Fr.: intégrale de Jacobi
The integral admitted by the equations of a body of infinitesimal mass moving under the → gravitational attractions of two massive bodies, which move in circles about their → center of gravity. The Jacobi integral is the only known conserved quantity for the circular → restricted three-body problem. In the co-rotating system it is expressed by the equation: (1/2) (x·2 + y·2 + z·2) = U - CJ, where the dotted coordinates represent velocities, U is potential energy, and CJ the constant of integration (→ zero-velocity surface). The Jacobi integral has been used for two different purposes: 1) to construct surfaces of zero velocity which limit the regions of space in which the small body, under given initial conditions, can move, and 2) to derive a criterion (→ Tisserand's parameter) for re-identification of a → comet whose orbit has suffered severe perturbations by a planet. Also known as Jacobi constant.
Named after Karl Gustav Jacobi (1804-1851), a German mathematician who did important work on elliptic functions, partial differential equations, and mechanics; → integral.