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

The two body body problem in general relativity is of great theoretical and observational interest, and can be studied in the post-Newtonian, post-Minkowskian and small mass ratio approximations, as well as with effective one body and fully numerical techniques. When gravitational wave dissipation is turned off, the motion is expected to form a Hamiltonian dynamical system. This had been established to various order in the post-Newtonian and post-Minkowskian approximations. In a previous work, we showed that the motion of a point particle under the conservative (time even) piece of the first-order self force is Hamiltonian in any stationary spacetime, and found an explicit expression for the Hamiltonian in terms of a Greens function. In this work we extend this result for spinning particles, to first order in mass and spin. We also show progress towards developing a new formalism to extend this result to second order in mass.