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.

Reparametrization Invariance and Some of the Key Properties of Physical Systems

“…The above equivalence of S 1 and S 2 could be demonstrated on a more complicated Lagrangian as a specific choice of parametrization such that g αβ (x) v α v β is constant [28]. Indeed, if one starts with the re-parametrization invariant Lagrangian L = qA α v α −m g αβ (x)v α v β and defines proper time gauge τ such that: dτ = g αβ dx α dx β ⇒ g αβ u α u β = 1, then one can effectively consider L = qA α u α − (m − χ) g αβ u α u β − χ as the model Lagrangian.…”

“…Since now S ′ 2 is reparametrization invariant then one can choose a gauge parametrization such that e = 1 and thus arriving at S 2 but under proper-time parametrization λ = τ where g αβ u α u β = 1. Having to choose the gauge such that g αβ v α v β is constant guarantees the equivalence of L 1 and L 2 [28].…”

Application of Local Gauge Theory to Fluid Mechanics - Part Ii: Example: Basic Model of Tollmien-Schlichting Waves and Analytical Solution

“…3.74 it is used that the expression is a tensor density and contains the information on the local momentum p of the system, which allows it to be interpreted as the root of a Lagrangian density. This is again equivalent to the interpretation as Lagrangian density L if trajectories along constant flow velocity are considered (Rizzuti, Vasconcelos & Resende 2019).…”

5-Dimensional $SO(1, 4)$-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal

“…We point out that the action of the group of covariance acts in a slightly non-linear way upon the physical sector of the models. Standard examples of such models are the free relativistic particle [15] and the electrodynamics Lagrangian action [12].…”

“…These quantities are not chosen randomly. We simply follow the same steps taken on the standard analysis of the free relativistic particle model [15], or, by the same token, the semiclassical spinning particle model [17], in order to remove the so discussed arbitrariness our model possesses. First of all, we point out that these quantities remain unaltered under the local symmetries,…”