Nonlinear Lissajous orbits and particular superintegrability

Abstract

We investigate the geometry of classical trajectories generated by separable two-dimensional polynomial potentials of the form $V(x,y)=\tfrac{1}{2}\big(x^{2N}+A\,y^{2N}\big)$, where $N=1,2,\ldots,$ and $A>0$. Special emphasis is placed on the emergence of nonlinear Lissajous figures and on the distinction between global and particular superintegrability in the Liouville sense. In the harmonic case ($N=1$) closed periodic orbits are a consequence of an additional \emph{global} integral of motion whenever the frequency ratio is rational, rendering the system maximally superintegrable. In contrast, for anharmonic oscillators, already in the quartic case ($N=2$), the oscillation frequencies depend on the partial energies, so periodic Lissajous-type trajectories occur only under nonlinear resonance conditions fixed by the initial data. Accordingly, the extra conserved quantities that characterize these closed orbits are not global invariants but \emph{particular} (trajectory-dependent) integrals that emerge only on the resonant trajectories. For higher-degree potentials $N\geq3$, the resonant trajectories are naturally described by hyperelliptic phase constraints rather than by a universal polynomial orbit equation.