Newtonian Equations of Motion

Let the particle be of mass $m_0$ and charge $q_0$. Then the dynamics of the particle is governed by the Newtonian equation of motion

$$m_0 \ddot{\vec{r}} \; = \; q_0\left( \vec{E} + \dot{\vec{r}} \times \vec{B} \right)$$ (1.2)

with the vector $\vec{r}=(x,y,z)^t$ in position space. Using the electromagnetic field (1.1) this yields for the components of $\vec{r}$:

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

$$m_0 \dot{y} \; = \; -q_0B_0x + \mbox{const.},$$ (1.2)

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

$$\ddot{x} + \omega_0^2x \; = \; \frac{q_0E_0T}{m_0} \sin kx \sum_{n=-\infty}^\infty \delta(t-nT),$$ (1.3)

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

$$\omega_0 \; := \; \frac{q_0B_0}{m_0},$$ (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 $x$ of the driving force is specified by the kick function which in equation (1.7) is proportional to $\sin kx$. 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 $nT$ 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.