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arXiv · 1311.6061

Periodic Orbits in the Kepler-Heisenberg Problem

Corey Shanbrom
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Abstract

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.

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Corey Shanbrom. 2013-11-23. Periodic Orbits in the Kepler-Heisenberg Problem. https://doi.org/10.3934/jgm.2014.6.261

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