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. (15[1])For a vector field, X, defined on a symplectic manifold, this theorem says that the Lie derivative along X of the symplectic (15[1])form is zero, or, equivalently, that symplectomorphisms preserve the symplectic volume. If the microstates are uniformly distributed at t = 0, (-5[1])this theorem implies that the system obeys the (*) ergodic 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, (10[1])what theorem states that the distribution function is constant along all trajectories in phase space? (10[1])■END■