Autoparallels and the Inverse Problem of the Calculus of Variations

This paper presents an analysis that systematically solves the inverse problem of the calculus of variations and the associated Helmholtz conditions. It demonstrates that the dynamics of autoparallels can admit a consistent variational formulation even in the presence of non-metricity. The findings establish a robust variational framework for particle motion within metric-affine geometries, thereby making a crucial contribution to the mathematical foundations of the geodesic principle in relativistic gravity. This work is fundamental for theoretical understanding of gravity beyond standard General Relativity.