Physics 110A: Hints on Chap 8 probs, part 1
3. Only one dimension matters, so follow the lecture/book development for the senter of mass and reduced mass approach, using Y to track the center of mass and r to track y1 − y2, which is initially at L. Develop initial position and velocity of Y based on the center of mass formula. Solve the Y and r Lagrangians separately (the coordinates don't mix).
12. Once you have found an equilibrium position, r0,
and shown stability, the task is to show that the oscillation period is the
same as the orbital period (thus the orbit will be closed). It is handy to
express the angular momentum in terms of r0, and also
note that the second derivative of the potential acts like a spring
constant, k, so that the radial oscillation has frequency-squared
of k/μ. The orbital period can be attained by setting
centripetal equal to gravity, in the usual way.