This is the first part of a series of work aiming at tackling the issue of
integrability, or lack thereof, for spinning particles around black hole
spacetimes in general relativity. In this article, we lay the foundations of
this program, and present a Hamiltonian system that describes the evolution
equations for a dipolar particle moving in a background spacetime in general
relativity. First, we construct a set of symplectic variables that covers the
phase space of such system. The formulae are valid irrespective of the choice
of background spacetime, of orthonormal tetrad that defines the spin variables,
and, most importantly, do not rely on any spin supplementary conditions (or
whether one has been imposed or not). Second, this formalism is simplified with
the help of the Tulczyjew-Dixon spin supplementary condition, which is argued
to be the most convenient one in the present context. This allows us to present
a 5-dimensional, fully covariant Hamiltonian that generates the linear-in-spin
dynamics. Third, as an application of our general results, we prove the
integrability, in the sense of Liouville, of the motion of a particle around a
Schwarzschild black hole, at linear order in spin. Our formalism opens the door
to a number of applications and extensions that are surveyed in a dedicated
section.
