From Spray to Metric: The Geometric Construction of the Jacobi Metric

TL;DR

This paper presents a geometric framework for converting dynamical equations into metric structures, unifying various known metrics and introducing new perspectives for analyzing dynamical systems geometrically.

Contribution

It introduces a systematic method to derive metric structures from equations of motion, including the Jacobi and optical metrics, without relying on traditional variational principles.

Findings

Recovered the optical and Jacobi metrics for static spacetimes.

Derived a Randers-type Finsler metric in the three-body problem.

Provided a geometric approach to dynamical systems independent of variational methods.

Abstract

This paper develops a systematic approach to the geometrization of dynamics from the viewpoint of the geodesic equation. The method promotes a semispray to a spray through the imposition of suitable dynamical constraints, and the associated metric structure is extracted via reparameterization. When applied to static spacetimes, this spray-to-metric framework recovers the optical metric, the Jacobi metric for massive particles, and its generalization for charged particles in electromagnetic fields. We further show that a Randers-type Finsler metric arises naturally in the planar circular restricted three-body problem. By establishing a direct pathway from equations of motion to metric structures, this work offers a geometric perspective, independent of the traditional variational framework, may provide a basis for further studies on dynamical systems.