Question

5: A practical application of Liouville's Theorem (40 points) When solving a differential equation with a computer, the basic task is to approximate the continuous behavior of a system with a discrete set of variables measured at discrete times: (2(0), u(0), 2(?t), u( t)) (2(2? ), u(2?t), etc.. A very good algorithm for doing this is called Velocity Verlet, and works as follow: Znew old+old) (At)2 Unew = Uold t. 2m We are going to check whether this algorithm conserves volume in phase space. In order to do this you'll need to use the Jacobian formulation of Liouville's Theorem, which requires that the determinant of the Jacobian be 1: Told Hint: Remember that when you see Znew in the formula for vnew, Tnew most certainly does depend on Told and also vold, which means that Fx(Tnew) depends on xold. If you aren't careful about using the chain rule when taking derivatives of F(Cnew) you will get something that fails to conserve phase space volume Also, you will have to make one approximation (because we always do): Assume that while Fx (x) will change as x changes, assume that F () does not change substantially, i.e. treat as roughly constant. This will enable you to cancel some terms

Answer 1

x_new = x_old + v_old Delta t + 1/2 F_x (x_old)/m (Delta t)^2 Therefore dx_new = dx_old + dv_old Delta t + 1/2m (dF_x/dx_old (x_old) dx_old) (Delta t)^2 Thus, partial differential x_new/partial differential x_old = 1 + 1/2 m dF_x (x_old)/dx_old (Delta t)^2, partial differential x_new/partial differential v_old = Delta t v_new = v_old + F_x (x_old) + F_x (x_new)/2m Delta t dv_new = dv_old + 1/2m (dF_x (x_old)/dx_old dx_old) Delta t + 1/2m (dF_x (x_new)/dx_new dx_new) Delta t = dv_old + 1/2m (dF_x (x_old)/dx_old dx_old) Delta t + 1/2m dF_x (x_new)/dx_new dx_old Delta t + 1/2m dF_x (x_new)/dx_new dv_old (Delta t)^2 + 1/4m^2 dF_x (x_new)/dx_new dF_x (x_old)/dx_old dx_old (Delta t)^3 partial differential v_new/partial differential x_old = 1/2m dF_x (x_old)/dx_old Delta t + 1/2m dF_x (x_new)/dx_new Delta t + 1/4m^2 dF_x (x_new)/dx_new dF_x (x_old)/dx_old (Delta)^3 \r\n partial differential v_new/partial differential v_old 1 + 1/2m dF_x (x_new)/dx_new (Delta t)^2 Therefore = Vertical-bar 1 + 1/2m dF_x (x_old)/dx_old (Delta t)^2 Delta t 1/2m dF_x (x_old)/dx_old Delta t + 1/2m dF_x (x_new)/dx_new Delta t 1 + 1/2m dF_x (x_new)/dx_new (Delta t)^2