Journal of Geometric Mechanics
2009 , Volume 1 , Issue 3
Special Issue on New Trends in the Hamilton-Jacobi Theory
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The Hamilton-Jacobi theory is a classical subject that was extensively developed in the last two centuries. The Hamilton-Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. Indeed, the powerful of this method is that, in spite of the difficulties to solve a partial differential equation instead of an ordinary differential one, in many cases it works, being an extremely useful tool, usually more than Hamilton’s equations. Indeed, in these cases the method provides an immediate way to integrate the equations of motion. The modern interpretation relating the Hamilton-Jacobi procedure with the theory of lagrangian submanifolds is an important source of new results and insights.
In addition, the Hamilton-Jacobi-Bellman equation is a partial differential equation which is central to optimal control theory. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton-Jacobi equation by Hamilton and Jacobi.
This special issue on Hamilton-Jacobi theory wants to bring specialists coming from different areas of research and show how the Hamilton-Jacobi theory is so useful in their domains: completely integrable systems, nonholonomic mechanics, Schrödinger equation, optimal control theory, and, in particular, applications in engineering and economics.
For a Hamiltonian enjoying rather weak regularity assumptions, we provide necessary and sufficient conditions for the existence of a global viscosity solution to the corresponding stationary Hamilton-Jacobi equation at a fixed level $a$, taking a prescribed value on a given closed subset of the ground space. The analysis also includes the case where $a$ is the Mañé critical value. Our results are based on a metric method extending Maupertuis approach.
For general underlying spaces, compact or noncompact, we give a global version of the classical characteristic method based on the notion of $a$-characteristic. In the compact case, we propose an inf-sup formula producing the minimal solution of the problem, where the generalized Aubry set is involved.
We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system.
We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi ($HJ$) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the $HJ$ theory on the tangent bundle of a configuration manifold, the quantum $HJ$ theory, $HJ$ problems for general differential operators and the $HJ$ problem for Lie groups.
We provide a Hop-Lax formula for variational problems with non-constant discount and deduce a dynamic programming equation. We also study some regularity properties of the value function.
We show how a minimal deformation of the geometry of the classical Hamilton-Jacobi equation provides a probabilistic theory whose cornerstone is the Hamilton-Jacobi-Bellman equation. This is the basis for a novel dynamical system approach to Stochastic Analysis.
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