Classical Relativistic Extension of Kanai’s Frictional Lagrangian

Abstract

Working in an arbitrary Lorentz frame, we address the question of formulating the covariant variational principle for classical, single-particle, dissipative, relativistic mechanics. First, within a Minkowskian geometry, the basic properties of the proper time τ and the covariant velocity u_μ are recapitulated. Next, using a scalar function ψ(x) and its negative derivatives ϕμ’s, we construct a covariant Lagrangian Λ that generalizes the famous Bateman-Caldirola-Kanai Lagrangian of nonrelativistic frictional mechanics. Finally, we propose a deterministic model for ψ (involving the drag coefficient A) whose explicit solution leads to relativistic damped Rayleigh motion in the rest frame of the medium.