  The Ermakov-Milne-Pinney equation is ubiquitous in many areas of physics that
have an explicit time-dependence, including quantum systems with time dependent
Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator
dynamics, etc. The Eliezer and Gray physical interpretation of the
Ermakov-Lewis invariant is applied as a guiding principle for the derivation of
the special relativistic analog of the Ermakov-Milne-Pinney equation and
associated first integral. The special relativistic extension of the Ray-Reid
system and invariant is obtained. General properties of the relativistic
Ermakov-Milne-Pinney are analyzed. The conservative case of the relativistic
Ermakov-Milne-Pinney equation is described in terms of a pseudo-potential,
reducing the problem to an effective Newtonian form. The non-relativistic limit
is considered as well. A relativistic nonlinear superposition law for
relativistic Ermakov systems is identified. The generalized
Ermakov-Milne-Pinney equation has additional nonlinearities, due to the
relativistic effects.
