122 Microcanonical ensemble
et al., 2005). The one example for which the shadow Hamiltonian is known is,not sur-
prisingly, the harmonic oscillator. Recall that the Hamiltonian for a harmonic oscillator
of massmand frequencyωis
H(x,p) =p^2
2 m+
1
2
mω^2 x^2. (3.13.1)If the equations of motion ̇x=p/mp ̇=−mω^2 xare integrated via the velocity Verlet
algorithm
x(∆t) =x(0) + ∆t
p(0)
m−
1
2
∆t^2 ω^2 x(0)p(∆t) =p(0)−mω^2 ∆t
2[x(0) +x(∆t)], (3.13.2)then it can be shown that the Hamiltonian
H ̃(x,p; ∆t) = p2
2 m(1−ω^2 ∆t^2 /4)+
1
2
mω^2 x^2 (3.13.3)is exactly preserved by eqns. (3.13.2). Of course, the form of theshadow Hamiltonian
will depend on the particular symplectic solver used to integrate theequations of
motion. A phase space plot ofH ̃vs. that ofHis provided in Fig. 3.4. In this case, the
eccentricity of the ellipse increases as the time step increases. Thecurves in Fig. 3.4
are exaggerated for illustrative purposes. For any reasonable (small) time step, the
difference between the true phase space and that of the shadow Hamiltonian would
be almost indistinguishable. As eqn. (3.13.3) indicates, if ∆t= 2/ω,H ̃ becomes ill-
defined, and for ∆t > 2 /ω, the trajectories are no longer bounded. Thus, the existence
ofH ̃ can only guarantee long-time stability for small ∆t.
Although it is possible to envision developing novel simulation techniques based on
a knowledge ofH ̃(Izaguirre and Hampton, 2004), the mere existence ofH ̃is sufficient
to guarantee that the error in a symplectic map is bounded. That is,given that we
have generated a trajectory ̃xn∆t,n= 0, 1 , 2 ,...using a symplectic integrator, if we
then evaluateH( ̃xn∆t) at each point along the trajectory, it should not drift away from
the true conserved value ofH(xt) evaluated along the exact (but, generally, unknown)
trajectory xt. Note, this does not mean that the numerical and true trajectories will
follow each other. It simply means that ̃xn∆twill remain on a constant energy hyper-
surface that is close to the true constant energy hypersurface. This is an important
fact in developing molecular dynamics codes. If one uses a symplecticintegrator and
finds that the total energy exhibits a dramatic drift, the integrator cannot be blamed,
and one should search for other causes!
In order to understand why the shadow Hamiltonian exists, let us consider the
Trotter factorization in eqn. (3.10.22). The factorization schemeis not an exact rep-
resentation of the propagator exp(iL∆t); however, a formally exact relation
exp[
iL 2∆t
2]
exp [iL 1 ∆t] exp[
iL 2∆t
2]
= exp[
∆t(
iL+∑∞
k=1∆t^2 kCk