Generalized Hamiltonian structures for Ermakov systems

TL;DR

This paper develops Poisson structures for Ermakov systems using the invariant as Hamiltonian, identifying conditions for superintegrability and linearization, thus advancing the geometric understanding of these dynamical systems.

Contribution

It introduces new Poisson structures for Ermakov systems, including degenerate cases with Casimir functions, and characterizes when these systems can be linearized.

Findings

Identified two classes of Poisson structures for Ermakov systems.

Derived Casimir functions in degenerate Poisson structures.

Characterized conditions for linearization of Ermakov equations.

Abstract

We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.