Making use of a minimal action principle, in this work we derive the dynamics of a test rigid body moving in a curved spacetime by means of a parametric invariant Lagrangian formalism. In doing so we complete a line of research due to Bailey-Israel and Anandan-Dadhich-Singh. This is accomplished through the following new contributions: by fixing the Lagrangian of the system, the elaboration of a complete variational procedure, the formulation of a rigidity constraint and the derivation of conserved quantities, already found, but in a very different form, in other approaches to the problem. The dynamics and the equations obtained are also generalized to all orders in the metric expansion by means of new mathematical tools. Besides, by the selection of an appropriate spatial section of the body world-tube, we obtain the simple Papapetru expression of the canonical momentum which, remaining unchanged to all orders, contributes to some reductions of complexity of the dynamics. Finally, in the quadrupolar approximation, applications of our results are presented in the form of useful observables in the context of ideal tests in general relativity.
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