4 - SCATTERING AND LINEAR OSCILLATIONS

Summary

CHAPTER OVERVIEW

This chapter contains material that grows out of and sometimes away from Lagrangian mechanics. Some of it makes use of the Lagrangian formulation (e.g., scattering by a magnetic dipole, linear oscillations), and some does not (e.g., scattering by a central potential, chaotic scattering). Much of this material is intended to prepare the reader for topics to be discussed later: chaotic scattering off hard disks is a particularly understandable example of chaos, and linear oscillators are the starting point for the discussion of nonlinear ones, of perturbation theory, and field theory.

SCATTERING

SCATTERING BY CENTRAL FORCES

So far we have mostly concentrated on bounded orbits of central-force systems. Yet unbounded orbits are very common for both attractive and repulsive central forces. Indeed, for repulsive ones all orbits are unbounded. For attractive ones, the outer turning point of bounded orbits – the maximum distance of the particle from the attracting center – increases with increasing energy. If the force is weak enough at large distances, the particle may escape from the force center even at finite energies: the orbit may break open and the outer turning point may move off to infinity. Then the orbit becomes unbounded, as in the hyperbolic orbits in the Coulomb potential. The result is what is called scattering.

GENERAL CONSIDERATIONS

Consider a central-force dynamical system that possesses unbounded orbits. Let the initial conditions be that a particle is approaching the force center from far away, so far away that it may be thought of as coming from infinity, which means that it is moving on one of the unbounded orbits.