On potentials whose level sets are orbits (Q6583651)

The authors answer a question raised by Mark Levi in 2003 in a talk at the Workshop ``Variational Methods in Celestial Mechanics'' at the American Institute of Mathematics. Levi asked for a characterization of the smooth potential energy functions on the plane having the property that any point in the plane lies on a level orbit. The authors call functions like this Levi potentials. A level orbit of a Hamiltonian system is a solution of Newton's equation that lies entirely in a level set of the potential energy.\N\NThey prove three main theorems: First, any Levi potential \(U: \mathbb{R}^2 \rightarrow \mathbb{R}\) with a totally path-disconnected critical set has a unique critical point and is radial. Second, any analytic Levi potential is radial. Third, for any non-empty compact convex subset \(C \subset \mathbb{R}^2\), there is a Levi potential \(U: \mathbb{R}^2 \rightarrow \mathbb{R}\) with critical set \(C\).\N\NRelevant to all three theorems is the crucial fact that outside the critical set, the family of level sets of a Levi potential can be parametrized into a solution of the inverse curvature flow.