Complete conservative dynamics for inspiralling compact binaries with spins at fourth post-Newtonian order

In this work we complete the spin dependent conservative dynamics of inspiralling compact binaries at the fourth post-Newtonian order, and in particular the recent derivation of the next-to-next-to-leading order spin-squared interaction potential. We derive the physical equations of motion of the position and the spin from a direct variation of the action. Further, we derive the quadratic in spin Hamiltonians, as well as their expressions in the center of mass frame. We construct the conserved integrals of motion, which form the Poincare algebra. This construction provided a consistency check for the validity of our result, which is crucial in particular in the current absence of another independent derivation of the next-to-next-to-leading order spin-squared interaction. Finally, we provide here the complete gauge invariant relations among the binding energy, angular momentum, and orbital frequency of an inspiralling binary with generic compact spinning components to the fourth post-Newtonian order. These high post-Newtonian orders, in particular taking into account the spins of the binary constituents, will enable to gain more accurate information on the constituents from even more sensitive gravitational wave detections to come