A generalization of Szebehely's inverse problem of dynamics

TL;DR

This paper broadens the inverse problem of dynamics by integrating concepts from the inverse calculus of variations, offering a more general framework and clarifying existing approaches mainly for planar curves.

Contribution

It introduces a generalized formulation of Szebehely's inverse problem by leveraging inverse calculus of variations, expanding the scope beyond traditional planar curve cases.

Findings

Develops a more general approach to the inverse problem of dynamics.

Clarifies current methods and their limitations for planar curves.

Proposes a unified framework connecting inverse problems of dynamics and calculus of variations.

Abstract

The so-called inverse problem of dynamics is about constructing a potential for a given family of curves. We observe that there is a more general way of posing the problem by making use of ideas of another inverse problem, namely the inverse problem of the calculus of variations. We critically review and clarify different aspects of the current state of the art of the problem (mainly restricted to the case of planar curves), and then develop our more general approach.