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Review Article Volume 3 Issue 1
To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models
Mateus Antônio Resende, Guilherme F Vasconcelos Júnior
Department of Physics, Federal University of Juiz de Fora
Received: January 14, 2020 | Published: March 4, 2020
Citation: Rizzuti BF, Vasconcelos JGF, Resende MA. To square root the lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models. Open Acc J Math Theor Phy . 2020;3(1):8-15 DOI: 10.15406/oajmtp.2020.03.00056
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. They are taken as equivalent but it is not the case from the beginning. 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. The equivalence appears, however, after fixing a particular condition. 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.
Keywords: Hertz, Differential geometry, quantum phenomena, (semi-)riemannian manifold,second Newton law, classical mechanics, spinning particle
©2020 Rizzuti, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.