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It is shown that a nonrelativistic mechanical system involving a generalnonrelativistic potential V(|r1-r2|) between point particles at positions r1and r2 can be extended to a Lagrangian system which is invariant under Lorentztransformation through order v^2/c^2. However, this invariance requires theintroduction of velocity-dependent and acceleration-dependent forces betweenparticles. The textbook treatments of "relativistic mechanics" can bemisleading; the discussions usually deal with only one particle experiencingprescribed forces and so make no mention of these additional velocity- andacceleration-dependent forces. A simple example for a situation analogous to aparallel-plate capacitor is analyzed for all the conservation laws of Galileaninvariance or Lorentz invariance. For this system, Galilean invariance requiresthat the mechanical momentum is given by pmech=mv but places no restriction onthe position-dependent potential function. On the other hand, Lorentzinvariance requires that the mechanical momentum is given bypmech=mv/(1-v^2/c^2)^1/2, and in addition requires that the potential functionis exactly the Coulomb potential V(|r1-r2|)=k/|r1-r2|. It is also noted thatthe transmission of the interparticle-force signal at the speed of light againsuggests a special role for the Coulomb potential. A nonrelativistic particlesystem interacting through the Coulomb potential becomes the Darwin Lagrangianwhen extended to a system relativistic through order v^2/c^2, and then allowsextension to classical electrodynamics as a fully Lorentz-invariant theory ofinteracting particles.

Relativistic Mechanics and a Special Role for the Coulomb Potential