The Kicked Harmonic Oscillator

There are several different ways to motivate the discussion of the kicked harmonic oscillator. In each case it describes -- more or less approximately -- certain physical systems. In this section I discuss the most common of these model systems that finds its application for example in plasma physics [AS83]. ZASLAVSKY and co-workers discuss this approach to the problem in some detail as well [ZSUC91].

Consider the dynamics of a charged point particle in a homogeneous stationary magnetic field $\vec{B}$ and a time-dependent electric field $\vec{E}$ orthogonal to $\vec{B}$:

with real-valued constants $B_0,E_0,k,\omega$. The electric field can be interpreted as a wave packet that is periodic both in space and time, and that consists of FOURIER-like components $\sin(kx-n\omega t)$ each of which propagates in $x$-direction and contributes to the complete packet with equal weight.

Field configurations of this type can also be used, for instance, to describe the beams of charged particles in storage rings under the influence of beam-beam-interactions [Hel83,Ten83]. The applicability in cases like this becomes more obvious when the series in equation (1.1b) is reformulated in a certain way. Taking into account that

$$\sum_{n=-\infty}^\infty \sin(kx-n\omega t) \; = \; \sin kx \... ...mega t \; = \; T \sin kx \sum_{n=-\infty}^\infty \delta(t-nT),$$ (1.0)

with the period $T=2\pi/\omega$, one obtains

$$\vec{E}(x,t) \; = \; E_0T \sin kx \sum_{n=-\infty}^\infty \delta(t-nT) % \hat{x}. \, \vec{e}_x.$$ (1.1)

This demonstrates that the electric field, although it is written as a propagating wave packet in equation (1.1b), can in fact be interpreted as a standing harmonic wave that is switched on stroboscopically at discrete points of time given by $nT$. At these times the charged particle is submitted to an impulsive force, similar to the situation one encounters in a particle accelerator or storage ring, where the particle beams intersect with each other at certain discrete points, namely in the interaction regions.