Binary dynamics at the fifth and fifth-and-a-half post-Newtonian orders

Using the new methodology introduced in a recent Letter Phys. \ Rev. \ Lett. \ 123, 231104 (2019), we present the details of the computation of the conservative dynamics of gravitationally interacting binary systems at the fifth post-Newtonian (5PN) level, together with its extension at the fifth-and-a-half post-Newtonian (5. 5PN) level. We present also the sixth post-Newtonian (6PN) contribution to the third-post-Minkowskian (3PM) dynamics. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, effective one-body, and Delaunay averaging. We determine the full functional structure of the 5PN Hamiltonian (which involves 95 non-zero numerical coefficients), except for two undetermined coefficients proportional to the cube of the symmetric mass ratio, and to the fifth and sixth power of the gravitational constant, G. We present not only the 5PN-accurate, 3PM contribution to the scattering angle, but also its 6PN-accurate generalization. Both results agree with the corresponding truncations of the recent 3PM result of Bern et al. Phys. \ Rev. \ Lett. \ 122, 201603 (2019). We also compute the 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scattering angle, including its nonlocal contribution, thereby offering checks for future 4PM calculations. We point out a remarkable hidden simplicity of the gauge-invariant functional relation between the radial action and the effective-one-body energy and angular momentum.