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Using the new methodology introduced in a recent Letter [Phys.\ Rev.\ Lett.\{\bf 123}, 231104 (2019)], we present the details of the computation of theconservative dynamics of gravitationally interacting binary systems at thefifth post-Newtonian (5PN) level, together with its extension at thefifth-and-a-half post-Newtonian (5.5PN) level. We present also the sixthpost-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 functionalstructure of the 5PN Hamiltonian (which involves 95 non-zero numericalcoefficients), except for two undetermined coefficients proportional to thecube of the symmetric mass ratio, and to the fifth and sixth power of thegravitational constant, $G$. We present not only the 5PN-accurate, 3PMcontribution to the scattering angle, but also its 6PN-accurate generalization.Both results agree with the corresponding truncations of the recent 3PM resultof Bern et al. [Phys.\ Rev.\ Lett.\ {\bf 122}, 201603 (2019)]. We also computethe 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scatteringangle, including its nonlocal contribution, thereby offering checks for future4PM calculations. We point out a remarkable hidden simplicity of thegauge-invariant functional relation between the radial action and theeffective-one-body energy and angular momentum.

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