Lagrangian-Hamiltonian formalism for the gravitational two-body problem with spin and parametrized post-Newtonian parametersγandβ

We generalize the Lagrangian and the Hamiltonian of our previous work on the gravitational two-body problem with spin by including the parametrized post-Newtonian parameters $\gamma$ and $\beta$. By this procedure we are able to obtain the precession of the orbit as well as the precession of the spin. Equations of motion corresponding to an arbitrary spin supplementary condition are also given. Finally we show how the masses of the binary pulsar PSR 1913 + 16 and its companion are related to the orbit and spin precessions. Combining this with a result derivable from the second-order Doppler effect and the gravitational red-shift, we obtain a relation constraining the values that $\gamma$ and $\beta$ can take.