Constants of motion associated with alternative Hamiltonians

TL;DR

The paper demonstrates that for non-autonomous systems expressed in Hamiltonian form with different coordinate sets, specific determinants and traces related to Poisson brackets are conserved quantities.

Contribution

It introduces a novel class of constants of motion derived from the Poisson brackets between different Hamiltonian coordinate systems.

Findings

Determinant and trace of matrix formed by Poisson brackets are conserved.

Constants of motion are associated with alternative Hamiltonian formulations.

Applicable to non-autonomous Hamiltonian systems.

Abstract

It is shown that if a non-autonomous system of $2n$ first-order ordinary differential equations is expressed in the form of the Hamilton equations in terms of two different sets of coordinates, $(q_{i}, p_{i})$ and $(Q_{i}, P_{i})$, then the determinant and the trace of any power of a certain matrix formed by the Poisson brackets of the $Q_{i}, P_{i}$ with respect to $q_{i}, p_{i}$, are constants of motion.