Any conservative force can be related to potential energy by the following (in 1-D):
\begin{equation} F(x) = - \frac{dU}{dx} \end{equation}or in 3-D:
\begin{equation} \mathbf{F} = - \nabla U \end{equation}For example, mass on a spring: \begin{equation} F_x= - k x \end{equation} \begin{equation} U_\textrm{spring} = \frac{1}{2}k x^2 \end{equation}
Or, for gravity near the surface of the Earth: \begin{equation} F_y = - m g \end{equation} \begin{equation} U_\textrm{grav} = m g y \end{equation}
And, if we consider any distance away from the Earth (or central body), we would have: \begin{equation} F(r) = - G \frac{M_E m}{r^2} \end{equation} \begin{equation} U(r) = - G \frac{M_E m}{r} \end{equation}
Now, if we have 2 source bodies, the functions have more features. Now, we ask what happens to a 3rd body in this landscape.
A different type of problem
\begin{equation} \mathbf{F}_\textrm{rot} = \mathbf{F}_\textrm{in}\; \underbrace{- m \boldsymbol{\omega} \times \left( \boldsymbol{\omega } \times \mathbf{r} \right)_\textrm{rot} }_\textrm{centrifugal}\; \underbrace{- 2 m \left( \boldsymbol{\omega} \times \mathbf{v}_\textrm{rot} \right)_\textrm{rot}}_\textrm{Coriolis} \; \underbrace{- m \left( \dot{\boldsymbol{\omega}} \times \mathbf{r} \right)_\textrm{rot}}_\textrm{Euler} \end{equation}
For the case of gravity: \begin{equation} U_G(r) = - \frac{G m_1 m_2}{r} \end{equation} and so our effective potential is: \begin{equation} U_\textrm{eff} = \frac{l^2}{2 \mu r^2} -\frac{G m_1 m_2}{r} \end{equation}