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 form is zero, or, equivalently, that symplectomorphisms preserve the symplectic volume. If the (15[1])microstates are uniformly (15[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 “for a density, rho, 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 states that the (10[1])distribution function is constant (10[2]0[1])along all trajectories in phase space? ■END■