1549380323-Statistical Mechanics Theory and Molecular Simulation

Rigid body motion 37

where the double dot-product notation in the expression∇∇σ··r ̇r ̇indicates a full
contraction of the two tensors∇∇σandr ̇r ̇. The first of eqns. (1.10.1) is then used to
substitute in for the second time derivative appearing in eqn. (1.10.5) to yield:

∇σ·

[

F

m

+

λ∇σ

m

]

+∇∇σ··r ̇r ̇= 0. (1.10.6)

We can now solve for the Lagrange multiplierλto yield the analytical expression

λ=−

∇∇σ··r ̇r ̇+∇σ·F/m

|∇σ|^2 /m

. (1.10.7)

Finally, substituting eqn. (1.10.7) back into the first of eqns. (1.10.1) yields the equa-
tion of motion:

m ̈r=F−

∇∇σ··r ̇r ̇+∇σ·F/m

|∇σ|^2 /m

∇σ, (1.10.8)

which is known asGauss’s equation of motion. Note that whenr ̇= 0, the total force
appearing on the right is

F−

(∇σ·F)∇σ

|∇σ|^2

=F−

(

∇σ

|∇σ|

·F

)

∇σ

|∇σ|

, (1.10.9)

which is just the projected force in eqn. (1.10.4). Forr ̇ 6 = 0, the additional force term
involves a curvature term∇∇σcontracted with the velocity–vector dyadr ̇r ̇. Since this
term would be present even ifF= 0, this term clearly corrects for free motion off the
surface of constraint.
Having eliminatedλfrom the equation of motion, eqn. (1.10.8) becomes an equa-
tion involving a velocity-dependent force. This equation, alone, generates motion on
the correct constraint surface, has a conserved energy,E=mr ̇^2 /2 +U(r), and, by
construction, conservesσ(r) in the sense that dσ/dt= 0 along a trajectory. However,
this equation cannot be derived from a Lagrangian or a Hamiltonian and, therefore,
constitutes an example ofnon-Hamiltoniandynamical system (see also Section 1.12).
Gauss’s procedure for obtaining constrained equations of motion can be generalized
to an arbitrary number of particles or constraints satisfying the proper differential
constraints relations.

1.11 Rigid body motion: Euler angles and quaterions

The discussion of constraints leads naturally to the topic of rigid body motion. Rigid
body techniques can be particularly useful in treating small molecules such as water
or ammonia or large, approximately rigid subdomains of large molecules, in that these
techniques circumvent the need to treat large numbers of explicit degrees of freedom.
Imagine a collection ofnparticles with all interparticle distances constrained. Such a
system, known as arigid body, has numerous applications in mechanics and statistical
mechanics. An example in chemistry is the approximate treatment ofsmall molecules
with very high frequency internal vibrations. A water molecule (H 2 O) could be treated
as a rigid isosceles triangle by constraining the two OH bond lengths and the distance