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. (15[1])For a vector field, X, defined on a symplectic manifold, this theorem says that (-5[1])the Lie derivative along X of the symplectic (15[1]-5[1])form is zero, (15[1]-5[1])or, equivalently, (15[1])that symplectomorphisms preserve the symplectic (15[2])volume. If the microstates are uniformly (15[1])distributed (15[2])at t (-5[1])= 0, (-5[2])this theorem implies that the (15[1])system obeys the (*) ergodic (-5[1])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, (10[1])what theorem states that the distribution function is constant along (-5[1])all (-5[1])trajectories in phase space? (10[1])■END■ (10[7]0[4])