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Section 21.17 Everything Else

There are many areas left to explore if you are interested: questions of the stability of orbits under perturbations, the precession of the orbit, and whether it is open or closed. There are many interesting examples, even within our solar system, that show the varied and unique outcomes of central force interactions: Lagrange points, resonant orbits, horseshoe orbits, to name a few. There are also other types of central forces. The repulsive inverse square force was very important to early atomic experiments. Rutherford bombarded a lattice of gold with alpha particles (helium nuclei). The repulsive electrostatic interaction can be handled easily by our preceding analysis. The theory fit experiment well until the alpha particle energies became high enough to overcome the effective potential and hit the nucleus head-on.

Many of the ideas in our analysis are handled nicely by the Lagrangian formalism. Lagrangian mechanics provides yet another starting point for obtaining the equations of motion. The ideas of symmetry and conservation are more easily recognized and handled within that context, which proves to be very powerful in more complicated situations. When you reach that point, remember some of the techniques we used here and then appreciate the simplicity and beauty provided by the new viewpoint.