Canonical Formulation

In principle, the Newtonian equation of motion (1.7) allows a complete analysis of the classical dynamics of the kicked harmonic oscillator. Nevertheless, it is advantageous to use the Hamiltonian formulation of the problem instead, because many classical results can be derived in this formulation with much more ease (cf. section 1.2). What is more, for the discussion of the corresponding quantum problem the Hamiltonian operator is indispensable anyway.

The dynamics of a charged particle in an electromagnetic field can be described by the Hamiltonian [Gol80]

$$H(\vec{r},\vec{p},t) \; = \; \frac{1}{2m_0} \left\{ \vec{p}-q_0\vec{A}(\vec{r},t) \right\}^2 + q_0\phi(\vec{r},t),$$ (1.5)

with the momentum $\vec{p}=(p_x,p_y,p_z)^t$ of the particle and the vector potential $\vec{A}$ and the scalar potential $\phi$ for the electromagnetic field. The potentials have to be chosen in such a way that the magnetic and electric fields are obtained via $\vec{B} = \vec{\nabla} \times \vec{A}$ and $\vec{E} = -\vec{\nabla} \phi-\partial/\partial t\,\vec{A}$. For the fields of equation (1.1), this is achieved, for example, by choosing

such that the corresponding Hamiltonian becomes

$$H(\vec{r},\vec{p},t) \; = \; \frac{1}{2m_0} \left\{ p_x^2 +... ...rac{q_0E_0T}{k} \cos kx \sum_{n=-\infty}^\infty \delta(t-nT).$$

(1.5)

$H$ is cyclic in $y$ and $z$. As in the previous subsection it is clear that the dynamics in $z$-direction is that of a free particle and separates from the rest of the dynamics; therefore in the following I drop the $z$-dynamics altogether. Cyclicity of $H$ in $y$ means that $p_y$ is a constant of motion. Since this constant enters the Hamiltonian via the term $p_y-q_0B_0x$ only, changing the value of $p_y$ just results in a shift of the origin of the $x$-axis (cf. with the constant in equation (1.6)). Hence $p_y$ can be set to zero without loss of generality. The remaining ``essential'' part of the Hamiltonian is

$$H(x,p_x,t) \; = \; \frac{1}{2m_0}p_x^2 + \frac{1}{2}m_0\omega_0^2x^2 + V_0T \cos kx \sum_{n=-\infty}^\infty \delta(t-nT),$$ (1.6)

with the parameter

$$V_0 \; := \; \frac{q_0E_0}{k}$$ (1.7)

which controls the amplitude of the kick; $V_0$ has the dimension of an energy. As from here on only the momentum $p_x$, conjugate to $x$, is of importance and no confusion with other momenta can arise, I now drop the index $x$.

Naturally, the two canonical equations that follow from equation (1.12),

$$\begin{array}{rcrcl} \dot{x} & = & \displaystyle \frac{\par... ...+kV_0T\sin kx\sum_{n=-\infty}^\infty \delta(t-nT), \end{array}$$ (1.8)

can be combined to obtain once again the Newtonian equation of motion (1.7).

In order to minimize the number of parameters of the system I introduce dimensionless variables by the scaling transformation

Similarly, the parameters $V_0$, $T$ are replaced with dimensionless versions:

$$\begin{array}{rcl} \displaystyle \frac{k^2T}{m_0\omega_0} \... ...m] \displaystyle \omega_0 \: T & \longmapsto & T. \end{array}$$ (1.8)

The Hamiltonian under investigation then reads

$$\hspace*{-0.2cm} \fbox{$ \displaystyle \rule{0.0cm}{0.75cm... ...os x \sum_{n=-\infty}^\infty \delta(t-nT). \hspace*{0.1cm} $}$$ (1.9)

For simplicity, the function

$$V(x) \; = \; V_0 \cos x$$ (1.10)

is often referred to as the kick potential.^1.1It is one of the advantages of the scaling used here that the two remaining parameters of the system, the (scaled) period $T$ of the perturbation and its (scaled) amplitude $V_0$, concern only the kick part of the Hamiltonian. The unperturbed part of $H$ after scaling is just a harmonic oscillator with -- formally -- unit mass and unit frequency.

Note that, while equation (1.12) reduces to the Hamiltonian of the CHIRIKOV-TAYLOR map [Chi79] in the limit $\omega_0\to 0$, no analogous statement holds for the scaled version (1.17), since $\omega_0\neq 0$ is a necessary condition for the scaling (1.15). The CHIRIKOV-TAYLOR map is an important standard example of classical kicked dynamical systems. I return to this map in subsection 5.1.1, where its bifurcation scenario and diffusive dynamics are described.

Footnotes

... potential.^1.1

The full driving term $V_0 \cos x \sum_{n=-\infty}^\infty \delta(t-nT)$ in equation (1.17) is correctly referred to as a potential only after scaling, i.e. using dimensionless time, as in the present case. Before scaling, this term has the dimension of energy over time, rather than energy.