30 Classical mechanics
Q
Q
(Q(t 1 ), Q(t 1 ))(Q(t 2 ), Q(t 2 ))Fig. 1.9 Two proposed paths joining the fixed endpoints. The actual path followed is a
stationary path of the action integral in eqn. (1.8.1.)
be viewed as a “function” of trajectories that satisfy the initial and final conditions.
However, this is not a function in the usual sense since the action is really a “function
of a function.” In mathematical terminology, we say that the actionis afunctional
of the trajectory. A functional is a quantity that depends on all values of a function
between two points of its domain. Here, the action is a functional oftrajectoriesQ(t)
leading fromQ 1 toQ 2 in timet 2 −t 1. In order to express the functional dependence,
the notationA[Q] is commonly used. Also, since at eacht,L(Q(t),Q ̇(t)) only depends
ont(and not on other times),A[Q] is known as alocal functionalin time. Functionals
will appear from time to time throughout the book, so it is important to become
familiar with these objects.
Stationarity of the action means that the action does not change to first order if
a small variation of a path is made keeping the endpoints fixed. In order to see that
the true classical path of the system is a stationary point ofA, we need to consider a
pathQ(t) between points 1 and 2 and a second path,Q(t) +δQ(t), between points 1
and 2 that is only slightly different fromQ(t). If a pathQ(t) rendersA[Q] stationary,
then to first order inδQ(t), the variationδAof the action must vanish. This can be
shown by first noting that the pathQ(t) satisfies the initial and final conditions:
Q(t 1 ) =Q 1 , Q ̇(t 1 ) =Q ̇ 1 , Q(t 2 ) =Q 2 , Q ̇(t 2 ) =Q ̇ 2. (1.8.2)Since all paths begin atQ 1 and end atQ 2 , the pathQ(t) +δQ(t) must also satisfy
these conditions, and sinceQ(t) already satisfies eqn. (1.8.2), the functionδQ(t) must