On the Hamiltonian structure of Ermakov systems

TL;DR

This paper develops a Hamiltonian formalism for Ermakov systems, enabling exact solutions for specific cases like the Calogero system and noncentral potentials, and explores their generalizations.

Contribution

It introduces a canonical Hamiltonian framework for a broad class of Ermakov systems, including all known Hamiltonian cases, facilitating their reduction to quadratures and exact solutions.

Findings

Hamiltonian structure derived for various Ermakov systems

Exact solutions obtained for Calogero and noncentral potential systems

Identification of generalized systems with solvable dynamics

Abstract

A canonical Hamiltonian formalism is derived for a class of Ermakov systems specified by several different frequency functions. This class of systems comprises all known cases of Hamiltonian Ermakov systems and can always be reduced to quadratures. The Hamiltonian structure is explored to find exact solutions for the Calogero system and for a noncentral potential with dynamic symmetry. Some generalizations of these systems possessing exact solutions are also identified and solved.