Centrifugal force in the two body problem? In the two body problem, the Effective radial potential energy in general relativity is given by
$$
V(r)=-\frac{G M m}{r}+\frac{L^{2}}{2\mu r^{2}}-\frac{G(M+m)L^{2}}{c^{2}\mu r^{3}}
$$
where the second term is the centrifugal potential energy.


*

*Firstly, I am a little bit confused about the second term because from my study of classical physics the centrifugal term only arises if we solve the equations in a rotated frame reference however the second term seem to appear even without that assumption. For example when one finds the potential energy in a schwartzchild metric the second term seems to appear too. Am I missing something here?

*Secondly, In a rotated frame of reference two "fictitious" forces appear the centrifugal force 
$$
F_{cent}=-m\,\Omega\times(\Omega\times r)
$$
and the Coriolis force
$$
F_{Cor}=-m\,\Omega\times\frac{dr}{dt}
$$
Is $F_{cent}$ equal to the derivative of the second term in $V(r)$ and if so then how because they look very different to me?

 A: *

*The centrifugal potential is nothing but the angular part of the kinetic energy of the (reduced mass) particle. To see this, write the Lagrangian in polar coordinates,
$$L=T-V=\frac{\mu}{2}\left(\dot r^2+r^2\dot\phi^2\right)-V(r).$$
Since $\phi$ is a cyclic coordinate its conjugated momentum (the angular momentum) is conserved
$$p_\phi=\frac{\partial L}{\partial \dot\phi}=\mu r^2\dot\phi=\mbox{constant}.$$
This relation eliminates $\dot\phi$ from the Lagrangian,
$$L=\frac{\mu}{2}\dot r^2+\frac{p_\phi^2}{2\mu r^2}-V(r).$$
Note that although the second term on the RHS is a kinetic term, it does not explicitly depend on velocities so it may be reinterpreted as a potential term,
$$L=\frac{\mu}{2}\dot r^2-V(r)_{\mathrm{ef}},$$
where the effective potential $V(r)_{\mathrm{ef}}$ is the sum of the so-called centrifugal potential and the interaction potential,
$$V(r)_{\mathrm{ef}}=\frac{p_\phi^2}{2\mu r^2}-V(r).$$
That is a general procedure, whenever one has a cyclic variable one can write its associated kinetic terms without explicit velocities and thus interpret them as potential terms. 

*The equation of motion for the radial coordinate is
$$\mu\ddot r=-\frac{\partial V(r)_{\mathrm{ef}}}{\partial r}=\frac{p_\phi^2}{\mu r^3}-\frac{\partial V(r)}{\partial r}.$$
The second term on the RHS is an repulsive term and it equals the centrifugal force,
$$\frac{p_\phi^2}{\mu r^3}=\mu r \dot\phi^2=|-\mu\vec\Omega\times(\vec\Omega\times\vec r)|,$$
since $\vec\Omega=\dot\phi \vec e_k$
