1549380323-Statistical Mechanics Theory and Molecular Simulation

Constraints 33

1

2

∑N

i=1

mir ̇^2 i−C= 0, (1.9.3)

whereCis a constant.
Since constraints reduce the number of degrees of freedom in a system, it is often
possible to choose a new system of 3N−Ncgeneralized coordinates, known as a
minimal setof coordinates, that eliminates the constraints. For example, consider the
motion of a particle on the surface of a sphere. If the motion is described in terms
of Cartesian coordinates (x,y,z), then a constraint condition of the formx^2 +y^2 +
z^2 −R^2 = 0, whereRis the radius of the sphere, must be imposed at all times.
This constraint could be eliminated by choosing the spherical polar angleθandφas
generalized coordinates. However, it is not always convenient to work in such minimal
coordinate frames, particularly when there is a large number of coupled constraints.
An example of this is a long hydrocarbon chain in which all carbon–carbon bonds are
held rigid (an approximation, as noted earlier, that is often made to eliminate the high
frequency vibrational motion). Thus, it is important to consider how the framework
of classical mechanics is affected by the imposition of constraints. We will now show
that the Lagrangian formulation of mechanics allows the influence ofconstraints to be
incorporated into its framework in a transparent way.
In general, it would seem that the imposition of constraints no longerallows the
equations of motion to be obtained from the stationarity of the action, since the
coordinates (and/or velocities) are no longer independent. More specifically, the path
displacementsδqα(cf. eqn. (1.8.6)) are no longer independent. In fact, the constraints
can be built into the action formalism using the method ofLagrange undetermined
multipliers. However, in order to apply this method, the constraint conditionsmust
be expressible in a differential form as:

∑^3 N

α=1

akαdqα+aktdt= 0, k= 1,...,Nc, (1.9.4)

whereakαis a set of coefficients for the displacements dqα. For a holonomic constraint
as in eqn. (1.9.1), it is clear that the coefficients can be obtained by differentiating the
constraint condition
∑N

α=1

∂σk

∂qα

dqα+

∂σk

∂t

dt= 0 (1.9.5)

so that

akα=

∂σk

∂qα

, akt=

∂σk

∂t

. (1.9.6)

Nonholonomic constraints cannot always be expressed in the form of eqn. (1.9.4). A
notable exception is the kinetic energy constraint of eqn. (1.9.3):

∑N

i=1

1

2

mir ̇^2 i−C= 0