One dimensional relativistic free particle in a quadratic dissipative medium

TL;DR

This paper derives a constant of motion, Lagrangian, and Hamiltonian for a relativistic particle in a quadratic dissipative medium, revealing expected trajectories in (x,v) space but unusual behavior in (x,p) space.

Contribution

It introduces a novel formulation of relativistic dissipative systems by explicitly deriving the constant of motion, Lagrangian, and Hamiltonian for quadratic velocity-dependent damping.

Findings

Trajectories in (x,v) space behave as expected.

Trajectories in (x,p) space exhibit odd behavior.

Provides a framework for analyzing relativistic particles with quadratic dissipation.

Abstract

The deduction of a constant of motion, a Lagrangian, and a Hamiltonian for relativistic particle moving in a dissipative medium characterized by a force which depends on the square of the velocity of the particle is done. It is shown that meanwhile the trajectories in the space (x,v), defined by the constant of motion, look as one might expected, the trajectories in the space (x,p), defined by the Hamiltonian, have a odd behavior.