Activities

1. Use the polar coordinates r and φ of the mass m to write the Lagrangian.
2. Check that φ is cyclic and that the corresponding constant of motion is L. Use the latter to write the one-dimensional problem describing the radial evolution. Check the expression of the effective potential energy.
3. How many circular orbits (for m) there exist for each value of L? Are they stable? Use the simulation to check your answers.
4. Is there any unbounded orbit?
5. Find a periodic non-circular orbit around E = 1.9
6. If the minimum and maximum distances between the hole and the mass m are, respectively, u_L/2 and u_L, which is M/m? Use the program (after enabling Limits) to check your computation.
7. What happens if they are 3u_L/8 and 3u_L/4?