Do the L4 and L5 points travel in a geodesic?

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In a two-body system like the sun and the earth, the L4 and L5 points do indeed travel along geodesics. While the two-body problem in General Relativity (GR) is complex and typically requires numerical solutions, test particles co-orbiting at these points experience free fall. Consequently, these particles follow geodesic paths as dictated by GR. This highlights the unique behavior of L4 and L5 points within the framework of gravitational dynamics. Overall, the discussion emphasizes the geodesic nature of co-orbiting bodies in a gravitational field.
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Given a two body system like the sun and the earth:

Do the L4 and L5 points travel in a geodesic, or is it just the sun and the Earth which do so?
 
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The two body problem in GR is fiendishly complicated and can only be solved numerically.

However, having said that, test-particles co-orbiting in the L4 or L5 points are in free fall, and in GR they would travel along geodesics.

Garth
 

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