32 Classical mechanics
particular initial condition, and in fact, initial conditions for Hamilton’sequations are
generally chosen at random (e.g., random velocities). Typically, we are interested in
the behavior of large numbers of trajectories all seeded differently. Similarly, we are
rarely interested in paths leading from one specific point in phase space to another as
much as paths that evolve from oneregionof phase space to another. Therefore, the
initial-value and endpoint formulations of classical trajectories canoften be two routes
to the solution of a particular problem.
The action principle suggests the intriguing possibility that classical trajectories
could be computed from an optimization procedure performed on the action given
knowledge of the endpoints of the trajectory. This idea has been exploited by various
researchers to study complex processes such as protein folding.As formulated, however,
stationarity of the action does not imply that the action is minimum along a classical
trajectory, and, indeed, the action is bounded neither from above nor below. In order
to overcome this difficulty, alternative formulations of the action principle have been
proposed which employ an action or a variational principle that leads to a minimization
problem. The most well known of these is Hamilton’sprinciple of least action. The
least action principle involves a somewhat different type of variational principle in
which the variations are not required to vanish at the endpoints. A detailed discussion
of this type of variation, which is beyond the scope of this book, canbe found in
Goldstein’sClassical Mechanics(1980).
1.9 Lagrangian mechanics and systems with constraints
In mechanics, it is often necessary to treat a system that is subject to a set of externally
imposed constraints. These constraints can be imposed as a matter of convenience, e.g.
constraining high-frequency chemical bonds in a molecule at fixed bond lengths, or as
true constraints that might be due, for example, to the physical boundaries of a system
or the presence of thermal or barostatic control mechanisms.
Constraints are expressible as mathematical relations among the phase space vari-
ables. Thus, a system withNcconstraints will have 3N−Ncdegrees of freedom and
a set ofNcfunctions of the coordinates and velocities that must be satisfied by the
motion of the system. Constraints are divided into two types. If the relationships
that must be satisfied along a trajectory are functions of only theparticle coordinates
q 1 ,...,q 3 N and possibly time, then the constraints are calledholonomicand can be
expressed asNcconditions of the form
σk(q 1 ,...,q 3 N,t) = 0, k= 1,...,Nc. (1.9.1)If they cannot be expressed in this manner, the constraints are said to benonholo-
nomic. A class of a nonholonomic constraints consists of conditions involving both the
particle positions and velocities:
ζ(q 1 ,...,q 3 N,q ̇ 1 ,...,q ̇ 3 N) = 0. (1.9.2)An example of a nonholonomic constraint is a system whose total kinetic energy is kept
fixed (thermodynamically, this would be a way of fixing the temperature of the system).
The nonholonomic constraint in Cartesian coordinates would then beexpressed as