Lagrange points of the circular restricted three-body problem

What you are seeing: two heavy bodies on a circular orbit (yellow: the primary, like the Sun; blue: the secondary, like Jupiter or the Moon), in a frame that rotates with them so they sit still. A small test particle (red) is dropped at a chosen starting location with a chosen velocity and the playground integrates its motion. Trails fade with age. The five labeled black dots are the Lagrange points: locations where the test particle can in principle sit still in the rotating frame.

Equations of motion in the synodic frame : $\ddot x - 2\dot y - x = -\frac{(1 - \mu)(x + \mu)}{r_1^3} - \frac{\mu (x - 1 + \mu)}{r_2^3}$, $\ddot y + 2\dot x - y = -\frac{(1 - \mu) y}{r_1^3} - \frac{\mu y}{r_2^3}$. Mass parameter $\mu = m_2 / (m_1 + m_2)$. L1, L2, L3 are on the $x$-axis and always unstable. L4 and L5 form equilateral triangles with the primaries; they are linearly stable if $\mu \lt \mu_R \approx 0.0385$ (Routh's stability criterion). The famous Trojan asteroid clouds live around Jupiter's L4 and L5 ($\mu \approx 10^{-3}$).

Click anywhere on the plot to drop a test particle there at zero velocity. Use the buttons to drop one exactly at L4 or L5 with a tiny perturbation.