Hamilton-Jacobi theory for one dimensional autonomous systems using parametric transformations

TL;DR

This paper introduces a method using parametric transformations to solve the Hamilton-Jacobi equation for one-dimensional autonomous systems, providing a necessary and sufficient condition for energy invariance and demonstrating the approach with an example.

Contribution

It presents a novel parametric transformation technique that simplifies solving the Hamilton-Jacobi equation for specific dynamical systems.

Findings

Derived a condition for energy-invariant parameter transformations.

Solved the Hamilton-Jacobi equation via quadrature using the proposed method.

Provided an illustrative example demonstrating the approach.

Abstract

A necessary and sufficient condition for a parameter transformation that leaves invariant the energy of a one dimensional autonomous system is obtained. Using a parameter transformation the Hamilton-Jacobi equation is solved by a quadrature. An example of this approach is given.