This is a numerical calculation illustrating the stability of L4. This is the code which is a specialized version of some more general code, and here is the yield.

There are two masses with ratio 999, in circular orbits about each other. The test particle follows the yellow line starting near L4. White dots are drawn on the curve each full revolution of the two masses. The calculation proceeds for 160/π revolutions.

In particular, the camera that captures the motion is fixed relative to the two masses and rotates with them. The distance between the masses is 2. L4 is in the lower left of the picture. I am sorry that I didn’t make the program plot it. The horizontal extent of the picture is about .025 . The test particle starts displaced from L4 by .001 in the direction away from the line between the masses.

Here is an interesting series by Ron Steinke:
An introduction to the Lagrangian
Rotational Motion
Central Force Motion
Lagrange points & Stability
The Stable Lagrange Points and the Coriolis Force

The last article shows that test particles near L4 or L5 vibrate about that Lagrange point with two distinct characteristic frequencies. This is just what the picture shows. The article also confirms that L4 and L5 are unstable when the two masses are more equal.