120 Microcanonical ensemble
Eqn. (3.12.4) leads to a slightly modified set of equations for the angular velocity
components. Instead of those in eqn. (1.11.44), one obtains
ω ̇ 1 =ω^21
|q|^2ω ̇x=
ω 1 ωx
|q|^2+
τx
Ixx+
(Iyy−Izz)
Ixxωyωzω ̇y=
ω 1 ωy
|q|^2+
τy
Iyy+
(Izz−Ixx)
Iyyωzωxω ̇z=ω 1 ωz
|q|^2+
τz
Izz+
(Ixx−Iyy)
Izz
ωxωy. (3.12.6)Ifω 1 (0) = 0 and eqn. (3.12.1) is satisfied by the initial quaternionq(0), then the
new equations of motion will yield rigid body dynamics in which eqn. (3.12.1) is
satisfied implicitly, thereby eliminating the need for an explicit constraint. This is
accomplished through the extra terms in the angular velocity equations. Recall that
implicit treatment of constraints can also be achieved via Gauss’s principle of least
constraint discussed in Section 1.10, which also leads to extra termsin the equations
of motion. The difference here is that, unlike in Gauss’s equations of motion, the extra
terms are derived directly from a Hamiltonian and, therefore, the modified equations
of motion are symplectic.
All that is needed now is an integrator for the new equations of motion. Milleret al.
showed that the Hamiltonian could be decomposed into five contributions that are par-
ticularly convenient for the development of a symplectic solver. Defining four vectors
c 1 ,..,c 4 as the columns of the matrixS(q),c 1 = (q 1 ,q 2 ,q 3 ,q 4 ),c 2 = (−q 2 ,q 1 ,q 4 ,−q 3 ),
c 3 = (−q 3 ,−q 4 ,q 1 ,q 2 ), andc 4 = (−q 4 ,q 3 ,−q 2 ,q 1 ), the Hamiltonian can be written as
H(q,p) =∑^4
k=1hk(q,p) +U(q)hk(q,p) =1
8 Ik(p·ck)^2 , (3.12.7)whereI 1 =I 11 ,I 2 =Ixx,I 3 =Iyy, andI 4 =Izz. Note that ifω 1 (0) = 0, then
h 1 (q,p) = 0 for all time. In terms of the Hamiltonian contributionshk(q,p), Liouville
operator contributionsiLk ={...,hk}are introduced, with an additional Liouville
operatoriL 5 =−(∂U/∂q)·(∂/∂p), and a RESPA factorization scheme is introduced
for the propagator