Instantaneous Interaction Relativistic Dynamics for Two Particles in One Dimension

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Abstract
Differential conditions which guarantee the Lorentz invariance of instantaneous action‐at‐a‐distance relativistic dynamics have been given by Currie and by Hill. The present paper obtains the general solution of these conditions for the special case of two particles in one dimension. The resulting equations of motion are integrated to obtain the world lines. World‐line invariance is explicitly demonstrated. The equations of motion are cast into Hamiltonian form with the transformations of the inhomogeneous Lorentz group canonical. The Hamiltonian formulation is made unique up to canonical transformation for those forces which fall off faster than the inverse square of the interparticle separation by the demand of asymptotic reduction to free particle form. The special case of the inverse‐square‐law forces of electrodynamics is considered; the constant of the motion associated with Lorentz invariance is found to have an interaction piece which survives asymptotically as in the relativistic mechanics of Van Dam and Wigner. The Poisson bracket [x1, x2] between the physical coordinates also has an interaction piece which survives asymptotically for electrodynamics.