  It is shown that a nonrelativistic mechanical system involving a general
nonrelativistic potential V(|r1-r2|) between point particles at positions r1
and r2 can be extended to a Lagrangian system which is invariant under Lorentz
transformation through order v^2/c^2. However, this invariance requires the
introduction of velocity-dependent and acceleration-dependent forces between
particles. The textbook treatments of "relativistic mechanics" can be
misleading; the discussions usually deal with only one particle experiencing
prescribed forces and so make no mention of these additional velocity- and
acceleration-dependent forces. A simple example for a situation analogous to a
parallel-plate capacitor is analyzed for all the conservation laws of Galilean
invariance or Lorentz invariance. For this system, Galilean invariance requires
that the mechanical momentum is given by pmech=mv but places no restriction on
the position-dependent potential function. On the other hand, Lorentz
invariance requires that the mechanical momentum is given by
pmech=mv/(1-v^2/c^2)^1/2, and in addition requires that the potential function
is exactly the Coulomb potential V(|r1-r2|)=k/|r1-r2|. It is also noted that
the transmission of the interparticle-force signal at the speed of light again
suggests a special role for the Coulomb potential. A nonrelativistic particle
system interacting through the Coulomb potential becomes the Darwin Lagrangian
when extended to a system relativistic through order v^2/c^2, and then allows
extension to classical electrodynamics as a fully Lorentz-invariant theory of
interacting particles.
