Complete conservative dynamics for inspiralling compact binaries with spinsat the fourth post-Newtonian order

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In this work we complete the spin-dependent conservativedynamics of inspiralling compact binaries at the fourth post-Newtonianorder, and in particular the derivation of thenext-to-next-to-leading order spin-squared interaction potential. Wederive the physical equations of motion of the position and the spinfrom a direct variation of the action. Further, we derive thequadratic-in-spin Hamiltonians, as well as their expressions in thecenter-of-mass frame. We construct the conserved integrals of motion,which form the Poincaré algebra. This construction provided aconsistency check for the validity of our result, which is crucial inparticular in the current absence of another independent derivation ofthe next-to-next-to-leading order spin-squared interaction. Finally, weprovide here the complete gauge-invariant relations among the bindingenergy, angular momentum, and orbital frequency of an inspirallingbinary with generic compact spinning components to the fourthpost-Newtonian order. These high post-Newtonian orders, in particulartaking into account the spins of the binary constituents, will enable togain more accurate information on the constituents from even moresensitive gravitational-wave detections to come.