An analytic solution for weak-field Schwarzschild geodesics
  It is well known that the classical gravitational two body problem can be
transformed into a spherical harmonic oscillator by regularization. We find
that a modification of the regularization transformation has a similar result
to leading order in general relativity. In the resulting harmonic oscillator,
the leading-order relativistic perturbation is formally a negative centrifugal
force. The net centrifugal force changes sign at three Schwarzschild radii,
which interestingly mimics the innermost stable circular orbit (ISCO) of the
full Schwarzschild problem. Transforming the harmonic-oscillator solution back
to spatial coordinates yields, for both timelike and null weak-field
Schwarzschild geodesics, a solution for $t,r,\phi$ in terms of elementary
functions of a variable that can be interpreted as a generalized eccentric
anomaly. The textbook expressions for relativistic precession and light
deflection are easily recovered. We suggest how this solution could be combined
with additional perturbations into numerical methods suitable for applications
such as relativistic accretion or dynamics of the Galactic-centre stars.
