The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle T∗Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{*}Q$$\end{document} of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.


"Les systèmes différentiels extérieurs" of Elie Cartan
First of all, we bring this already classic theme, from the work of Cartan. Thus, in [5,(Chapter 3)], his essential objective is the reduction in the number of variables of an exterior differential system. This matter, in turn, is at the base of our procedure.
Let Q be a manifold of dimension n. An exterior differential system of rank k on Q is a choice of a k-dimensional subspace W q of T * q Q for every q ∈ Q, in such a way that there is a neighborhood U of q and k linearly independent 1-forms w 1 , . . . , w k on U which form a basis of W x for every x ∈ U. We say that U is a trivializing neighborhood for W and that w 1 , . . . , w k are sections of W on U ; we shall denote by W(U ) the submodule of ∧ 1 (U ) generated by these sections.
In a similar manner, a distribution D of rank r on M choose an rdimensional subspace D q of T q Q for each q ∈ Q, in such a way that in some neighborhood U there exists sections X 1 , . . . , X r ∈ D(U ) which form a basis of D q for every q ∈ U.
We shall call orthogonal distribution of the exterior differential system W, denoted by W 0 , to the distribution on Q defined by: We shall call characteristic distribution of the exterior differential system W to the distribution W 0 of vector fields on Q,: for every trivializing neighborhood U of W, where L stands for the Lie derivative.
It is then obvious that: Definition 1. The exterior differential system W of rank k is said to be completely integrable, if for every point q ∈ Q, there exists a submanifold N ⊂ Q through q such that:

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The Lagrange-Charpit Theory Page 3 of 10 8 Theorem 2. Let Q be a manifold of dimension n and W an exterior differential system of rank k on Q. The following conditions are equivalent: such that: One may also consult [6] or [8] .

The Lagrange-Charpit Theory on T * Q
The good fit of the Lagrange-Charpit method of PDE integration with the different geometric aspects of the symplectic structure on the cotangent bundle, allows to make a presentation in this context, thus allowing a brief, effective and self-contained writing.

2.1.
Let us consider the classical phase space of a mechanical system as the cotangent bundle T * Q of the configuration manifold Q and let π Q : T * Q → Q be the canonical projection. Let U be a neighborhood in Q coordinated by (q 1 , . . . , q n ) and let us consider the corresponding coordinates (q 1 , . . . , q n ; p 1 , . . . , p n ) on π −1 Q (U ). The Liouville form is defined locally by w Q = i p i dq i . In this way, setting Ω Q = −dw Q , we obtain the local expression Ω Q = i dq i ∧ dp i of the canonical symplectic form on T * Q.
For every F ∈ C ∞ (T * Q), its Hamiltonian vector field is the unique vector field X F on T * Q such that i XF Ω Q = dF. In this way, the Poisson bracket on C ∞ (T * Q) is defined by: The local expression of the Hamiltonian vector field X F is: When {F, G} = 0, we say that F and G are in involution. A characteristic result, classically attributed to Jacobi and Lie, settles that if F 1 , . . . , F r ∈ C ∞ (T * Q) is a system of functionally independent functions in involution, then r ≤ n and we can extend it to a system {F 1 , . . . , F n , G 1 , . . . , G n } in such a way: both isotropic and coisotropic we say that N is a Lagrangian submanifold of T * Q.
The geometric structure of the Hamilton-Jacobi theory on the ndimensional configuration space Q is the symplectic phase space of momenta (T * Q, Ω Q ). In this way, for a given Hamiltonian function H ∈ C ∞ (T * Q), there exists a vector field X H determined by equation: i XH Ω Q = dH so that its integral curves are the trajectories of the system (v.g.r. see [7] ). The classic formulation of the Hamilton-Jacobi problem is to find a function S(t, q) satisfying: If we put S(t, q) = W − tE for a constant E, then W satisfies: In our geometric context, we can write (dW ) * H = E, where dW is understood as a section of T * Q. In this way, as a closed form is locally exact, we seek for closed 1-forms α : Q → T * Q such that: Thus, our objective is to find submanifolds V ⊂ T * Q on which H is constant.

Theorem 3 (Hamilton-Jacobi for isotropic submanifolds). Let us consider the Hamiltonian dynamical equation:
and let V be an isotropic submanifold of T * Q.
(ii) If H| V is constant, then the restriction of the left-hand side of (a) to V vanishes, hence X H ∈ X(V ) ⊥ . For the Lagrangian counterpart see [4] or the smart discussion [7].

2.2.
Our purpose now is to construct a completely integrable exterior differential system in such a way that, roughly, a generator system of exact differentials parameterizes isotropic submanifolds on (T * Q, Ω Q ). In this way, to establish and solve our problem, we need to extend the previous phase space to T * Q × R. In this new frame, the consideration of contact geometry will be essential in our purpose Thus, a contact manifold C is a (2n+1)-manifold endowed with a 1-form α such that: in such a way that dα is nondegenerate on ker(α).
The Reeb vector field is the unique R ∈ X(C) such that α(R) = 1, i R dα = 0.
Every differentiable function H : C −→ R has associated a vector field X H , called the Hamiltonian vector field defined by the relations: We can take local Darboux coordinates (q i , p j , z) such that: Then R = ∂/∂z and the vector field X H has the expression: (see, for example, [9] or [1]). Instead of X H we will consider here the reduced Hamiltonian vector field X H red , X H red = X H + HR (c) which, as we will see, has interesting geometric properties for us.
On the (2n + 1)-dimensional manifold T * Q × R, where R is supposed to be coordinated by the variable z, on which we define the extended Liouville form: Let us denote by j the inclusion j : T * M → T * M × R : p q → (p q , 0).

Theorem 4. Let us consider the Hamilton equation on
is completely integrable. Accordingly, there is a local basis of exact differentials for W and hence there are functions g 0 , . . . , g n , F n defined on a local domain (which we denote in the same way) such that: Proof. The argument relies on two key facts. The first one is that the orthogonal distribution W ⊥ is (locally) generated by the vector fields: The second one is that both, the orthogonal and the characteristic distribution coincide, that is W 0 = W 0 .
In this way, is it easy to see that: Hence, as rank(W 0 ) = (2n + 1) − rank(W) = n, we obtain that the local basis in (e) generate W 0 .
Having disposed of this first step, a brief computation gives us: Thus, form (f) and (g) we obtain W 0 ⊆ W 0 , which is precisely the assertion of the Theorem.

Corollary 5. With the previous notations, the equations:
H = a 0 , F 1 = a 1 , . . . , j * (F n ) = a n for a 0 , . . . , a n constants, determine an isotropic submanifold in U ⊂ T * Q on which H is constant. Example 6. Let us consider the canonical coordinates (q 1 , q 2 ; p 1 , p 2 ) on T * (R 2 ). The extended Liouville form on T * (R 2 )×R is w = p 1 dq 1 +p 2 dq 2 −dz. Let us consider the Hamilton-Jacobi equation i XH Ω = dH with H = pq. Then

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The Lagrange-Charpit Theory Page 7 of 10 8 As a first integral of X H red is F 1 = p 1 , we take the following exterior differential system on T * (R 2 ) × R: Thus, there are functions F 2 ,g 0 , g 1 , g 2 such that: By restricting to {H = a, F 1 = b} (for a, b constants), we get: In this way, as w = bdq 1 + (a/b)dq 2 − dz = d(bq 1 + (a/b)q 2 − z) and j * (bq 1 + (a/b)q 2 − z) = bq 1 + (a/b)q 2 , we obtain that: are the equations of an isotropic submanifold of T * (R 2 ) on which H is constant.

Appendix: Liouville vs. Reeb Dynamics
The objective of this section is to indicate some research perspectives, which in parallel to the lines of this article, have gained a great recent boom and they share, as an essential starting point, some of the concepts used in this work.
In the previous section we have searched for isotropic submanifolds of T * Q as solutions of {w Q = 0}. This sufficient condition is not, of course, necessary. However, there are situations in which both conditions coincide.

Definition 7.
We shall call Liouville vector field on T * Q to the unique vector field satisfying: i Ω Q = −w Q . Proposition 8. We have, (i) L Ω Q = Ω Q .
(ii) The Lie derivative, L preserve the set of Hamiltonian vector fields.

Proof. (i) is trivial. For (ii), if
We call M ⊂ T * Q a Liouville submanifold if ∈ X(M ). We can state