1549380323-Statistical Mechanics Theory and Molecular Simulation

46 Classical mechanics

For a rigid body in three dimensions, we require four variables, (q 1 ,q 2 ,q 3 ,q 4 ), the
quaternions, that must satisfy

∑ 4

i=1q

2
i= 1 and are, by convention, formally related
to the three Euler angles by

q 1 = cos

(

θ

2

)

cos

(

φ+ψ

2

)

q 2 = sin

(

θ

2

)

cos

(

φ−ψ

2

)

q 3 = sin

(

θ

2

)

sin

(

φ−ψ

2

)

q 4 = cos

(

θ

2

)

sin

(

φ+ψ

2

)

. (1.11.40)

From eqn. (1.11.40), it is straightforward to verify that

∑

iq
2
i= 1. The advantage of
the quaternion structure is that it leads to a simplification of the rigid-body motion
problem. First, note that at any time, a Cartesian coordinate vector in the space fixed
frame can be transformed into the body-fixed frame via a rotationmatrix involving
the quaternions. The relations are

r(body)=A(θ,φ,ψ)r(space) r(space)=AT(θ,φ,ψ)r(body). (1.11.41)

The rotation matrix is the product of individual rotations about thethree axes, which
yields

A(θ,φ,ψ) =







cosψcosφ−cosθsinφsinψ cosψsinφ+ cosθcosφsinψ sinθsinψ

−sinψcosφ−cosθsinφcosψ −sinψsinφ+ cosθcosφcosψ −sinθcosψ

sinθsinφ −sinθcosφ cosθ





.

(1.11.42)
In terms of quaterions, the matrix can be expressed in a simpler-looking form as

A(q) =











q 12 +q 22 −q 32 −q^24 2(q 2 q 3 +q 1 q 4 ) 2(q 2 q 4 −q 1 q 3 )

2(q 2 q 3 −q 1 q 4 ) q^21 −q 22 +q^23 −q 42 2(q 3 q 4 +q 1 q 2 )

2(q 2 q 4 +q 1 q 3 ) 2(q 3 q 4 −q 1 q 2 ) q 12 −q^22 −q 32 +q^24











. (1.11.43)

It should be noted that in the body-fixed coordinate, the moment of inertia tensor is
diagonal. The rigid-body equations of motion, eqns. (1.11.35), can now be transformed
into a set of equations of motion involving the quaternions. Direct transformation of
these equations leads to a new set of equations of motion given by

q ̇=

1

2

S(q)ω

ω ̇x=

τx

Ixx

+

(Iyy−Izz)

Ixx

ωyωz