  This PhD thesis is the conclusion of some of my works carried out at the
Centre de Physique Th\'eorique, under the supervision of Serge Lazzarini. Two
subjects are presented in the manuscript, both aiming at studying the effect of
the spin of elementary particles on otherwise known theories.
  First is a study of the L\'evy-Leblond-Newton (LLN) equation, which is used
to describe the evolution of a quantum system with spin 1/2 that is coupled to
its own gravitational potential. After reviewing some symmetries of non
relativistic Quantum Mechanics, and how to geometrize them with the help of
Bargmann structures, we recall what is the L\'evy-Leblond equation. In this
chapter, the LLN equation is described in a fully covariant way on Bargmann
structures. This covariant formulation then helps to derive the dynamical
symmetries of the equation, and conserved quantities. The symmetry group of
this equation turns out to be the already known Schr\"odinger-Newton group.
  The second part deals with the trajectory of particles with spin in General
Relativity. We first highlight Souriau's geometric method to derive the
Mathisson-Papapetrou-Dixon (MPD) equations, and then discuss the different
possible Spin Supplementary Conditions (SSC) that exist to close the system of
MPD equations. In this chapter, we assume the Tulczyjew SSC to describe the
trajectory of photons, leading to the Souriau-Saturnini equations. We then
present some works where we have applied these equations, in a Schwarzschild
spacetime, and in a spacetime deformed by a gravitational wave. In the first
work, we found gravitational birefringence: the trajectory deviates from the
geodesic plane, depending on the helicity of the photon and its wavelength. The
second work presented also found birefringence in gravitational interferometry
experiments, however the effect is many orders of magnitude lower than what we
can detect.
