Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

Rashkovskiy Sergey

doi:10.3934/jgm.2020024

Article Contents

2020, Volume 12, Issue 4: 563-583. Doi: 10.3934/jgm.2020024

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Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

Rashkovskiy Sergey

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia

Received: March 31, 2018

Revised: June 30, 2020

Early access: August 2020

Published: December 2020

This work was done on the theme of the State Task No. AAAA-A20-120011690135-5

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Abstract

A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the Schrödinger form to classical mechanics.

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Mathematics Subject Classification: Primary: 70H20, 70S05; Secondary: 37J05.

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