To square root the Lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models
Abstract
The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We analyze a basic but misinterpreted difference between the Lagrangian that gives the arc length of a curve and the one that describes the motion of a free particle in curved space. Although they provide the same formal equations of motion, they are not equivalent. We explore this difference from a geometrical point of view, where we observe that the nonequivalence is nothing more than a matter of symmetry. As applications, some distinct models are studied. In particular, we explore the standard free relativistic particle, a couple of spinning particle models and also the forceless mechanics formulated by Hertz.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.01177
 Bibcode:
 2019arXiv190501177R
 Keywords:

 Physics  Classical Physics;
 Mathematical Physics;
 53Z05
 EPrint:
 19 pages, no figures, geometrical perspective in reparametrization invariance and forceless mechanics of Hertz. Some formulae were corrected from the previous v1 version