Adiabatic damping violates the conditions required for this theorem to hold, causing the Courant-Snyder invariant to decrease. In particle accelerators, this theorem implies that focusing or defocusing lenses cannot change the beam emittance. For a vector field, X, defined on a symplectic manifold, this theorem says that the Lie derivative along X of the symplectic form is zero, or, equivalently, that symplectomorphisms preserve the symplectic volume. If the microstates are uniformly distributed (15[1])at t = 0, this theorem implies that the (15[1])system obeys the (*) ergodic (-5[1])hypothesis. This theorem can be stated as “for a density, rho, the time derivative of rho equals the negative of the Poisson bracket of rho and the Hamiltonian.” For 10 points, what theorem states that the distribution function is constant along all trajectories in phase space? ■END■ (10[1])