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 (-5[1])form is zero, (15[1])or, equivalently, that symplectomorphisms preserve the symplectic volume. If the microstates are uniformly distributed (15[1])at t (-5[1])= 0, this theorem implies that the system obeys the (*) ergodic hypothesis. (10[1])This theorem can be stated as “for a density, (10[1])rho, the (-5[1])time derivative of rho equals (10[1]-5[1])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])