Two-dimensional classical superintegrable systems: polynomial algebra of integrals

TL;DR

This paper establishes that 2D classical superintegrable systems generate a four-dimensional polynomial algebra of integrals, providing a new algebraic framework applicable to various physically relevant systems without relying on separability.

Contribution

It introduces a novel construction of polynomial algebra of integrals for 2D superintegrable systems independent of Hamilton-Jacobi separability and polynomial integrals, extending previous algebraic approaches.

Findings

Derived explicit algebraic forms of classical trajectories.

Identified conditions for special trajectories with I_1=I_2=0.

Applied the algebraic method to multiple physical systems.

Abstract

In this work, we investigate generic classical two-dimensional (2D) superintegrable Hamiltonian systems H, characterized by the existence of three functionally independent integrals of motion (I_0=H,I_1,I_2). Our main result, formulated and proved as a theorem, establishes that the set (I_0,I_1,I_2,I_12={I_1,I_2}) generates a four-dimensional polynomial algebra under the Poisson bracket. Unlike previous studies, this study describes a construction that neither depends on the additive separability of the Hamilton-Jacobi equation nor presupposes polynomial integrals of motion in the canonical momenta. Specifically, we prove an instrumental observation presented in [D. Bonatsos et al., PRA 50, 3700 (1994)] concerning deformed oscillator algebras in superintegrable systems. We apply the method to a variety of physically relevant examples, including the Kepler system, Holt potential, Smorodinsky-Winternitz potential, Fokas-Lagerstrom potential, the Higgs oscillator, and the non-separable Post-Winternitz system. In several cases, we explicitly derive the form of the classical trajectories y=y(x;I_0,I_1,I_2) using purely algebraic means. Moreover, by examining the conditions under which I_1=I_2=0, we identify and characterize special classes of trajectories.