2025/06/20 by Escobar-Ruiz, A. M., Azuaje, R., Gordiano, J. C.
#FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2506.17519
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 (I0=H,I1,I2). Our main result, formulated and proved as a theorem, establishes that the set (I0,I1,I2,I12=I1,I2) 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;I0,I1,I2) using purely algebraic means. Moreover, by examining the conditions under which I1=I2=0, we identify and characterize special classes of trajectories.