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 (15[1])field, X, defined on a symplectic manifold, this theorem says that (-5[1])the Lie derivative along X of the symplectic form (15[1])is zero, (-5[1])or, equivalently, (15[1])that symplectomorphisms preserve the symplectic volume. If the (15[1])microstates are uniformly (15[5]-5[1])distributed at t = (10[1])0, this theorem implies that the system obeys (10[1])the (10[1])(*) ergodic hypothesis. This theorem can be stated as (-5[1])“for (10[1])a density, rho, (10[2])the time derivative of rho equals the negative of the Poisson bracket of rho and the Hamiltonian.” (-5[1])For (10[1])10 points, what theorem (10[1])states that the (10[1])distribution function is constant (10[2]0[1])along (-5[1])all (10[1])trajectories in phase space? ■END■ (10[3]0[3])