Franco Cardin, Claude Viterbo
We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend some results of Barles and Tourin on multi-time Hamilton-Jacobi equations to a more general setting.
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Let G be an algebraic group over a field F of characteristic zero, with Lie algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a Poisson variety and each irreducible G-invariant subvariety X\subset g^* carries the induced Poisson structure. We prove that there is a family of algebraically independent polynomial functions {f_1,...f_l} on X, which pairwise commute with respect to the Poisson bracket and such that l=(dim X+tr.deg F(X)^G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varie…
For non convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton-Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution.
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of Abraham and Marsden in [1], such that we can prove two types of geometric Hamilton-Jacobi theorem for a Hamiltonian system on the cotangent bundle of a configuration manifold, by using the symplectic structure and dynamical vector field. Then these results are generalized to the regular reducible Hamiltonian system with symmetry and momentum map, by using the re…
Let $(M,J,ω)$ be a quantizable compact Kähler manifold, with quantizing Hermitian line bundle $(A,h)$, and associated Hardy space $H(X)$, where $X$ is the unit circle bundle. Given a collection of $r$ Poisson commuting quantizable Hamiltonian functions $f_j$ on $M$, there is an induced Abelian unitary action on $H(X)$, generated by certain Töplitz operators naturally induced by the $f_j$'s. As a multi-dimensional analogue of the usual Weyl law and trace formula, we consider the problem of describing the asymptotic clustering of the joint eigenvalues of these Töplitz operators along a given ray…
We develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton-Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton-Jacobi equation associated with the Hamiltonized system also solve the…
In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton-Jacobi problem for a Čaplygin system to the Hamilton-Jacobi problem for a forced Lagrangian system.
We give a construction of completely integrable 4-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at pairs of birational 4-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb R^4$, and possess two independent integrals of motion, which are perturbations of the original Hamilton functions. Thus, these maps are comple…
We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmet…
In order to describe the impact of different geometric structures and constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we first drive precisely the geometric constraint conditions of magnetic symplectic form for the magnetic Hamiltonian vector field.which are called the Type I and Type II of Hamilton-Jacobi equation. Secondly, for the magnetic Hamiltonian system with nonholonomic constraint, we first define a distributional magnetic Hamiltonian system, then derive its two types of Hamilton-Jacobi equat…
Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional consistency. In the present work, we follow this line of research and develop a Lagrangian theory of integrable one-dimensional systems. We give a complete solution of the following problem: one looks for a function of several variables (interpreted as multi-time) which delivers critical points to the action functionals obtained by integrating a Lagrangian 1-form along any smooth curve in the multi-time. The Lagrangian 1-form is supposed …
In this paper we propose a Hamilton-Jacobi theory for implicit contact Hamiltonian systems in two different ways. One is the understanding of implicit contact Hamiltonian dynamics as a Legendrian submanifold of the tangent contact space, and another is as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. In these two scenarios, we propose a Hamilton-Jacobi theory specifically derived with the aid of Herglotz Lagrangian dynamics generated by non-regular Lagrangian functions.
Following the analysis we have presented in a previous paper (that we refer to as [I]), we describe a Noether theorem related to symmetries, with the associated reduction procedures, for classical dynamics within the Lagrangian and the Hamilton-Jacobi formalism.
Jacobi Hamiltonian Integrators: construction and applications
本文提出了一种构造保持结构的JacobiHamilton积分器的系统框架，并展示了其在接触Hamilton系统和经典模型中的应用及优势。
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