"A Solvable Hamiltonian System" Integrability and Action-Angle Variables

TL;DR

This paper proves that a specific Calogero-type Hamiltonian system is equivalent to non-interacting harmonic oscillators, providing explicit conserved currents, action-angle variables, and solutions to the initial value problem.

Contribution

It establishes the integrability of the system by explicitly constructing conserved quantities and action-angle variables, simplifying its analysis.

Findings

System is equivalent to non-interacting harmonic oscillators

Explicit conserved currents in involution are derived

Initial value problem is solved explicitly

Abstract

We prove that the dynamical system charaterized by the Hamiltonian $ H = \lambda N \sum_{j}^{N} p_j + \mu \sum_{j,k}^{N} {{(p_j p_k)}^{1\over 2}} \{ cos [ \nu ( q_j - q_k)] \} $ proposed and studied by Calogero [1,2] is equivalent to a system of {\it non-interacting} harmonic oscillators. We find the explicit form of the conserved currents which are in involution. We also find the action-angle variables and solve the initial value problem in simple form.