Autoparallels and the Inverse Problem of the Calculus of Variations

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.