Next: Canonical Formulation Up: The Kicked Harmonic Oscillator Previous: The Kicked Harmonic Oscillator   Contents

## Newtonian Equations of Motion

Let the particle be of mass and charge . Then the dynamics of the particle is governed by the Newtonian equation of motion

 (1.2)

with the vector in position space. Using the electromagnetic field (1.1) this yields for the components of :

The dynamics in -direction is trivial (rectilinear and uniform) and can thus be neglected. From equation (1.5b) one obtains

 (1.2)

where the constant can be set to zero without loss of generality (by an appropriate choice of the origin of the -axis). Substitution into equation (1.5a) finally yields
 (1.3)

the equation of motion of a kicked harmonic oscillator with eigenfrequency equal to the cyclotron frequency,
 (1.4)

which is essentially given by the magnetic field. The right hand side of equation (1.7) describes the impulsive force that is driving the oscillator; the strength of this driving is essentially determined by the amplitude of the electric field. The functional dependence on of the driving force is specified by the kick function which in equation (1.7) is proportional to . In subsection 1.1.4 I briefly return to the issue of choosing the kick function in a variety of other cases.

At the times the impulse -- the kick -- changes the momentum of the particle instantaneously, whereas its position remains unchanged. For all other times the dynamics is just that of an unperturbed (i.e. free) harmonic oscillator.

Next: Canonical Formulation Up: The Kicked Harmonic Oscillator Previous: The Kicked Harmonic Oscillator   Contents
Martin Engel 2004-01-01