First integrals of some two-dimensional integrable Hamiltonian systems

TL;DR

The paper studies first integrals of two-dimensional, Liouville‑integrable Hamiltonian systems, focusing on the 2D harmonic oscillator and its (super)integrability through a canonical anisotropic-to-isotropic mapping. It introduces the Jacobi last multiplier as a practical tool to construct additional first integrals and applies it to physical models—the Landau problem with a hyperbolic mode, the planar Kepler problem, and linear curl forces. Key results include explicit integrals that generate the underlying symmetries (e.g., I0,I1,I2,I3), the action-angle interpretation of Θ, and Runge–Lenz/Curl-Force invariants in the listed systems. The findings deepen understanding of multidimensional integrability and provide concrete, locally-defined invariants that can classify trajectories under these classical dynamics, with potential implications for related quantum and geometrical analyses.

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.