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

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 $p_{\theta}$ reasonably playing the role of a deformation parameter.

These algebraic structures remain to be properly understood, and we hope that this short work 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.