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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?

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@Mathphysicist: I have merged two of your tags in one, hoping to be more descriptive of the content. –  Giuseppe Tortorella Jul 9 '11 at 8:23
    
@Giuseppe: there's really no point for an geometric-theory-of-pdes tag. If you must, you should use the already existing geometric-analysis tag. –  Willie Wong Jul 16 '11 at 2:04
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3 Answers

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'.

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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?

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Dear Michael, I stated that historical attribution not for having read the original papers of Lie on transformation groups written in the seventies of ninenteenth century, but I learned it from Ch.5 §2.2 "The Geometry of Differential Equations" in "Geometry I, EMS 28" of Alekseevskij, Vinogradov, Lychagin. –  Giuseppe Tortorella Jul 9 '11 at 13:41
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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.

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Dear Pait, I appreciate very much your thoughtful responce. I have given a look at the corresponding sections of the book of Agrachev and Sachkov, but I have not found the lagrangian submanifolds not transversal to the fibers considered as solution generalized for the H.-J. eqn for the optimal cost. Where could I look for such objects in the context of control theory? I would like to learn if considering generalized solutions is possible obtain more information rather than using only usual solution. Thank you. –  Giuseppe Tortorella Jul 9 '11 at 7:50
    
Would you mind helping me with (or pointing to) explanations for the terms "lagrangian submanifolds not transversal to the fibers" and "generalized solutions"? That would help me translate between the two sides of the literature, the mathematical and the engineering. Thanks! –  Pait Jul 10 '11 at 21:38
    
I described this notion already in the text of my question. Please I woulde like to know the points of my question that are not enough clear, or are not written in proper english, so that I could correct them. Thank you in advance. –  Giuseppe Tortorella Jul 11 '11 at 13:37
    
Given the HJ eqn $H\circ dS=0$ in the unknown function $S$ on the smooth manifold $M$, where $H\in C^\infty(T^\ast M)$. A generalized solution is defined to be a submanifold of $T^\ast M$, the cotangent space of $M$, which is included in $H^{−1}(0)$ and lagrangian w.r.t. the canonical symplectic form $dθ_M$. Here $θ_M$ is the tautological, or Liouville, 1-form on $T^\ast M$. –  Giuseppe Tortorella Jul 11 '11 at 17:46
    
I think I wanted a reference, maybe to a book, with a more leisurely explanation. It's not that your text is in any way unclear, it's just that I have a different background and need to do my homework to learn the language better. –  Pait Jul 13 '11 at 15:07
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