Approximately Relativistic Lagrangians for Classical Interacting Point Particles

Abstract

In classical special-relativistic dynamics, equations of motion for interacting point particles can be derived from Lorentz-invariant variational principles of the Fokker type. Similarly, approximately relativistic equations of motion for such particles, obtained by expansion of the exact equations in inverse powers of the speed of light, follow from variational principles involving approximately relativistic Lagrangians of the type first found by Darwin for electrodynamics. Here the general form of such Lagrangians is established by directly approximating Lorentz-invariant variational principles describing point particles interacting through two-body forces. Only interactions which possess a static Newtonian limit are considered; the interaction is not assumed to be symmetric in the particles' variables. The exact variational principle is assumed to depend at most on velocities and thus leads to at most acceleration-dependent forces. The same is found to be true of the approximate variational principles. The general approximate Lagrangian obtained is characterized by the absence of terms of order $c^{-1\mathrm{}}$ and by the possible presence, for each relativistic particle interaction, of three new functions of the Euclidean interparticle separation which may be independent of the static Newtonian potential. The form of the usual ten approximate conservation theorems is established using the invariance properties of the approximate Lagrangian. Two examples of approximate Lagrangians are evaluated explicitly. The first establishes the form of the approximate Lagrangians associated with relativistic particle interactions which allow definition of "adjunct fields," while the second establishes this form for a particular interaction connecting pairs of points on the world lines of particles with space-like separation. Possible applications of the results in the classical and quantum domains are discussed.