Rigid Body Rotors in Planar Potentials: A Novel type of Superintegrable Mechanical Systems in the Plane

TL;DR

This work introduces a novel class of planar superintegrable systems obtained by coupling a rigid rotor to a two-dimensional translational oscillator. By treating the center-of-mass motion in the plane together with an internal rotational degree of freedom, the authors show that resonance between the orbital frequency and rotor rotation yields additional integrals of motion, expanding the hidden symmetry algebra beyond the standard planar cases. In the harmonic case, a set of five functionally independent integrals emerges, and complex ladder constructions reveal a resonant tower of higher-order integrals, while the familiar $\mathfrak{su}(2)$ structure persists as a subalgebra, with the Casimir modified by the resonance. The vertical-plane and gravity-augmented variant demonstrates that the algebraic framework remains intact under translations of the equilibrium point, suggesting broad applicability, higher-dimensional generalizations, and potential quantum realizations. Overall, the work positions rotor-extended models as a natural source of new families of (resonant) superintegrable systems with rich symmetry algebras and promising avenues for future exploration.

Abstract

We investigate the superintegrability of rigid body rotors coupled to planar systems. In particular, we study the isotropic harmonic oscillator in two dimensions, with its (central) force acting on the rotor's center of mass constrained to move in the plane. By including an internal rotational degree of freedom described by a rigid rotor, the resulting planar system possesses three degrees of freedom: two translational and one rotational. When the orbital motion and the internal rotation are tuned to resonance, additional integrals of motion arise, extending the hidden symmetry algebras of the underlying models. For the oscillator, the well-known $\mathfrak{su}(2)$ symmetry algebra can be enlarged by the presence of the rotor, with the conserved momentum $p_θ$ reasonably playing the role of a deformation parameter. These algebraic structures remain to be properly understood, and we hope that this short letter will serve as an invitation to further investigate these interesting models. To close the work, we also examine the oscillator in a vertical plane, in the presence of a rotor, under the effect of a uniform gravitational field, showing that the algebraic structure persists as a translated version of the isotropic case, as expected. In all these settings, the extended dynamics admits five functionally independent integrals, thereby confirming maximal superintegrability. Our simple yet nontrivial results suggest that rigid-body rotors provide a natural mechanism for generating new families of (resonant) superintegrable systems, along with their associated symmetry algebras, an outcome that aligns with the main objective of this work.

Figures (3)

• Figure 1: A homogeneous rod of mass $M$ completes one full revolution over a single orbital cycle. The forces acting on the rotor’s center of mass are purely elastic, so the resulting trajectory is an ellipse, as expected.
• Figure 2: On the left it is reported a schematic realization of the physical problem described by the Hamiltonian \ref{['eq:hamk']}, whereas on the right a schematic representation of an orbit in the plane of the same system is provided. Here, we have taken $M=1$, $k=1$ and $L=1$.
• Figure 3: Schematic representation of the physical trajectory described by the Hamiltonian of an anisotropic oscillator, with frequency ratio $\omega_x:\omega_y=3:5$, coupled to a rotor. Here we still consider a homogeneous rod with mass $M=1$ and length $L=1$.