On First Integrals of Hamiltonian System with Holonomic Hamiltonian

TL;DR

This paper investigates the solutions of the Hamilton-Jacobi equation with holonomic Hamiltonians, linking first integrals to Pfaffian systems and providing a finite-dimensional algebraic characterization.

Contribution

It introduces a finite-dimensional algebraic framework for identifying first integrals of Hamiltonian systems with holonomic Hamiltonians.

Findings

Finite-dimensional algebraic equations characterize first integrals.

Solution space of the HJE can be analyzed via Pfaffian systems.

Numerical example demonstrates the practical applicability.

Abstract

In this study, the solution of the Hamilton-Jacobi equation (HJE) with holonomic Hamiltonian is investigated in terms of the first integrals of the corresponding Hamiltonian system. Holonomic functions are related to a specific type of partial differential equations called Pfaffian systems, whose solution space can be regarded as a finite-dimensional real vector space. In the finite-dimensional solution space, the existence of first integrals that define a solution of the HJE is characterized by a finite number of algebraic equations for finite-dimensional vectors, which can be easily solved and verified. The derived characterization was illustrated through a numerical example.