We provide a covariant and geometric derivation of the first law of binary mechanics for two (extended-)spinning compact objects on a circular orbit. To this end, we consider a spacetime endowed with a helical Killing vector field and two compact objects described as multipolar particles up to quadrupolar order (i.e. including spin and finte size effects such as spin-induced and tidally-induced quadrupoles). 
First I will briefly explain the derivation, then compare and check the result at dipolar order to existing litterature and finally present some preliminary results at quadrupolar order.