= Jacobi integral of a rigidly rotating potential
{c}
{title2=$E_J=E-\Omega L_z$}
For $\Phi(R,\phi-\Omega t,z)$ with constant pattern angular velocity $\Omega$, ordinary specific <energy> satisfies $\dot E=\partial_t\Phi=-\Omega\partial_\phi\Phi$ and specific axial <angular momentum> satisfies $\dot L_z=-\partial_\phi\Phi$. Their combination $E_J$ is conserved. This is the rotating-frame <integral of motion>; in the <circular restricted three-body problem>, the usual <Jacobi constant> is an affine rescaling of it. Thus time dependence in an inertial frame does not alone imply the absence of all integrals.
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