Jacobi integral

The Jacobi integral is a conserved quantity in the circular restricted three-body problem, combining a test particle’s kinetic energy with the effective gravitational and centrifugal potential in a rotating frame.

Circular restricted three-body problem: A model of two massive bodies on circular orbits and a third body whose mass is negligible. The Jacobi integral is conserved under this model’s circular-orbit and negligible-mass assumptions.
Conservation law: A quantity that remains constant during a system’s evolution because of its dynamics or symmetries. The Jacobi integral is the conserved quantity that organizes trajectories in the rotating problem.
Interplanetary Transport Network: A set of low-energy pathways through the gravitational dynamics of the Solar System. Jacobi constraints help identify transitions among the regions that form these pathways.
Mechanical energy: The sum of kinetic and potential energy in a mechanical system. Mechanical energy is generally not conserved in the rotating description, while the Jacobi integral is.
Rotating reference frame: A coordinate system that rotates relative to an inertial frame. The integral arises when the equations of motion are expressed in a frame rotating with the primaries.
Lagrangian mechanics: A formulation of classical mechanics based on kinetic energy minus potential energy. Rotating-frame equations and their conserved quantities can be derived from a Lagrangian.
Low-energy transfer: A spacecraft trajectory designed to reduce propulsion needs by exploiting gravitational dynamics. The integral helps locate energy levels at which routes through necks near equilibrium points open.
Tisserand parameter: An approximate invariant used to compare orbits in the circular restricted three-body problem. It resembles a Jacobi-based orbital constraint but is expressed using orbital elements and approximations.
Effective potential: A potential that combines physical potential energy with terms representing constraints or inertial effects. The rotating-frame centrifugal contribution joins gravitational potential in the Jacobi integral.
Angular velocity: The rate and axis of rotation of a body or reference frame. The primaries’ angular velocity sets the rotating frame and contributes to the integral’s centrifugal term.

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