Concave Toric Domains in Stark-type Mechanical Systems

TL;DR

This paper investigates the conditions under which energy surfaces in planar Stark-type mechanical systems form concave toric domains, using action-variable computations to establish criteria for these geometric structures.

Contribution

It introduces a new approach based on action-variables to determine when energy surfaces are concave toric domains in integrable mechanical problems.

Findings

Criteria for potentials leading to concave toric domains

Application of criteria to specific mechanical problems

New method for analyzing energy surface geometry

Abstract

It has been shown in [Frauenfelder, 2023] that the bounded component of the energy surface of the planer Stark problem after the Levi-Civita transformation are concave toric domains. In this paper, we present a different approach on the problem of determining concave toric domains in a family of integrable natural mechanical problems in the plane based on the computation of action-variables. We give criteria on the potentials for the bounded components of the energy hypersurfaces to be concave toric domains and apply these criteria to a class of problems.