## Lagrange Points

An interesting result of the restricted three-body problem, in which two orbiting masses are much more massive than the third, is the existence of five Lagrange points. At each of the Lagrange points, the forces due to the gravity of the two larger bodies and the centrifugal pseudoforce effectively balance out. Therefore, in the co-rotating frame of the two massive bodies, a less massive object placed at any of the Lagrange points will remain stationary (however, the stability at each of the Lagrange points requires more careful consideration of the Coriolis pseudoforce).

This animation shows the 3D surface of the effective potential felt by a massless particle in the rotating frame of the Earth-Moon system. The mathematical form of the effective potential is $$\phi_{eff}(\vec{r}) = -\frac{G M_1}{|\vec{r} - \vec{r}_1|} - \frac{G M_2}{|\vec{r} - \vec{r}_2|} -\frac{1}{2} \frac{G(M_1 + M_2)}{|\vec{r}_2 - \vec{r}_1|^3} |\vec{r}|^2$$ where $M_1$, $\vec{r}_1$ and $M_2$, $\vec{r}_2$ are the masses and positions of the Earth and Moon respectively and $\vec{r}$ is measured from the center of mass. The first two terms are the gravitational potential and the third term is the centrifugal pseudopotential. Here, the effective potential is plotted in log space for contrast, and a 2D contour plot of the potential is projected onto the bottom of the figure. The larger and smaller black circles represent the positions of the Earth and Moon, and the blue points show the five Lagrange points (i.e., where the gradient of the potential is zero).