Journal of Nonlinear Mathematical Physics

Volume 26, Issue 4, July 2019, Pages 650 - 658

Systems of Hamilton-Jacobi equations

Authors
Julio Cambronero
Departamento de Matemáticas Fundamentales, UNED C/ Senda del Rey 9, 28040 Madrid, Spain, jcambronero@alumno.uned.es
Javier Pérez Álvarez
Departamento de Matemáticas Fundamentales, UNED C/ Senda del Rey 9, 28040 Madrid, Spain, jperez@mat.uned.es
Received 11 April 2019, Accepted 7 June 2019, Available Online 9 July 2019.
DOI
https://doi.org/10.1080/14029251.2019.1640473How to use a DOI?
Keywords
Hamilton-Jacobi equation, Lagrangian submanifolds
Abstract

In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.

Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Hamilton-Jacobi theory arises with William Hamilton in the 1820’s of the XIX century, who carries his unification purpose of particle and wave concepts of light, in the geometric optics, to crystallize (towards 1835) in the method of canonical transformations to determine the trajectories of systems. Later on Carl Gustav Jacobi interprets the dynamics of mechanical systems in terms of the complete solutions of the associated Partial Differential Equation.

Beyond classical mechanics, the Hamilton-Jacobi theory lets feel its influence in Quantum mechanics, not only under the principle that a classical system should be obtained as an appropriate limit of the quantum one, (v.g.r. [1] or [2], where it is considered how the equations of the characteristics, in the short-wave limit of evolutionary wave equation on the configuration space, produce the Hamilton equation on the cotangent bundle), but the consideration of Hamilton-Jacobi theory as a tool itself in Quantum Systems. Thus, a complete solution of the Hamilton-Jacobi equation for the Hamiltonian H(t, qi, pi), is a function S(t, qi, xi) satisfying the Hamilton-Jacobi equation

St(t,qi,xi)+H(t,qi,Sqi(t,qi,xi))=0.(1.1)

Under the non-degeneracy condition det (2S/∂qi∂xj) ≠ 0, we can define the canonical transformation by

Sqi=pi,Sxi=yi
in such a way that in the new coordinates (t, xi, yi), the dynamical system governed by the Hamiltonian H turns into a trivial dynamical system. Nevertheless, canonical transformations do not preserve neither the quantum Hilbert space nor the phase space path integral of Feynman’s formulations of quantum mechanics. This fact has aroused a great interest establishing the frame which can be called the quantum Hamilton-Jacobi equation (v.g.r. see [3] for a discrete version of classical transformations in the path integral formulation, [4] for the determination of the quantum mechanical amplitude by means of a single momentum integration form a complete solution of the classical Hamilton-Jacobi equation, or [5] for a modification of the Hamilton-Jacobi equation that has suitable covariance properties in such a way that the function S is related to solutions of the Schrödinger equations).

Thus, Hamilton-Jacobi theory is not only a stepping stone in our comprehension of the quantum theory from classical terms but also it aims to provide a powerful quantum tool.

In the present paper, that goes back to the classical theory, we consider systems of Hamilton-Jacobi equations submitted to a compatibility condition – the involutive character – and consider the classical geometric problem of finding foliations transverse to the fibers of T*Q and invariant under every dynamical evolution, this way extending the standard theory (see [6], [7], [8]). An argument that leads us to the consideration of Darboux coordinates, will allow us to raise this problem locally.

The importance of the Marsden-Weinstein reduction procedure along with the technique of generating functions motivated the consideration of the reduction of the Hamilton-Jacobi theory (see [9], [8] and [10] for a complete reference for Hamilton reduction). It is therefore, section 4 is devoted to extending the reduction and reconstruction procedures for the involutive systems frame.

2. Preliminaries

Important aspects of the modern geometric formulation of the Hamilton-Jacobi theory were estab- lished in [11] with the Poisson geometry of T*Q and more recently in [6] by considering Lagrangian foliations transverse to the fibers of T*Q that are invariant under the dynamical evolution associated to the symplectic structure.

Let us settle the fundaments of the theory in the most self-contained, brief and clear possible way.

Let (M,w2) be a 2n-dimensional symplectic manifold. A vector field X ∈ 𝔛(M) is called an infinitesimal symmetry of the symplectic structure if LXw2 = 0. This condition is equivalent to the fact that iXw2 is a closed 1-form. Even more, if iXw2 is exact, that is, of the form df for certain smooth function f defined on M, then X (usually written as Xf) is called the Hamiltonian vector field associated to f. The Poisson bracket of two smooth functions f and g on M is defined by

[f,g]=w2(Xf,Xg)
and endows 𝒞(M) with a Lie algebra structure.

Let 𝒳 be a submanifold of M. For every p𝒳, we write (Tp𝒳) for the w2-orthogonal complement of Tp𝒳 in TpM. We say that 𝒳 is a coisotropic submanifold of M dimension m, if for every p𝒳 , (Tp𝒳)Tp𝒳. It is clear that then 2nmm and hence nm. If dim𝒳 = n, then 𝒳 is called a Lagrangian submanifold of M. It is immediate that, in this case, the condition (Tp𝒳) = Tp𝒳 for every p𝒳 means w2|𝒳 = 0.

Proposition 2.1.

Let 𝒳 be a submanifold of M of dimension m = 2nr.

  1. (i)

    Let ℐ be the sheaf of ideals of 𝒳. Let f1,..., fr be local generators ofon the open set UM. Then the Hamiltonian vector fields Xfi(1 ≤ ir) constitute a basis for (Tp𝒳) for every pU𝒳.

  2. (ii)

    𝒳 is coisotropic if and only if ℐ is stable by Poisson bracket.

Proof.

  1. (i)

    It suffices to take into account that if XTp𝒳 then

    0=Xfi=dfi,p(X)=w2,p(Xfi,X),1ir.

  2. (ii)

    By (i), 𝒳 is coisotropic if and only if Xfi ∈ 𝔛(𝒳U), for 1 ≤ ir. This means that

    Xfi(fk)=[fk,fi]=0,1i,kr.

We say that the symplectic structure (M,w2) is homogeneous (or, also, exact) if there exists a 1-form w1 on M such that w2 = dw1. The vector field X ∈ 𝔛(M) is Hamiltonian with respect to this homogeneous structure if and only if LXw1 is exact. This fact trivially follows from the relation

LXw1=d(iXw1)+iXdw1=dw1(X)+iXw2.

Let G be a connected Lie group and 𝒢 the corresponding Lie algebra. Let us suppose that there is a free and proper left action on (M, w2). We say that this action is symplectic if for each element gG, we have

g*w2=w2(2.1)
(where g* denotes the action of g by pull-back on differential forms on M). If for each element A𝒢 we denote by A* its fundamental vector field on M (that is, the infinitesimal generator of the 1-parameter group of transformations of M : γ(t) = exp(tA)), then (2.1) implies
LA*w2=0(iA*w2isaclosed1form).

Moreover, if for each A𝒢, we have

iA*w2=dfA*,forsomesmoothfunctionfA*onQ
we say that the action of G on M is Hamiltonian. In that case, we define the momentum mapping
J:Q𝒢*:J(x)(A)=fA*(x),A(𝒢)(xQ).

Let us now denote with Q any smooth manifold, set M = T*Q for its cotangent bundle, and let π : MQ be the natural projection. There is an intrinsic way of define a 1-form w1 on M as follows: let qQ, pTq*Q and XTpM, then

w1:Xp,πX
where π′ is the tangent linear map π′ : TpMTqQ. It is easily seen that w2 = −dw1 is a non-degenerate 2-form, thus defining the so called canonical symplectic structure on T*Q. The 1-form w1 is canonically defined and hence invariant under the induced action of any diffeomorphism of Q. Consequently, if there is an action of a Lie group G on Q then under the induced action, we have g*w1 = w1 for every gG. In this way, for every A𝒢 we have
LA*w1=0.(2.2)

Hence, the action of G on T*G is Hamiltonian, and in fact, it follows from (2.2)

iA*w2=d(w1(A*)).

There is an explicit and intrinsic way of expressing the function w1(A*). Given X ∈ 𝔛(Q) we define PX : T*Q → ℝ: αqαq(X). Now

w1(A*)(αq)=αq,πA*=αq(A*)=PA*(αq)
(we use the same notation A* for the fundamental vector field associated to the action of G on Q and on T*Q and that the latter projects onto the former). In this way, the momentum mapping J : T*Q𝒢* is given by
J(αq)(A)=PA*(αq).

It is not difficult to see the G-equivariance of J, that is, J(g*αq)=adg1*J(αq) and that for any μ𝒢*, J−1(μ) is a submanifold of T*Q. So, if μ𝒢* is a fixed point for the coadjoint action of G, the canonical symplectic form w2 defines a 2-form w¯2 on the quotient manifold J−1(μ)/G by

w2(Xp,Yp)=w¯2(X¯p,Y¯p),
where X¯p, Y¯p are the respective classes of Xp and Yp in TpJ−1(μ)/Tp(G · p). The definition of w¯2 makes sense, since TpJ−1(μ) and Tp(G · p) are orthogonal complements in Tp(T*Q), as it is not difficult to see.

The geometric frame of the Hamilton-Jacobi theory on the n-dimensional configuration space Q is the phase space of momenta T*Q and its canonical exact symplectic structure w2. Thus, given a Hamiltonian function H𝒞 (T*Q), there exists a vector field XH provided by the dynamical equation

iXHw2=dH(2.3)
whose integral curves are the trajectories of the system (v.g.r. see [6], [12] or [9]). In the classics formulation, the Hamilton-Jacobi problem consists in finding a function S(t,q) that satisfies the partial differential equation
St+H(q,Sq)=0.

If we write S(t,q) = WtE for a constant E, then the function W satisfies

H(t,q,Wq)=0.

In geometric terms, this equation means (dW)* H = E, where dW is understood as a section of T*Q. In a more adequate context, as a closed form is locally exact, we seek for closed 1-forms α : QT*Q such that

H|Ima=E.

Since the closed character of α implies that Imα to be a Lagrangian submanifold of T*Q, our aim is to find Lagrangian submanifolds LT*Q and vector fields ZH ∈ 𝔛(T*Q) whose integral curves contained in L be the trajectories of the system.

The essence of the geometrical character is gathered in the following central result.

Theorem 2.1 (Hamilton-Jacobi)

Let us consider the dynamical equation

iZHw2=dH.(2.4)

The following assertions are equivalent for a Lagrangian submanifold LT*Q

  1. (i)

    The vector field ZH is tangent to L.

  2. (ii)

    H|L is constant.

In any of these cases, we say that L is a solution of (2.4).

Proof.

If ZH ∈ 𝔛(L) then the equation (2.4) can be restricted to L, and then (ii) trivially follows as w2|L = 0. Conversely, if H|L is constant, then (2.4) says that ZH ∈ (𝔛(L)). But for a Lagrangian manifold (𝔛(L)) = 𝔛(L), which completes the proof of the theorem.

Some other proofs of this key fact can be consulted in [8], [6] or even in [9] for an interesting proof based on ℝ-actions on T*Q.

Now let (qi) the coordinates in M and (qi, pi) the induced coordinates on T*M. If α=i=1nαidqi is a closed 1-form, since the functions

fi=αipi,1in
generate the Lagrangian submanifold Imα in T*Q, the Hamiltonian vector fields
Xfi,1in
span, by Proposition 2.1, the orthogonal complement (Tx(Im α)) (x ∈ Imα). Hence a necessary and sufficient condition for Im α (or α) to be a solution of the Hamilton-Jacobi equation is that
wx(Xfi,XH)=0,xL,1in,
fact that we state as follows.

Proposition 2.2.

The closed 1-form α=i=1nαidqi on Q is a solution of the Hamilton-Jacobi equation (2.4) if and only if

XH,x(αipi)=0,xL,1in.

3. Involutive systems

Let us consider the system of Hamilton-Jacobi equations

{iXH1w2=dH1iXHkw2=dHk(3.1)
where the functions Hi are pairwise in involution, that is, we have
[Hi,Hj]=0,1i,jk
and whose differentials are linearly independent in every point of T*Q. In this case, we say the system (3.1) is involutive.

In the framework of Hamilton-Jacobi theory, the question which this definition implies is to find Lagrangian submanifolds L of T*Q invariant under the flows of the vector fields XHi, 1 ≤ ik.

Here we will give a local solution using an easy argument based on a classical result of Jacobi-Lie in relation to the extension of a set of functions, pairwise in involution and functionally independent to a complete set of canonical symplectic coordinates.

In precise terms, we have:

Theorem 3.1.

Let us consider the involutive system (3.1). The submanifold M of T*Q defined by the equations

Hi=0,i=1,,k.
is coisotropic and kn. Let us denote with w2 and XHi the respective restrictions of the symplectic form w2 and the fields XHi to M. For each point pM there exists a neighborhood U in M in such a way that a family {NKj} of Lagrangian submanifolds contained in U is obtained by equaling to constants nk first integrals common to XHi, 1 ≤ ik, and such that XHi ∈ 𝔛(NKj), 1 ≤ ik.

Proof.

The first claim trivially follows form Proposition 2.1. In this way, the condition dimMn for a coisotropic submanifold says that 2nkn and hence kn. From the Carathéodory-Jacobi-Lie theorem (v.g.r., see [13]), in a neighborhood V of each point pT*Q there are another 2nk functions

Hk+1,,HnJ1,,Jn(3.2)
in such a way that [Hi,Hj] = 0, [Ji,Jj] = 0, [Hi,Jj] = δij.

In fact, these functions constitute a set of Darboux coordinates in which the symplectic form w2 on V is expressed as

w2=j=1ndHjdJj.

Consequently for the restriction w2 of w2 to U = VM, we have w2=j=1nkdHk+jdJk+j.

Now by Proposition 2.1, at each point pM, the tangent vectors XHi,p, (1 ≤ ik) generate the orthogonal complement (TpM) of TpM in Tp(T*Q). As M is a coisotropic manifold (TpM)TpM, with what the tangent vectors XHi,p form a basis for the space of tangent vectors XpTpM such that iXpw2 = 0.

In this way, the condition iXHi w2 = 0, means that Hk+j, Jk+j (1 ≤ jkn) are first integrals of the fields XHi.

Thus, the submanifold of M

Nkj={Hk+j=kj,kj,1jnk}
is a Lagrangian submanifold in U.

The argument above combined with the Hamilton-Jacobi theory provides XHi ∈ 𝔛(Nkj) for 1 ≤ ik, which completes the proof of the theorem.

4. Reduction and Reconstruction

In this section we address the reduction and reconstruction procedures for an involutive system of Hamilton-Jacobi equations with symmetries. Maybe one of the seeds in the consideration of the subjection of the Hamilton-Jacobi dynamics to the symplectic reduction might be [8, Th. 4.3.5]. Years later, this topic is approached together with the inverse problem of reconstruction in [9]. We tackle here this deep subject in a simpler way, without using the structural results of [9] neither the consideration of magnetic terms added to the canonical symplectic form of T*G.

Let G be a connected Lie group acting freely and properly on a manifold Q and let us consider its natural lifted action on T*G, which is also Hamiltonian with respect to the canonical symplectic structure w2 with momentum mapping J : T*G𝒢*. Let dimQ = n and dimG = m.

Proposition 4.1.

Let Hi : T*Q → ℝ (1 ≤ ik) smooth functions invariant under the G-action. Let μ𝒢* be a fixed point for the coadjoint action of G, and let us consider the symplectic structure w¯2 defined in the quotient by the projection

π:J1(μ)J1(μ)/G.

Let us assume that the projections H¯i of the functions Hi (1 ≤ ik) to the quotient manifold J−1(μ)/G are functionally independent. The Hamilton-Jacobi system

{iXH1w2=dH1iXHkw2=dHk(4.1)
with knm, determines a Hamilton-Jacobi system on the reduced symplectic manifold J−1(μ)/G,
{iX¯H1w¯2=dH¯1iX¯Hkw¯2=dH¯k(4.2)
in such a way that if LJ−1(μ) is a Lagrangian submanifold of T*Q solution of the system (4.1) then π(L) is a solution of the system (4.2).

Conversely if L¯ is a Lagrangian submanifold of J−1(μ)/G solution of the system (4.2), then L=π1(L¯)J1(μ) is a Lagrangian submanifold of T*Q solution of the system (4.1).

Proof.

First of all, we must show that every vector field XHi is tangent to J−1(μ). It suffices to see that if pJ−1(μ) then J*,p (XH) = 0. But

J*,p(XH),A=w2,p(XH,A*)=Ap*H=0,A𝒢,(4.3)
by the G-invariance of H. Now, the G-invariance of both w2 and Hi defines XHi as a G-invariant vector field on J−1(μ) hence inducing a vector field X¯H𝔛(J1(μ)/G). In this manner, each of the equations of the system (4.1) provides an equation of the reduced dynamics
iX¯Hiw¯2=dH¯i,1ik,inJ1(μ)/G.(4.4)

Let LJ−1(μ) a Lagrangian submanifold of T*Q. A consideration similar to (4.3) proves that L is G-invariant. In fact, it suffices to see that for every fundamental vector field A* we have A* ∈ 𝔛(L). Let pL and XTp(L), then

w2,p(X,A*)=J*,p(X)(A)=0.

Thus, L¯=π(L) is a Lagrangian submanifold of J−1(μ)/G: in fact as dimJ−1(μ) = 2nm (and hence dimJ−1(μ)/G = 2(nm)), as dimL¯=nm it suffices to see that w¯2|L¯=0, which is guaranteed by the fact

w¯2|L¯=w2|L(4.5)
in a self-explanatory notation. Finally L¯ is a solution of the system (4.2), since by the G-invariance
H¯i|L¯=Hi|L=cifor1ik.(4.6)

Conversely, if the Lagrangian submanifold L¯ of J−1(μ)/G is a solution of the system (4.2), then the above argument on dimensions and (4.5) say that L=π1(L¯) is a Lagrangian submanifold of J−1(μ) which, again by (4.6), is a solution of the system (4.1).

References

[8]R. Abraham and J.E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[10]J.E. Marsden, G. Misiołek, J.P. Ortega, M. Perlmutter, and T.S. Ratiu, Hamiltonian reduction by stages, Springer, Berlin, 2007. Lecture Notes in Mathematics 1913
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 4
Pages
650 - 658
Publication Date
2019/07
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2019.1640473How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Julio Cambronero
AU  - Javier Pérez Álvarez
PY  - 2019
DA  - 2019/07
TI  - Systems of Hamilton-Jacobi equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 650
EP  - 658
VL  - 26
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1640473
DO  - https://doi.org/10.1080/14029251.2019.1640473
ID  - Cambronero2019
ER  -