[Paper Review] To square root the Lagrangian or not: an underlying geometrical analysis on classical and relativistic mechanical models

B. F. Rizzuti, G. F. Vasconcelos Júnior+1 more|arXiv (Cornell University)|May 1, 2019

Dynamics and Control of Mechanical Systems4 citations

TL;DR

This paper analyzes the fundamental geometric distinction between two Lagrangian formulations in classical and relativistic mechanics: one minimizing arc length (S₁ = ∫√g(V,V)dτ) and another extremizing kinetic energy (S₂ = ∫½g(V,V)dτ). While both yield identical equations of motion under a fixed-velocity constraint, the former lacks physical time interpretation due to reparametrization invariance, revealing a hidden constraint structure. The key contribution is a geometric proof that auxiliary degrees of freedom—such as in Hertz's forceless mechanics—can be consistently removed via Killing vector field analysis, resolving non-uniqueness and symmetry-breaking issues.

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.

Motivation & Objective

• To clarify the non-equivalence between the square-root and non-square-root Lagrangians in curved space, despite yielding same equations of motion.
• To identify the role of reparametrization invariance and first-class constraints in distinguishing physical from unphysical degrees of freedom.
• To provide a geometric justification for eliminating auxiliary variables in models like Hertz's forceless mechanics using Killing vector fields.
• To demonstrate that forces in physical space correspond to symmetry breaking in the extended curved manifold, even when the system is described as free.

Proposed method

• Use of differential geometry on (pseudo-)Riemannian manifolds to compare two action functionals: S₁ = ∫√g(V,V)dτ and S₂ = ∫½g(V,V)dτ.
• Application of the Levi-Civita connection and geodesic equations to derive equations of motion from both actions.
• Identification of first-class constraints in the square-root action due to reparametrization invariance, indicating unphysical degrees of freedom.
• Use of Killing vector fields to analyze symmetries of the extended manifold in Hertz's formulation, particularly focusing on the auxiliary variable X.
• Demonstration that ∂X is not a Killing vector field when forces are present, implying symmetry breaking in the extended space.
• Projection of solutions from the extended manifold to physical space, showing non-uniqueness and justifying removal of auxiliary degrees of freedom.

Experimental results

Research questions

• RQ1 Under what conditions are the square-root and non-square-root Lagrangians equivalent, despite their geometric and dynamical differences?
• RQ2 How does reparametrization invariance in S₁ lead to unphysical degrees of freedom, and how is this resolved in S₂?
• RQ3 Why does the auxiliary variable X in Hertz's forceless mechanics not contribute to physical dynamics, and how can this be geometrically justified?
• RQ4 What is the role of Killing vector fields in identifying physical symmetries and eliminating unphysical degrees of freedom in extended manifolds?
• RQ5 How does the presence of a physical force in the original space manifest as a symmetry-breaking effect in the extended curved manifold?

Key findings

• The square-root action S₁ is reparametrization-invariant and contains a first-class constraint, implying unphysical degrees of freedom, while S₂ breaks this symmetry and assigns physical meaning to the evolution parameter.
• The two actions yield identical equations of motion only when the constraint g(V,V) = const is imposed, proving their equivalence is conditional.
• In Hertz's formulation, the auxiliary variable X is not physical because the vector field ∂X is not a Killing vector field, indicating it breaks isotropy.
• Rotations around fixed axes that include the X-direction are not isometries of the extended manifold, confirming that ∂X is privileged and should be excluded from physical dynamics.
• The presence of a physical force in the original system implies that ∂X is not a symmetry generator in the extended space, even though the particle is free there.
• A geometric proof is established that different initial values of X project to the same physical trajectory, justifying the removal of the auxiliary degree of freedom via projection.