Motion of a spinning particle under the conservative piece of the self-force is Hamiltonian to first order in mass and spin

Abstract

We consider the motion of a point particle with spin in a stationary spacetime. We define, following Witzany et al. [Classical Quantum Gravity 36, 075003 (2019)] and later Ramond [arXiv:2210.03866], a 12-dimensional Hamiltonian dynamical system whose orbits coincide with the solutions of the Mathisson-Papapetrou-Dixon equations of motion with the Tulczyjew-Dixon spin supplementary condition, to linear order in spin. We then perturb this system by adding the conservative pieces of the leading order gravitational self-force and self-torque sourced by the particle's mass and spin. We show that this perturbed system is Hamiltonian and derive expressions for the Hamiltonian function and symplectic form. This result extends a previous result for spinless point particles [F. M. Blanco and E. E. Flanagan, Phys. Rev. Lett. 130, 051201 (2023)].