1549380323-Statistical Mechanics Theory and Molecular Simulation

Rigid body motion 43

=−

∂U

∂θ

. (1.11.23)

Therefore, we see that the torque is simply the force on an angularcoordinate. The
equation of motion can thus be written in vector form as

dl

dt

=τ, (1.11.24)

which is analogous to Newton’s second law in Cartesian form

dp

dt

=F. (1.11.25)

A rigid diatomic, being a linear object, can be described by a single anglecoordinate
in two dimensions or by two angles in three dimensions. For a general rigid body
consisting ofnatoms in three dimensions, the number of constraints needed to make
it rigid is 3n−6 so that the number of remaining degrees of freedom is 3n−(3n−6) = 6.
After removing the three degrees of freedom associated with themotion of the body-
fixed frame, we are left with three degrees of freedom, implying that three angles
are needed to describe the motion of a general rigid body. These three angles are
known as theEuler angles. They describe the motion of the rigid body about three
independent axes. Although several conventions exist for defining these axes, any choice
is acceptable.
A particularly convenient choice of the axes can be obtained as follows: Consider
the total angular momentum of the rigid body, obtained as a sum of the individual
angular momentum vectors of the constituent particles:

l=

∑n

i=1

ri×pi=

∑n

i=1

miri×vi. (1.11.26)

Now,vi= dri/dtis measured in the body-fixed frame. From the analysis above, it
follows that the velocity is justω×riso that

l=

∑n

i=1

miri×(ω×ri). (1.11.27)

Expanding the double cross product, we find that

l=

∑n

i=1

mi

(

ωr^2 i−ri(ri·ω)

)

, (1.11.28)

which, in component form, becomes

lx=ωx

∑n

i=1

mi(r^2 i−x^2 i)−ωy

∑n

i=1

mixiyi−ωz

∑n

i=1

mixizi