^{1,a)}and Massimo Pin

^{1,b)}

### Abstract

We lay down the foundations of particle dynamics in mechanical theories that satisfy the relativity principle and whose kinematics can be formulated employing reference frames of the type usually adopted in special relativity. Such mechanics allow for the presence of anisotropy, both conventional (due to nonstandard synchronization protocols) and real (leading to detectable chronogeometrical effects, independent of the choice of synchronization). We give a general method for finding the fundamental dynamical quantities (Lagrangian, energy, and momentum) and write their explicit expression in all the kinematics compatible with the basic requirements. We also write the corresponding dispersion relations and outline a formulation of these theories in terms of a pseudo-Finslerian space-time geometry. Although the treatment is restricted to the case of one spatial dimension, an extension to three dimensions is almost straightforward.

It is a pleasure to thank Jean-Marc Lévy-Leblond and Abraham Ungar for correspondence, and an anonymous referee for suggesting a correction. S.S. is grateful to Stefano Liberati, Lorenzo Sindoni, and Matt Visser for stimulating discussions.

I. INTRODUCTION

II. ANISOTROPIC RELATIVISTIC KINEMATICS

A. Transformation law

1. Case ,

2. Case , (or )

3. Case ,

4. Case , ,

B. Comments

1. Velocity composition law

2. Conventional and real anisotropy

3. Three-dimensional case

III. FOUNDATIONS FOR DYNAMICS

A. Lagrangian

B. Momentum and energy

C. Compatibility

D. Mass, rest energy, rest momentum

IV. ANISOTROPIC RELATIVISTIC DYNAMICS

A. Case

1. Anisotropic Einstein’s dynamics

2. Anisotropic Newtonian dynamics

B. Case

V. HAMILTONIAN AND DISPERSION RELATION

A. Case

1. Anisotropic Einstein’s dynamics

2. Anisotropic Newtonian dynamics

B. Case

VI. GEOMETRICAL FORMULATION

VII. COMMENTS

### Key Topics

- Anisotropy
- 44.0
- Lagrangian mechanics
- 28.0
- Kinematics
- 21.0
- Theory of relativity
- 19.0
- Differential equations
- 13.0

## Figures

Comparison between the expressions given by Eqs. (4.5) and (4.6) for (solid lines) and (dashed lines). In both cases, the conventional anisotropy parameter has been set equal to zero, and the values of the quantities at rest have been chosen according to Eqs. (4.7) and (4.8). The plot on the left represents momentum and the one on the right represents energy.

Comparison between the expressions given by Eqs. (4.5) and (4.6) for (solid lines) and (dashed lines). In both cases, the conventional anisotropy parameter has been set equal to zero, and the values of the quantities at rest have been chosen according to Eqs. (4.7) and (4.8). The plot on the left represents momentum and the one on the right represents energy.

Comparison between the expressions given by Eqs. (4.13) and (4.14) for (solid lines) and (dashed lines). In both cases, the values of the quantities at rest have been chosen equal to zero. The plot on the left represents momentum and the one on the right represents energy. Note the horizontal asymptotes of and as (displayed as thin solid straight lines).

Comparison between the expressions given by Eqs. (4.13) and (4.14) for (solid lines) and (dashed lines). In both cases, the values of the quantities at rest have been chosen equal to zero. The plot on the left represents momentum and the one on the right represents energy. Note the horizontal asymptotes of and as (displayed as thin solid straight lines).

Comparison between the set of pairs that satisfy Eq. (5.10) for (thick solid line) and (dashed line). The conventional anisotropy parameter has been set equal to zero, and the asymptotes are also displayed (thin solid straight lines).

Comparison between the set of pairs that satisfy Eq. (5.10) for (thick solid line) and (dashed line). The conventional anisotropy parameter has been set equal to zero, and the asymptotes are also displayed (thin solid straight lines).

Comparison between the set of pairs that satisfy Eq. (5.13) for (solid line) and (dashed line). Note that the curve for a finite possesses an end point ( in the diagram), which corresponds to the finite asymptotic values of and as .

Comparison between the set of pairs that satisfy Eq. (5.13) for (solid line) and (dashed line). Note that the curve for a finite possesses an end point ( in the diagram), which corresponds to the finite asymptotic values of and as .

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