To see an example where Liouville's theorem does not apply, we can modify the equations of motion for the simple harmonic oscillator to account for the effects of friction or damping. Consider again the system of N particles each in a 3 -dimensional isotropic harmonic potential, the Hamiltonian for which is given in the previous example. This time, we add the condition that each particle experiences a frictional force − − γ γ p i , where γ γ is a positive constant dictating the amount of friction. As this is a non-conservative force, we need to extend Hamilton's equations as