Jeans's theorem

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Jeans’s theorem says that in a steady, collisionless system of stars moving in a fixed gravitational potential, the distribution of stars in phase space, f(x, v), depends only on the constants of motion (the integrals of motion) of the orbits in that potential. Put simply: every steady-state solution of the collisionless Boltzmann equation can be written as a function of these integrals, and any such function is a steady-state solution.

This is often discussed for potentials with three global integrals, where all orbits are regular (not chaotic). In more general potentials, some orbits are chaotic and have fewer integrals, but the theorem still applies in a broader sense: the phase-space density is constant within each well-connected region of phase space.

A well-connected region is a part of phase space that cannot be split into two finite pieces such that all trajectories stay in one piece forever. Regular, nonchaotic regions (invariant tori) are examples, but chaotic regions are included too. Therefore, having a steady state does not require complete integrability of the motion.