Hamiltonian systems

The new formalism for quantization of gauge systems based on the concept of the
dynamical Hamiltonian recently introduced as a basis for the canonical theory of
quantum gravity was considered in the context of general gauge theories. This and
other Hamiltonian methods, that include Dirac's theory of extended Hamiltonian
and the Hamiltonian reduction formalism were critically examined. It was established
that the classical theories of constrained gauge systems formulated within the
framework of either of the approaches are equivalent. The central to the proof of
equivalence was the fact that the gauge symmetries resuIt in the constraints of the
first class in Dirac's terminology that Iead to redundancy of equations of motion
for some of the canonica variables. Nevertheless, analysis of the quantum theories
showed that in general, the quantum theory of the dynamical Hamiltonian is inequivalent
to those of the extended Hamiltonian and the Hamiltonian reduction. The
new method of quantization was applied to a number of gauge systems, including
the theory of relativistic particle, the Bianchi type IX cosmological model and spinor electrodynamics along side with the traditional methods of quantization. In all of the
cases considered the quantum theory of the dynamical Hamiltonian was found to be
well-defined and to possess the appropriate classical limit. In particular, the quantization
procedure for the Bianchi type IX cosmological spacetime did not run into
any of the known problems with quantizing the theory of General Relativity. On the
other hand, in the case of the quantum electrodynamics the dynamical Hamiltonian
approach led to the quantum theory with the modified self-interaction in the matter
sector. The possible consequence of this for the quantization of the full theory of
General Relativity including the matter fields are discussed.

In this thesis we will discuss connections between Hamiltonian systems with a periodic kick and planar diffeomorphisms. After a brief overview of Hamiltonian theory we will focus, as an example, on derivations of the Hâenon map that can be obtained by considering kicked Hamiltonian systems. We will conclude with examples of Hâenon maps of interest.