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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]

(1.5) 
with the momentum
of the particle
and
the vector potential and the scalar potential for the
electromagnetic field.
The potentials
have to be chosen in such a way that the magnetic and electric fields
are obtained via
and
.
For the fields of equation (1.1), this is
achieved, for example, by
choosing
such that the corresponding Hamiltonian becomes
is cyclic in and . As in the previous subsection it is clear
that the dynamics in direction is that of a free particle and
separates from the rest of the dynamics; therefore in the following
I drop the dynamics altogether.
Cyclicity of in means that is a constant of motion.
Since this constant enters the Hamiltonian
via the term only,
changing the value of just results in a shift of the
origin of the axis (cf. with the constant in equation
(1.6)).
Hence can be set to zero without loss of generality.
The
remaining
``essential'' part of the Hamiltonian is

(1.6) 
with the parameter

(1.7) 
which
controls the
amplitude
of the kick; has the dimension of an energy.
As from here on only the momentum , conjugate to , is of importance
and no confusion with other momenta
can arise,
I now
drop the index .
Naturally, the two canonical equations that follow from equation
(1.12),

(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 , are replaced with dimensionless
versions:

(1.8) 
The Hamiltonian under investigation then reads

(1.9) 
For simplicity, the function

(1.10) 
is often referred to as the
kick potential.^{1.1}It is one of the advantages of the scaling used here that the two
remaining parameters of the system,
the (scaled) period of the perturbation and its (scaled) amplitude
,
concern
only
the kick part of the Hamiltonian.
The unperturbed part of
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 CHIRIKOVTAYLOR map
[Chi79] in the limit ,
no analogous statement holds for the scaled version
(1.17), since
is a necessary condition for the
scaling (1.15).
The CHIRIKOVTAYLOR 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
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.
Next: Discrete Dynamics
Up: The Kicked Harmonic Oscillator
Previous: Newtonian Equations of Motion
Contents
Martin Engel 20040101