Canonical Formulation For A Non-Relativistic Spinning Particle Coupled To Gravity

We systematically derive an action for a nonrelativistic spinning particle in flat background and discuss its canonical formulation in both Lagrangian and Hamiltonian approaches. This action is taken as the starting point for deriving the corresponding action in a curved background. It is achieved by following our recently developed technique of localising the flat space Galilean symmetry (Banerjee et al 2014 Phys. Lett. B 737 369; Banerjee et al 2015 Phys. Rev. D 91 084021; Banerjee et al 2015 Class. Quantum Grav. 32 045010). The coupling of the spinning particle to a Newton-Cartan background is obtained naturally. The equation of motion is found to differ from the geodesic equation, in agreement with earlier findings. Results for both the flat space limit and the spinless theory (in curved background) are reproduced. Specifically, the geodesic equation is also obtained in the latter case.