The form of the Lagrangian is not arbitrary. In fact, it can be derived
directly from quantum mechanics using the
Feynman path integral formulation (as discussed in statistical mechanics).
owever, the Lagrangian can be motivated
by introducing the so called action principle, which states that a
particular functional of all paths that a
particle can take between two points is extremized along the
correct classical solution. The classical action
is defined in terms of the Lagrangian as follows: Consider
a system described by generalized coordinates
and velocities 
.
Suppose the system
moves from point A to point B in time T. At t=0, the coordinates and
velocites have values
and at T, 
. The action is then defined to be
The notation S[q] indicates that the action is a function of the 3N
functions 
. That is, S depends on all values of these
functions between 0 and T. Hence, it is called a functional. Like an
ordinary function, functoinals can be differentiated and integrated. However,
these operations must be performed
with respect to the function(s) the functional depends on, which means
the function(s) must be evauated at
some particular point when doing the differentiation.
Likewise, when integrating a functional, it must be
integrated with respect to all values the function can
take on in its full range. We shall see, in particular,
how the operation of differentiation works in the course of our derivation.
We will now show that, of all possible paths that the system
may follow between A and B, the correct path is the
one that extremizes the action. In order to show this, let
us consider two paths q(t) and 
given by
where 
is a small deviation from the path q(t).
The path, is required to satisfy the
same initial and final conditions of q(t), i.e.
This means that the deviation satisfies
Given these conditions, the problem of extrimizing the action means that we must find where its first derivative is equal to 0. Using a finite difference representation, this means
Now, since we need to let 
, we can expand 
to first
order in 
. Remembering that q and are multidimensional,
the expansion can be written as
The first and last terms cancel leaving only
In order to have an expression that depends only on 
,
we integrate the second term by parts:
where the boundary term vanishes because 
. The deviation, is
defined so that it is not exactly 0 for all t.
Thus, in order that 
, the term in brackets must vanish, However,
this is just the Euler-Lagrange condition:
Thus, the path that extremizes the action is exactly that which solves the Euler-Lagrange equation, which is the classical path. Thus, the correct solution to the classical equations of motion also extremizes the action.
The action principle is more than just a formal device. It has been used by various groups to study rare events in chemical processes. The articles by R. Elber and coworkers and Passerone and Parrinello show how the action principle can be used in actual computational chemical studies.