Exact invariants for a class of three-dimensional time-dependent classical Hamiltonians

TL;DR

This paper derives an exact invariant for a class of three-dimensional time-dependent Hamiltonian systems, providing a new analytical tool for understanding complex dynamical behaviors in such systems.

Contribution

It introduces a novel exact invariant involving a time-dependent function satisfying a third-order differential equation for specific Hamiltonian systems.

Findings

Invariant applies to multi-particle systems in general potentials

Solution involves a third-order linear differential equation

Applicable to non-linear oscillators and Coulomb systems

Abstract

An exact invariant is derived for three-dimensional Hamiltonian systems of $N$ particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function $f_{2}(t)$, embodying a solution of a linear third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent non-linear oscillator, and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.