 Jacobi constant

The Jacobi constant is a conserved quantity for the circular restricted three-body problem. In barycentric coordinates rotating at constant angular frequency $n$, defineThe equation of motion in a rotating frame is ${\dot{\mathbf{v}}}_{\mathrm{rot}}+2\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{rot}}=\nabla U$. Taking its dot product with ${\mathbf{v}}_{\mathrm{rot}}$ proves $d(|{\mathbf{v}}_{\mathrm{rot}}{|}^2/2-U)/dt=0$: the Coriolis force does no work.

Only positions satisfying $2U-C\ge 0$ are accessible. The boundary gives a zero-velocity surface; openings near Lagrange points allow transit between regions but do not force it. Ordinary inertial-frame energy can change even while $C$ remains constant.^[ref]