1549380323-Statistical Mechanics Theory and Molecular Simulation

The action integral 29

w 1 :

w 2 :

w 3 :

Fig. 1.8 Normal modes of the harmonic polymer model forN= 3 particles.

1.8 The action integral

Having introduced the Lagrangian formulation of classical mechanics and derived the
Hamiltonian formalism from it using the Legendre transform, it is natural to ask if
there is a more fundamental principle that leads to the Euler–Lagrange equations.
In fact, we will show that the latter can be obtained from a variational principle
applied to a certain integral quantity, known as theaction integral. At this stage,
however, we shall introduce the action integral concept without motivation because in
Chapter 12, we will show that the action integral emerges naturallyand elegantly from
quantum mechanics. The variational principle to be laid out here has more than formal
significance. It has been adapted for actual trajectory calculations for large biological
macromolecules by Olender and Elber (1996) and by Passerone and Parrinello (2001).
In order to define the action integral, we consider a classical system with generalized
coordinatesq 1 ,...,q 3 Nand velocities ̇q 1 ,...,q ̇ 3 N. For notational simplicity, let us denote
byQthe full set of coordinatesQ≡ {q 1 ,...,q 3 N}andQ ̇ the full set of velocities
Q ̇ ≡ {q ̇ 1 ,...,q ̇ 3 N}. Suppose we follow the evolution of the system from timet 1 tot 2
with initial and final conditions (Q 1 ,Q ̇ 1 ) and (Q 2 ,Q ̇ 2 ), respectively, and we ask what
path the system will take between these two points (see Fig. 1.9). We will show that
the path followed renders stationary the following integral:

A=

∫t 2

t 1

L(Q(t),Q ̇(t)) dt. (1.8.1)

The integral in eqn. (1.8.1) is known as theaction integral. We see immediately that the
action integral depends on the entire trajectory of the system.Moreover, as specified,
the action integral does not refer to one particular trajectory but toanytrajectory
that takes the system from (Q 1 ,Q ̇ 1 ) to (Q 2 ,Q ̇ 2 ) in a timet 2 −t 1. Each trajectory
satisfying these conditions yields a different value of the action. Thus, the action can