Covariant variation for point-particle Lagrangians

Finnian Gray; Sebastian Murk; Daniel R. Terno

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arXiv · 2606.25266

Covariant variation for point-particle Lagrangians

Finnian Gray·Sebastian Murk·Daniel R. Terno

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Abstract

Structureless test particles in general relativity follow geodesics. For extended bodies, higher-order multipole moments lead to departures from geodesic motion; in particular, spinning test bodies obey the Mathisson--Papapetrou--Dixon (MPD) equations. Similarly, the leading correction to the eikonal approximation for electromagnetic-wave propagation can be formulated as the nongeodesic propagation of spinning null particles. When the resulting equations are treated as standalone worldline models, with the relevant dynamical quantities defined only along the representative worldline, their variational formulation requires particular care.Following DeWitt's construction, we distinguish several types of variation, define the corresponding covariant variation for each, and identify the role of parallel transport in models that couple worldline variables to tensor fields. This framework simplifies the variational treatment of the MPD equations and yields a simple Lagrangian for the null-particle model of light propagation.

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Finnian Gray, Sebastian Murk, Daniel R. Terno. 2026-06-24. Covariant variation for point-particle Lagrangians. https://arxiv.org/abs/2606.25266

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