This exercise explores the differences between a symplectic method and a method which conserves energy exactly. Recall that the equation
” =
(
) can be written as the first-order system given in (7.85) and (7.86). A Hamiltonian for this system is
Also recall that
corresponds to the total energy of the system.
(a) It is assumed that the trapezoidal method given in (7.87) can be modified to produce a method that conserves energy. In particular, it is assumed that one can be found of the form
Using the above formula for the Hamiltonian, show that
From this, conclude that the method is conservative if
It is possible to show that the resulting method is second-order (you do not need to prove this).
(b) What finite difference equations do you obtain when the method in part (a) is applied to the pendulum system in (7.83), (7.84)? The resulting method is implicit. Show that finding
j+1
and
j+1
reduces to solving
(
j+1) = 0, where
It is assumed here that
=
= 1.
(c) Show that in the case of when
j+1
is close to
j
, the solution of
(
j+1) = 0 is close to
(d) Compute the solution using the method from part (b) for 0 ≤
≤ 10000, using
= 1/2 and the initial conditions
(0) =
/4,
’(0) = 0. These are the same values used for Figures 7.10, 7.11, and 7.12. In your write-up, state what method you used to solve f(θj+1) = 0, including the stopping condition and what you used for the initial guess(es) for
j+1.
(e) With the computed solution from part (d), plot
for 0 ≤
≤ 40 and for 9060 ≤
≤ 10000. Also plot the energy
(
’ ) over the same time intervals. Comment on how these compare to the results obtained using RK4 and the velocity Verlet method. To help with this comparison, note that the energy values for velocity Verlet seen in Figure 7.12 oscillate between about 0.2755 and 0.2929.
(f) As stated in Section 7.6.3, using velocity Verlet the computed value of the period is about 6.47, while the exact value is about 6.53. Determine the period for your computed solution from part (d), and compare it to the velocity Verlet value.