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The geodesic has a fundamental role in physics and in mathematics: roughlyspeaking, it represents the curve that minimizes the arc length between twopoints on a manifold. We analyze a basic but misinterpreted difference betweenthe Lagrangian that gives the arc length of a curve and the one that describesthe motion of a free particle in curved space. Although they provide the sameformal equations of motion, they are not equivalent. We explore this differencefrom a geometrical point of view, where we observe that the non-equivalence isnothing more than a matter of symmetry. As applications, some distinct modelsare studied. In particular, we explore the standard free relativistic particle,a couple of spinning particle models and also the forceless mechanicsformulated by Hertz.

To square root the Lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models