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 non-equivalence 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.

arXiv comment19 pages, no figures, geometrical perspective in reparametrization invariance and forceless mechanics of Hertz. Some formulae were corrected from the previous v1 version

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Cited by in corpus (3)

• Reparametrization Invariance and Some of the Key Properties of Physical Systems
• Classical gauge principle -- From field theories to classical mechanics
• 5-Dimensional $SO(1,4)$-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal