m x (cid:19) , where d m x = d x ∧ . . . ∧ d x m and d m − x i = i (cid:16) ∂∂x i (cid:17) d m x . The Lagrangian function is regularif Ω L is a multisymplectic ( m + 1) -form (i.e., -nondegenerate). Then the couple ( J π, Ω L ) is a multisymplectic Lagrangian system . The Lagrangian problem consists in finding m -dimensional, ¯ π -transverse, and holonomic distributions D L in J π such that their integral sections ψ L ∈ Γ(¯ π ) are canonical liftings j φ of sections φ ∈ Γ( π ) that are solutions to the Lagrangian field equation ( j φ ) ∗ i ( X )Ω L = 0 , for every X ∈ X ( J π ) . (10)In coordinates, the components of j φ = (cid:16) x i , y α , ∂y α ∂x i (cid:17) satisfy the Euler–Lagrange equations ∂ L ∂y A ◦ j φ − ∂∂x µ (cid:16) ∂ L ∂y Aµ ◦ j φ (cid:17) = 0 . Definition 15
The generalized Lagrangian Hamilton–Jacobi problem for the multisym-plectic Lagrangian system ( J π, Ω L ) is to find a section Ψ ∈ Γ( π ) (which is called a jet field )and an m -dimensional integrable distribution D in E such that, if γ ∈ Γ( π ) is an integral sectionof D , then ψ L = Ψ ◦ γ ∈ Γ(¯ π ) is an integral section of D L ; that is, if T u Im( γ ) = D u , for every u ∈ Im( γ ) , then T ¯ u Im(Ψ ◦ γ ) = ( D L ) ¯ u , for every ¯ u ∈ Im(Ψ ◦ γ ) . Then, the couple (Ψ , D ) is a solution to the generalized Hamiltonian Hamilton–Jacobi problem . . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Remark 2
The Hamilton–Jacobi problem can also be stated associating the distributions D and D L with multivector fields . An m -multivector field , on a manifold M is a section of the bundle Λ m (T M ) / / M , where Λ m (T M ) = T M ∧ ( m ) ...... ∧ T M (i.e, a skew-symmetric contravariant tensorfield). If X is an m -multivector field in M then, for every p ∈ M , there is a neighbourhood U p ⊂ M and local vector fields X , . . . , X m ∈ X ( U p ) such that X | U p = X ∧ ( m ) ...... ∧ X m . Then,if D is an m -dimensional distribution in M , sections of Λ m D / / M are m -multivector fields in M , and a multivector field is integrable if its associated distribution is also.Now, if M = J π , let X and X L be the m -multivector fields associated with the distributions D and D L , respectively; then the Lagrangian Hamilton–Jacobi problem can be represented by thediagram Λ m T E Λ m TΨ Λ m T J π Λ m T π o o E X O O Ψ J π π o o X L O O (where Λ m TΨ and Λ m T π denote the natural extensions of the maps ψ and π to the multitan-gent bundles), and thus this problem can be considered as a special case of a slicing problem . Theorem 10
The following statements are equivalent:1. The couple (Ψ , D ) is a solution to the generalized Lagrangian Hamilton–Jacobi problem.2. The distributions D and D L are Ψ -related. As a consequence, D = T π ( D L | Im(Ψ) ) , and iscalled the distribution associated with Ψ .3. The distribution D L is tangent to the submanifold Im(Ψ) of J π .4. The integral sections of D L which have boundary conditions in Im(Ψ) project onto theintegral sections of D .5. If γ is an integral section of the distribution D associated with the jet field Ψ then, for every Y ∈ X ( E ) , the equation γ ∗ i ( Y )(Ψ ∗ Ω L ) = 0 holds for Ψ . ( Proof ) (Guidelines for the proof):The equivalence between 1 and 2 is a consequence of the Definition 15, the equivalencebetween distributions and multivector fields, and the definition of integral sections.Items 3 and 4 follow from 2.Item 5 is obtained from Definition 15 and using the field equation (10).
Definition 16
The
Lagrangian Hamilton–Jacobi problem for the multisymplectic Lagrangiansystem ( J π, Ω L ) is to find a jet field Ψ ∈ Γ( π ) such that it is solution to the generalized La-grangian Hamilton–Jacobi problem and satisfies that Ψ ∗ Ω L = 0 . Then, the jet field Ψ is a solution to the Lagrangian Hamilton–Jacobi problem . The condition Ψ ∗ Ω L = − d(Ψ ∗ Θ L ) = 0 is equivalent to ask that the form Ψ ∗ Θ L is closed andthen there exists a ( m − -form ω ∈ Ω m − ( U ) , with U ⊂ E , such that Ψ ∗ Θ L = d ω . Furthermore, ω is π -semibasic, since Θ L , and hence Ψ ∗ Θ L , are also.. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Theorem 11
The following statements are equivalent:1. The jet field Ψ is a solution to the Lagrangian Hamilton–Jacobi problem.2. Im(Ψ) is an m -Lagrangian submanifold of J π and the distribution D L is tangent to it.3. The form Ψ ∗ Θ L is closed. In coordinates, ω = W i d m − x i , and the Hamilton–Jacobi equation in the Lagrangian formal-ism has the form m X i =1 ∂W i ∂x i + ψ αi ∂W i ∂u α − L ( x i , u α , ψ αi ) = 0 , Definition 17
Let Λ ⊆ R mn . A family of solutions { Ψ λ ; λ ∈ Λ } , depending on n parameters λ ≡ ( λ , . . . , λ n ) ∈ Λ , is a complete solution to the Lagrangian Hamilton–Jacobi problem if the map Φ : E × Λ −→ J π ( p, λ ) Ψ λ ( p ) is a local diffeomorphism. A complete solution defines an ( m − n ) -dimensional foliation in J π which is transverse tothe fibers and such that the distribution D L is tangent to it. Then, all the sections which aresolutions to the Euler–Lagrange equations (that is, all the integral sections of the distribution D L ) are recovered from a complete solution. The Hamiltonian formalism for a regular first-order multisymplectic field theory is developed inthe so-called reduced dual jet bundle of J π , J π ∗ = Λ m (T ∗ E ) / Λ m (T ∗ E ) (where Λ m (T ∗ E ) isthe bundle of m -forms over E vanishing when they act on π -vertical bivectors). It is endowedwith the canonical projections π E : J π ∗ / / E and ¯ π E : J π ∗ / / M , and natural coordinatesin J π ∗ are denoted ( x i , y α , p iα ) . The physical information is given by a Hamiltonian section h of the natural projection µ : Λ m (T ∗ E ) / / J π ∗ , which is associated with a local Hamiltonianfunction H ∈ C ∞ ( J π ∗ ) such that h ( x i , y α , p iα ) = ( x i , y α , − H, p iα ) . Then, from the canonicalform Ω ∈ Ω m +1 (Λ m (T ∗ E )) , we construct the Hamilton-Cartan multisymplectic form Ω h = h ∗ Ω ∈ Ω m +1 ( J π ∗ ) whose coordinate expression is Ω h = − d p iα ∧ d y α ∧ d m − x i + d H ∧ d m x , and the couple ( J π ∗ , Ω h ) is a multisymplectic Hamiltonian system . Then, the Hamiltonianproblem consists in finding integrable m -dimensional ¯ π E -transverse distributions D h in J π ∗ such that their integral sections ψ h ∈ Γ(¯ π E ) are solutions to the Hamiltonian field equation ψ ∗ h i ( X )Ω h = 0 , for every X ∈ X ( J π ∗ ) . The existence of such distributions D h is assured. In coordinates, this equation gives the Hamilton–De Donder–Weyl equations ∂ ( y A ◦ ψ h ) ∂x ν = ∂ h ∂p νA ◦ ψ h , ∂ ( p νA ◦ ψ h ) ∂x ν = − ∂ h ∂y A ◦ ψ h . . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. Definition 18
The generalized Hamiltonian Hamilton–Jacobi problem for the multi-symplectic Hamiltonian system ( J π ∗ , Ω h ) is to find a section s ∈ Γ( π E ) and an integrable m -dimensional distribution D in E such that, if γ ∈ Γ( π ) is an integral section of D , then ψ h = s ◦ γ ∈ Γ(¯ π E ) is an integral section of D h ; that is, if T u Im( γ ) = D u , for every u ∈ Im( γ ) ,then T ¯ u Im( s ◦ γ ) = ( D h ) ¯ u , for every ¯ u ∈ Im( s ◦ γ ) . Then, the couple ( s, D ) is a solution to thegeneralized Hamiltonian Hamilton–Jacobi problem . Remark 3
As in the Lagrangian case, the Hamiltonian Hamilton–Jacobi problem can be con-sidered as a special case of the following slicing problem Λ m T E Λ m T s Λ m T J π ∗ Λ m T π E o o E X O O s J π ∗ π E o o X h O O where X and X h are m -multivector fields associated with the distributions D and D h , respec-tively.The following Theorems and Definitions are analogous to those of the Lagrangian case. Theorem 12
The following conditions are equivalent.1. The couple ( s, D ) is a solution to the generalized Hamiltonian Hamilton–Jacobi problem.2. The distributions D and D h are s -related. As a consequence, the distribution D is given by D = T π E ( D h | Im( s ) ) , and it is called the distribution associated with s .3. The distribution D h is tangent to the submanifold Im( s ) of J π ∗ .4. The integral sections of D h which have boundary conditions in Im( s ) project onto the inte-gral sections of D .5. If γ is an integral section of the distribution D associated with s then, for every Y ∈ X ( E ) ,the equation γ ∗ i ( Y )d( h ◦ s ) = 0 holds for s . Definition 19
The
Hamiltonian Hamilton–Jacobi problem for the multisymplectic Hamil-tonian system ( J π ∗ , Ω h ) is to find a section s ∈ Γ( π E ) such that it is a solution to the generalizedHamilton–Jacobi problem and satisfies that s ∗ Ω h = 0 . Then, the section s is a solution to theHamiltonianian Hamilton–Jacobi problem . Theorem 13
The following conditions are equivalent.1. The couple ( s, D ) is a solution to the generalized Hamiltonian Hamilton–Jacobi problem.2. Im( s ) is an m -Lagrangian submanifold of J π ∗ and the distribution D h is tangent to it.3. The form h ◦ s ∈ Ω m ( E ) is closed. . Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches. π E -semibasic m -form h ◦ s is closed, there exists a local π -semibasic ( m − -form ω ∈ Ω m − ( E ) , such that h ◦ s = d ω . In coordinates, if ω = W i d m − x i , where W i ∈ C ∞ ( E ) arelocal functions, we obtain that − H ( x i , y α , s iα ) = m X i =1 ∂W i ∂x i ; ∂W i ∂y α = s iα , from where we obtain the classical Hamiltonian Hamilton–Jacobi equation m X i =1 ∂W i ∂x i + H (cid:18) x i , y α , ∂W i ∂y α (cid:19) = 0 . The definition and the characteristics of complete solution are similar to those of the La-grangian case.
Finally, let
F L : J π / / J π ∗ be the Legendre transformation defined by the Lagrangian L ,which is locally given by F L ∗ x i = x i , F L ∗ y α = y α , F L ∗ p iα = ∂ L ∂y αi . If L is a regular or a hyperregular Lagrangian (i.e., F L is a local or global diffeomorphism), then
F L ∗ Θ h = Θ L and F L ∗ Ω h = Ω L . In addition, the integral sections of the distributions D L and D h , which are the solution to the Lagrangian and the Hamiltonian problems respectively, are inone-to-one correspondence through F L . (See [27] for definitions and details). Then we have:
Theorem 14
Let
L ∈ Ω m ( J π ) be a regular or a hyperregular Lagrangian. Then, if Ψ ∈ Γ( π ) is a jet field solution to the (generalized) Lagrangian Hamilton–Jacobi problem, then the section s = F L ◦ Ψ ∈ Γ( π E ) is a solution to the (generalized) Hamiltonian Hamilton–Jacobi problem.Conversely, if s ∈ Γ( π E ) is a solution to the (generalized) Hamiltonian Hamilton–Jacobi problem,then the jet field Ψ =
F L − ◦ s ∈ Γ( π ) is a solution to the (generalized) Lagrangian Hamilton–Jacobi problem. ( Proof ) (Guidelines for the proof): The proof follows the same patterns as Theorem 5, but usingmultivector fields.
Remark 4
As a final remark, notice that the Hamilton–Jacobi theory for non-autonomous (i.e., time-dependent ) dynamical systems can be recovered from the multisymplectic Hamilton–Jacobitheory as a particular case taking M = R and identifying the distributions D , D L , D h and theirassociated multivector fields X , X L , X h with time-dependent vector fields (see [27]). In this work, the Lagrangian and the Hamiltonian versions of the Hamilton–Jacobi theory hasbeen reviewed from a modern geometric perspective.First, this formulation is done for autonomous dynamical systems and, in particular, theHamiltonian case is compared with the “classical” Hamiltonian Hamilton–Jacobi theory which isbased in using canonical transformations.. Román-Roy , Hamilton–Jacobi theory: classical and geometrical approaches.
Acknowledgments
I acknowledge the financial support from project PGC2018-098265-B-C33 of the Spanish Minis-terio de Ciencia, Innovación y Universidades and the project 2017–SGR–932 of the Secretary ofUniversity and Research of the Ministry of Business and Knowledge of the Catalan Government
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