Frobenius theorem and invariants for Hamiltonian systems

TL;DR

This paper uses Frobenius integrability theorem to identify invariants and invariant foliations in one-dimensional Hamiltonian systems with time-dependent potentials, providing new classes of such potentials and explicit invariants.

Contribution

It introduces a novel application of Frobenius theorem to find invariants in Hamiltonian systems, including an infinite class of potentials and a method to construct invariants explicitly.

Findings

Identified classes of potentials with invariant foliations.

Derived explicit invariants for certain Hamiltonian systems.

Proved the inverse: known invariants can be obtained via Frobenius theorem.

Abstract

We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assurately restricted to a two-dimensional foliation. In some cases, Frobenius theorem allows the explicit construction of an associated invariant. It is proven the inverse result that, if an invariant is known, then it always can be furnished by Frobenius theorem.