Towards the fourth post-Newtonian Hamiltonian for two-point-mass systems

The article presents the conservative dynamics of gravitationally interacting two-point-mass systems up to the eight order in the inverse power of the velocity of light, i.e., fourth post-Newtonian (4PN) order and up to quadratic order in Newton's gravitational constant. Additionally, all logarithmic terms at the 4PN order are given as well as terms describing the test-mass limit. With the aid of the Poincare algebra, additional terms are obtained. The dynamics is presented in the form of an autonomous Hamiltonian derived within the formalism of Arnowitt, Deser, and Misner. Out of the 57 different terms of the 4PN Hamiltonian in the center-of-mass frame, the coefficients of 45 of them are derived. Reduction of the obtained results to circular orbits is performed resulting in the 4PN-accurate formula for energy expressed in terms of angular frequency in which two coefficients are obtained for the first time.