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

TL;DR

The paper introduces a geometry‑first framework to geometrize dynamics by promoting a semispray to a spray through dynamical constraints and then reconstructing a metric via a reparameterization. In static spacetimes, this approach recovers the optical metric for massless particles, the Jacobi metric for massive particles, and a Randers‑type Finsler metric for charged particles, with explicit constructions in the planar circular restricted three‑body problem. The method provides a direct link from equations of motion to metric structures without relying on variational principles, offering a unified tool for studying dynamics through geodesic and curvature ideas. This 정ties together spray geometry, metrizability, and classical gravitational physics, and points to extensions to constrained systems beyond the traditional Maupertuis–Jacobi framework and to broader dynamical contexts such as KCC theory.

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.