Autoparallels and the Inverse Problem of the Calculus of Variations

TL;DR

This paper establishes a variational principle for autoparallel curves in metric-affine geometries, even with non-metricity, enhancing the mathematical understanding of particle motion in relativistic gravity.

Contribution

It explicitly constructs an action functional for autoparallels in affine connections with non-metricity, solving the inverse calculus of variations problem.

Findings

Autoparallel curves can be derived from a variational principle.

The action functional is explicitly constructed for non-metric-compatible connections.

The variational formulation applies to metric-affine geometries with non-metricity.

Abstract

We prove that autoparallel curves associated with a torsion-free but not necessarily metric-compatible affine connection can be derived from an action principle. We explicitly construct the action functional and show by standard variational techniques that it produces the desired equations. Our analysis is based on systematically solving the inverse problem of the calculus of variation and the associated Helmholtz conditions. This demonstrates that the dynamics of autoparallels admit a consistent variational formulation even in the presence of non-metricity. Our results provide a variational framework for particle motion in metric-affine geometries and thereby contribute to the mathematical foundations of the geodesic principle in relativistic gravity.