In the classical setting, the derivative is the derivative with respect to J of the integral which defines J, so it is tautologically equal to 1. d d J ∫ ∫ 0 T P d X = 1 = ∫ ∫ 0 T d t ( d P d J d X d t + P d d J d X d t ) = ∫ ∫ 0 T d t ( d P d J d X d t − − d P d t d X d J ) where the derivatives dP / dJ and dX / dJ should be interpreted as differences with respect to J at corresponding times on nearby orbits, exactly what would be obtained if the Fourier coefficients of the orbital motion were differentiated. (These derivatives are symplectically orthogonal in phase space to the time derivatives dP / dt and dX / dt).