  We calculate the perihelion precession delta for nearly circular orbits in a
central potential V(r). Differently from other approaches to this problem, we
do not assume that the potential is close to the Newtonian one. The main idea
in the deduction is to apply the underlying symmetries of the system to show
that delta must be a function of r V''(r)/V'(r), and to use the transformation
behaviour of delta in a rotating system of reference. This is equivalent to
say, that the effective potential can be written in a one-parameter set of
possibilities as sum of centrifugal potential and potential of the central
force. We get a universal formula for delta. It has to be read as follows: a
circular orbit at this value r exists and is stable if and only if this delta
is a well-defined real; and if this is the case, then the angular difference
from one perihelion to the next one for nearly circular orbits at this r is
exactly 2 pi + delta. Then we show how to apply this result to examples of
recent interest like modified Newtonian gravity and linearized fourth-order
gravity. In the second part of the paper, we generalize this universal formula
to static spherically symmetric space-times. For the Schwarzschild black hole
with mass parameter m > 0 it leads to a surprisingly unknown formula. It
represents a strict result and is applicable for all values r > 6m and is in
good agreement with the fact that stable circular orbits exist for r > 6m only.
