A generalization of Szebehely's inverse problem of dynamics in dimension three

TL;DR

This paper generalizes Szebehely's inverse problem of dynamics in three dimensions by allowing non-Euclidean constant metrics, broadening the scope of potential functions that generate specified integral curves.

Contribution

It introduces a more flexible formulation of the inverse problem, incorporating general constant metrics beyond the standard Euclidean case, with detailed analysis and examples.

Findings

Generalized inverse problem formulation with non-Euclidean metrics

Clarified existing literature on the inverse problem

Provided multiple illustrative examples

Abstract

Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential V such that the Lagrangian L = T - V, where T is the standard Euclidean kinetic energy function, generates integral curves which include the given family of curves. Our more general way of posing the problem makes use of ideas of the inverse problem of the calculus of variations and essentially consists of allowing more general kinetic energy functions, with a metric which is still constant, but need not be the standard Euclidean one. In developing our generalization, we review and clarify different aspects of the existing literature on the problem and illustrate the relevance of the newly introduced additional freedom with many examples.