Invariance and first integrals of canonical Hamiltonian equations

TL;DR

This paper explores the connection between symmetries and first integrals in Hamiltonian systems, introducing a new identity that simplifies constructing conserved quantities, demonstrated through examples like Kepler motion.

Contribution

It introduces a novel identity linking symmetries and first integrals in Hamiltonian equations, enhancing the method for finding conserved quantities.

Findings

New identity simplifies the construction of first integrals

Application to Kepler motion demonstrates practical utility

Provides a clearer understanding of symmetry-invariant relations in Hamiltonian systems

Abstract

In this paper we consider the relation between symmetries and first integrals of canonical Hamiltonian equations. Based on a newly established identity (which is an analog of well known Noether's identity for Lagrangian approach), this approach provides a simple and clear way to construct first integrals with the help of symmetries of a Hamiltonian. The approach is illustrated by a number of examples, including equations of the three-dimensional Kepler motion.