Dynamical invariants for general relativistic two-body systems at the third post-Newtonian approximation

We extract all the invariants (i.e. all the functions which do not depend on the choice of phase-space coordinates) of the dynamics of two point masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by showing how a contact transformation can be used to reduce the 3PN higher-order Hamiltonian derived by Jaranowski and Schäfer to an ordinary Hamiltonian. The dynamical invariants for general orbits (considered in the center-of-mass frame) are then extracted by computing the radial action variable $\oint p_r\mathrm{dr}$ as a function of energy and angular momentum. The important case of circular orbits is given special consideration. We discuss in detail the plausible ranges of values of the two quantities ${\omega }_{\mathrm{static}},$ ${\omega }_{\mathrm{kinetic}}$ which parametrize the existence of ambiguities in the regularization of some of the divergent integrals making up the Hamiltonian. The physical applications of the invariant functions derived here (e.g. to the determination of the location of the last stable circular orbit) are left to subsequent work.