Adiabatic damping violates the conditions required for this theorem to hold, causing the Courant-Snyder invariant to decrease. (15[1])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 (-5[1])the Lie derivative along X of the symplectic form is zero, (-5[1])or, equivalently, (15[1])that symplectomorphisms preserve the symplectic volume. If the microstates are uniformly distributed at t = 0, 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, what theorem states that the distribution function is constant along (-5[1])all trajectories in phase space? ■END■ (10[2]0[2])