  Using a recently introduced method [Phys.\ Rev.\ Lett.\ {\bf 123}, 231104
(2019)], which splits the conservative dynamics of gravitationally interacting
binary systems into a non-local-in-time part and a local-in-time one, we
compute the local part of the dynamics at the sixth post-Newtonian (6PN)
accuracy. Our strategy combines several theoretical formalisms: post-Newtonian,
post-Minkowskian, multipolar-post-Minkowskian, effective-field-theory,
gravitational self-force, effective one-body, and Delaunay averaging. The full
functional structure of the local 6PN Hamiltonian (which involves 151 numerical
coefficients) is derived, but contains four undetermined numerical
coefficients. Our 6PN-accurate results are complete at orders $G^3$ and $G^4$,
and the derived $O(G^3)$ scattering angle agrees, within our 6PN accuracy, with
the computation of [Phys.\ Rev.\ Lett.\ {\bf 122}, no. 20, 201603 (2019)]. All
our results are expressed in several different gauge-invariant ways. We
highlight, and make a crucial use of, several aspects of the hidden simplicity
of the mass-ratio dependence of the two-body dynamics.
