First integrals of some two-dimensional integrable Hamiltonian systems

TL;DR

This paper explores integrable two-dimensional Hamiltonian systems, revisiting classical problems like the harmonic oscillator and introducing methods to compute first integrals using the Jacobi last multiplier, with applications to various physical models.

Contribution

It presents a novel approach to compute first integrals for 2D integrable systems using the Jacobi last multiplier, including new physical examples.

Findings

Canonical transformation links anisotropic and isotropic oscillators.

Jacobi last multiplier effectively computes first integrals.

New physical models analyzed with explicit first integrals.

Abstract

In this paper, we discuss some results on integrable Hamiltonian systems with two degrees of freedom. We revisit the much-studied problem of the two-dimensional harmonic oscillator and discuss its (super)integrability in the light of a canonical transformation which can map the anisotropic oscillator to a corresponding isotropic one. Following this, we discuss the computation of first integrals for integrable two-dimensional systems using the framework of the Jacobi last multiplier. Using the latter, we describe some novel physical examples, namely, the classical Landau problem with a scalar-potential-induced hyperbolic mode, the two-dimensional Kepler problem, and a problem involving a linear curl force.