This problem has been addressed before under https://www.physicsforums.com/threads/trouble-understanding-coordinates-for-the-lagrangian.1006528/ I also copied the following problem statement with Landau's very sketchy solution from this old post, because I don't have the English edition of... Read more

Landau, volume 1, chapter 1, problem 4: Lagrangian of a somewhat complex system
Rick16
