Discussion on the equivalence of two relativistic point-particle Lagrangians

This paper rigorously demonstrates that the two common relativistic point-particle Lagrangians are not generally equivalent for arbitrary external potentials, as they yield different dynamical behaviors (chaotic vs. integrable) and Hamiltonian structures, though they coincide for electromagnetic interactions or vanishing potentials.

The Gist

Imagine you are trying to describe how a car drives through a city. You have two different maps (or rulebooks) to choose from: Map A and Map B.

For a long time, physicists thought these two maps were just different ways of drawing the same route. They believed that whether you used the complex, winding instructions of Map A or the simple, straight-line instructions of Map B, the car would end up in the exact same place, driving in the exact same way.

A recent study by Lei and colleagues claimed these two maps were always equivalent. However, in this new paper, Liubin Wang and Xin Wu act like detective mechanics who say, "Hold on a minute. That's only true if the city is empty or if the car is running on a very specific type of fuel. If you add a different kind of obstacle, the two maps tell the car to do completely different things."

Here is the breakdown of their discovery using simple analogies:

1. The Two Maps (The Lagrangians)

In physics, a Lagrangian is essentially a mathematical recipe that tells a particle how to move.

• Map A (The "Square-Root" Recipe): This is the classic, "hardcore" physics recipe. It's mathematically heavy and complex (it involves a square root). Think of it as a GPS that calculates the absolute shortest, most direct path based on the fundamental laws of the universe. It is very strict and never makes mistakes about the laws of physics.
• Map B (The "Quadratic" Recipe): This is a simplified, "lazy" version. It drops the complex square root and uses a simpler formula. Think of this as a simplified navigation app that approximates the route. It's much easier to calculate and faster for computers to run, but it cuts corners.

2. The "Speed Limit" Rule (The Mass Shell Constraint)

In the universe of relativity, there is a fundamental rule: Massive objects (like cars or planets) must always travel at a specific relationship between their speed and energy. Physicists call this the "mass shell constraint."

• Map A has this speed limit hardwired into its code. No matter what happens, the car cannot break the rule. It is built into the foundation.
• Map B does not have this rule hardwired. It's like a car with no speedometer. You have to manually tell the driver, "Hey, don't go over the speed limit!" If you forget to add that instruction, the car might drive in a way that violates the laws of physics.

3. The Experiment: When Do They Agree?

The authors tested these two maps under three different "traffic conditions" (external potentials):

• Condition 1: The Empty Highway (No Potential)

  • Result: Both maps work perfectly. They give the same route.
  • Why? When there are no obstacles, the simplified Map B happens to follow the rules by accident.

• Condition 2: The Electric Highway (Electromagnetic Potential)

  • Result: Both maps still agree!
  • Why? This is the special case where the "simplified" Map B can be easily tweaked to include the speed limit rule. If you force Map B to obey the rules (by adding a specific constraint), it becomes identical to Map A. This is why many previous studies on charged particles near black holes got away with using the simpler Map B.

• Condition 3: The Bumpy, Weird Road (Generic Mechanical Potential)

  • Result: The maps diverge! They tell the car to take completely different paths.
  • The Discovery: When the "obstacle" is a weird, non-electric force (like a mechanical spring or a custom gravity field), Map B fails. It loses the speed limit rule.
  • The Chaos:
    • Map A (The strict GPS) predicts that the car will get stuck in a chaotic loop, bouncing around unpredictably. This is "real" physics; the system is too complex to predict perfectly.
    • Map B (The simplified app) predicts the car will drive in perfect, smooth circles. It claims the system is "integrable" (predictable).
  • The Problem: Map B is lying! It's giving a smooth, pretty picture that doesn't match reality. It's like a GPS that ignores traffic jams and tells you the road is clear, leading you into a crash.

4. The Verdict: Which Map Should You Use?

The authors conclude that Map A is the "Universal Truth." It is the only one that is always correct, whether the gravity is weak or strong, and whether the particle has mass or not (like a photon).

However, Map B isn't useless!

• When to use Map B: If you are dealing with electrically charged particles near black holes (like in many astrophysics simulations), Map B is fantastic. It's computationally cheap, fast, and if you add the "speed limit" rule manually, it gives the right answer. It's the "sports car" of simulations: fast and efficient for specific tracks.
• When to use Map A: If you are dealing with any other kind of force (like a mechanical potential) or if you need to be 100% sure you aren't breaking the laws of physics, you must use Map A. It is the "heavy-duty truck" that never breaks down, even if it's slower to drive.

The Big Takeaway

The paper is a warning to physicists: "Don't assume the shortcut works just because it worked yesterday."

For decades, people used the simplified math (Map B) for everything because it was easier. This paper proves that while it works for electricity, it fails for other forces, potentially hiding chaotic behavior that actually exists in the universe. If you want to understand the true, messy, chaotic dance of particles in complex gravity fields, you need the rigorous, square-root recipe (Map A).

Technical Summary

Technical Summary: Equivalence of Two Relativistic Point-Particle Lagrangians

Paper Title: Discussion on the equivalence of two relativistic point-particle Lagrangians
Authors: Liubin Wang and Xin Wu
Context: The paper addresses a claim by Lei et al. (2021) that two specific Lagrangians, $L_1$ (square-root form) and $L_2$ (quadratic form), are dynamically equivalent for describing particle motion in combined gravitational and matter fields. The authors rigorously refute this general equivalence, demonstrating that it depends critically on the nature of the external potential $V$.

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1. Problem Statement

In relativistic mechanics, two common Lagrangians are used to describe a particle of mass $m$ in a metric $g_{\mu\nu}$ with an external potential $V$:

1. Square-root Lagrangian ($L_1$): $L_1 = -mc\sqrt{-g_{\mu\nu}\dot{x}^\mu \dot{x}^\nu} - V$
2. Quadratic Lagrangian ($L_2$): $L_2 = \frac{1}{2}mg_{\mu\nu}\dot{x}^\mu \dot{x}^\nu - V$

While these are known to be equivalent in pure gravity ($V=0$) or for electromagnetic potentials (where $V \propto A_\mu \dot{x}^\mu$), Lei et al. claimed they remain equivalent for generic external potentials $V(x)$. The authors of this paper argue that this claim is incorrect for non-electromagnetic potentials, leading to fundamentally different dynamical systems (integrable vs. non-integrable).

2. Methodology

The authors employ a Hamiltonian formalism rather than the Euler-Lagrange equations to analyze the systems. This approach is chosen because:

• Hamiltonian dynamics are independent of coordinate choices.
• They explicitly reveal the mass shell constraint ($p_\mu p^\mu = -m^2$), which is crucial for relativistic consistency.
• They allow for the application of Liouville's theorem to determine integrability (existence of conserved quantities) and the potential for chaos.

The analysis proceeds in three stages:

1. Theoretical Derivation: Deriving the Hamiltonians ($H_1$ and $H_2$) for $L_1$ and $L_2$ under three cases of external potentials:
   • Electromagnetic potential ($V = qA_\mu \dot{x}^\mu$).
   • Non-electromagnetic potential ($V = \psi(r, \theta)$).
   • Combined potentials.
2. Constraint Analysis: Examining whether the derived Hamiltonians inherently satisfy the mass shell constraint or require artificial imposition.
3. Numerical and Analytical Toy Model: Using the Schwarzschild metric coupled with an artificial mechanical potential ($V = \omega r^{-2}[(r-r_c)^2 + r_g^2\theta^2]$) to test for integrability and chaos via Poincaré sections and the existence of a fourth constant of motion.

3. Key Contributions and Results

A. Theoretical Equivalence Analysis

• Case 1: Electromagnetic Potential ($V = qA_\mu \dot{x}^\mu$)
  • Result: $L_1$ and $L_2$ are equivalent.
  • Reasoning: Both Hamiltonians can be transformed into one another via time reparametrization. Crucially, the redefined momenta in both systems naturally satisfy the mass shell constraint ($\tilde{p}_\mu \tilde{p}^\mu = -m^2$). The Hamiltonian $K$ derived from $L_2$ becomes equivalent to $H_1$ from $L_1$ when the constraint $K = -m/2$ is imposed.
• Case 2: Non-Electromagnetic Potential ($V = \psi(x)$)
  • Result: $L_1$ and $L_2$ are not equivalent.
  • Reasoning:
    • $L_1$: The Hamiltonian $H_1$ is derived directly from the mass shell constraint. It inherently enforces the physical dispersion relation.
    • $L_2$: The Hamiltonian $K$ derived via Legendre transformation does not satisfy the mass shell constraint unless an artificial constraint is forced. Even then, the resulting energy expression differs from $H_1$. The system described by $L_2$ fails to correctly couple the particle's gravitational and matter energies in a way that preserves the relativistic mass shell condition.

B. Dynamical Behavior (Toy Model)

Using a Schwarzschild black hole with an artificial mechanical potential:

• $L_2$ Dynamics: The system is analytically integrable. The Hamiltonian $K$ allows for variable separation, yielding a fourth conserved quantity (analogous to the Carter constant). Numerical simulations show only regular KAM tori; no chaos is observed.
• $L_1$ Dynamics: The system is non-integrable and chaotic. The Hamiltonian $H_1$ does not allow variable separation. Numerical simulations (Poincaré sections) reveal the onset of chaotic orbits as energy increases, indicating the absence of a fourth integral of motion.

C. Implications for Massless Particles

• $L_1$: Remains valid for massless particles ($m=0$) even with external potentials, as the Hamiltonian structure (derived from the constraint) adapts correctly.
• $L_2$: Becomes invalid for massless particles interacting with non-electromagnetic potentials because the Hamiltonian fails to incorporate the external potential correctly without violating the mass shell condition.

4. Significance and Recommendations

The paper resolves a significant ambiguity in relativistic dynamics literature regarding the choice of Lagrangian:

1. Universality of $L_1$: The square-root Lagrangian ($L_1$) is the physically superior and universal choice. It correctly enforces the mass shell constraint for all types of external potentials (gravitational, electromagnetic, and generic matter fields) and for both massive and massless particles. It is recommended for strong-field problems and general relativistic studies.
2. Utility of $L_2$: The quadratic Lagrangian ($L_2$) is not generally equivalent to $L_1$.
   • Limitation: It produces non-physical, integrable dynamics for potentials where the true system should be chaotic (and vice versa).
   • Niche Application: $L_2$ is only valid and preferred for charged particles in electromagnetic fields (where equivalence holds). In this specific case, $L_2$ is advantageous due to its mathematical simplicity and computational efficiency, particularly for constructing explicit symplectic integrators for chaotic orbits near black holes.
3. Correction of Previous Literature: The authors clarify that while Lei et al. were correct about the equivalence in electromagnetic fields, their generalization to arbitrary potentials was flawed. Studies using $L_2$ for non-electromagnetic potentials (e.g., artificial mechanical potentials) may have yielded spurious results regarding integrability and chaos.

Conclusion: The equivalence of $L_1$ and $L_2$ is conditional. It holds strictly for electromagnetic interactions but breaks down for generic external potentials, where $L_1$ is the only formulation that preserves the fundamental relativistic mass shell constraint and yields the correct chaotic/non-integrable dynamics.