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To square root the Lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models
Classical Physics · Submitted 2 May 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.04 | 0.03 | asserted 0.94 |
| s1 | 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. | contribution 0.92 | 0.82 | 0.13 | asserted 0.99 |
| s2 | Although they provide the same formal equations of motion, they are not equivalent. | result 0.92 | 0.54 | 0.59 | asserted 1.00 |
| s3 | 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. | interpretation 0.67 | 0.88 | 0.71 | asserted 1.00 |
| s4 | As applications, some distinct models are studied. | contribution 0.84 | 0.80 | 0.03 | asserted 0.89 |
| s5 | In particular, we explore the standard free relativistic particle, a couple of spinning particle models and also the forceless mechanics formulated by Hertz. | contribution 0.88 | 0.86 | 0.05 | asserted 0.95 |
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The paper
Original 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.
Human review
How the literature uses it
4 arXiv papers cite this in 6 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
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We simply follow the same steps taken on the standard analysis of the free relativistic particle model [ref] , or, by the same token, the semiclassical spinning particle model [ref] , in order to remove the so discussed arbitrariness our model possesses.
5-Dimensional $SO(1, 4)$-Invariant Action as an Origin to the Magueijo-Smolin Doubly Special Relativity Proposal · Physical Sector · 0.98
Cites it as support 1 paper
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Having to choose the gauge such that $g_αβv^αv^β$ is constant guarantees the equivalence of $L_1$ and $L_2$ [ref] .
Reparametrization Invariance and Some of the Key Properties of Physical Systems · Relativistic Particle Lagrangian · 0.96
Cites it as background 1 paper
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Its arms extends to different areas, ranging from classical mechanics [ref] to the description of constant-curvature spaces of a homogeneous isotropic and static spacetimes [ref] passing through, recently, to their meaning in comprehending fluid flows on curved surfaces [ref] .
From local isometries to global symmetries: Bridging Killing vectors and Lie algebras through induced vector fields · Introduction · 0.88
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