  The principal subject of this thesis is the gravitational two-body problem in
the extreme-mass-ratio regime---that is, where one mass is significantly
smaller than the other---in the full context of our contemporary theory of
gravity, general relativity. We divide this work into two broad parts: the
first provides an overview of the theory of general relativity along with the
basic mathematical methods underlying it, focusing on its canonical formulation
and perturbation techniques; the second presents our novel work in these areas,
focusing on the problems of entropy, motion and the self-force in general
relativity. We begin here with a study of entropy theorems in classical
Hamiltonian systems, and in particular, the issue of the second law of
thermodynamics in classical mechanics and general relativity. Then, we develop
a general approach based on conservation laws for calculating the correction to
the motion of a sufficiently small object due to gravitational perturbations in
general relativity. When the perturbations are attributed to the small object
itself, this effect is known as the gravitational self-force. It is what drives
the orbital evolution of extreme-mass-ratio inspirals: compact binary systems
where one mass is much smaller than---thus effectively orbiting and eventually
spiralling into---the other, expected to be among the main sources for the
future space-based gravitational wave detector LISA. Finally, we present some
work on the numerical computation of the scalar self-force using an approach
called the Particle-without-Particle method, as well as the generalization of
this method to general partial differential equations and applications to other
areas of applied mathematics.
