The optical-mechanical analogy in general relativity: new methods for the paths of light and of the planets

The optical-mechanical analogy involves the expression of geometrical optics and particle mechanics in the same mathematical language. In this paper, an especially simple version of the optical-mechanical analogy is extended to general relativity. A variational principle for the trajectories of photons and particles is obtained which is applicable to a broad class of metrics. This permits us to cast the exact equations of motion for both massive and massless particles into the form of Newtonian mechanics. The new equations of motion are illustrated by applications to the Schwarzschild metric