How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?

Question

Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.

For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$.

If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties:

1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$,
2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$,
3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective.

Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant.

But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally.

The idea of Sophus Lie was to give up both conditions 2 and 3.

Adopting this point of view, we define a generalized (or geometric) solution of $H\cic dS=0$ to be any lagrangian submanifold $\Lambda$ of $(T^\ast M,d\theta_M)$ which is included in $H^{-1}(0)$.

I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions.

But I am hoping to find "more" practical applications which illustrate the meaningfulness of geometric solutions. I would like to learn if ther is some physical or geometrical problem involving an H.-J. eqn, whose comprehension is sensibly augmented by the consideration of generalized solutions. So my question is:

What are the possible arguments and applications that motivate and help to interpret the notion of geometric solutions for an Hamilton-Jacobi equation?

Answer 1

The famous KAM tori arose out of HJ considerations. They are Lagrangian torii. They were found by attempting to solve the HJ equation generally, and then finding one can only solve it when certain appropriately irrational frequency conditions hold. They occur in perturbations of integrable systems, or near `typical' linearly stable periodic orbits in a fixed Hamiltonian systems. You can read about them in an Appendix to Arnol'd's Classical Mechanics, and also get some idea from Chris Golé's book 'Symplectic Twist Maps', or from Siegel and Moser's 'Stable and Random Motion'.

Answer 2

A very interesting practical application is the problem of state estimation - for linear systems the answer is called the Kalman filter. Given a vector field $\dot{x} = a(x,v)$ and a measurement equation $y=c(x,w)$, compute the initial condition $x(t_0)$, the perturbation $v(t)$, and the measurement error $w(t)$ that minimize a cost function $J$. The cost is usually expressed as an integral over time of some function of $v$ and $w$.

Using Pontryagin's maximum principle or Bellman's dynamic programming, one arrives at a HJ equation which is used to find $v$. The additional step needed is to determine $x(t_0)$. It is a static minimization problem, which however needs to be repeated at each instant $t$ in the interval of interest. This is not a very practical answer. For linear systems with quadratic costs, the Kalman filter provides a recursive solution to the complete problem. In more general cases, the problem is much less studied either by engineers or by mathematicians. This is unlike the optimal control problem which has been studied extensively.

I think the geometry of the solutions is crucial. My understanding is that the filter equation is a particular symmetry of the Hamilton-Jacobi-Bellman partial differential equation - at least when everything is smooth. Meanwhile, the Hamiltonian vector field is a characteristic of the partial differential equation - also a particular symmetry, but not the one that gives a recursive solution to the estimation problem.

Answer 3

As you suspect these generalized solutions and their apparent singularities (=points of the Lagrangian submanifold where condition 2. fails) are unavoidable.

First observe that any Lagrangian submanifold contained in $H^{-1}(0)$ must be tangent to the Hamiltonian field $X_H$ (this is the method of characteristics). I assume here that $H^{-1}(0)$ is smoot and $2n-1$ dimensional. Now start with some non-characteristic classical initial data (= an $n-1$ dimensional submanifold in $H^{-1}(0)$ transversal to $\tau^*_M$ and transverslat to $X_M$). If you let the initial datum flow with $X_H$ this will swipe out the unique solution in $T^*M$. For short times this Lagrangian manifold will be transversal but at some point it can start to bend so that condition 2. fails. The projection to $M$ of points where transversality fails are called caustics in the literature.

Here's the classical physics example which you'll find for example in Arnolds books (his PDE course but I think also in his mechanics book): in the particle picture, light particles all move along straight lines with the same speed $c$ in possibly different directions (but they don't interact). An initial data would be given by a surface in the room and a direction field along this surface giving the initial direction of light rays. Initially the light rays don't intersect, but after some times they might start to intersect. The solution S(q) of HJ in this example describes the time after which the wave front arrives at a point q in space. If light rays intersect this function becomes multivalued.

By the way I'd be interested in the original source of Lie, could you add that to your question?