Let us first consider the real time quantum propagator. The quantity appearing in the exponential is an integral of
which is known as the Lagrangian in classical mechanics. We can ask, which paths will contribute most to the integral
known as the action integral. Since we are integrating over a complex
exponential 
, which is oscillatory, those paths away from which
small deviations cause no change in S (at least to first order) will
give rise to the dominant contribution. Other paths that cause
to oscillate rapidly as we change from one path to another will give rise to
phase decoherence and will ultimately cancel when integrated over. Thus,
we consider two paths x(s) and a nearby one constructed from it
and demand that the change in S between these paths be 0
Note that, since x(0)=x and x(t)=x', 
, since all
paths must begin at x and end at x'. The change in S is
Expanding the first term to first order in 
, we obtain
The term proportional to 
can be handled by an integration by parts:
because vanishes at 0 and t, the surface term is 0, leaving us
with
Since the variation itself is arbitrary, the only way the integral can vanish, in general, is if the term in brackets vanishes:
This is known as the Euler-Lagrange equation in classical mechanics. For the
case that 
, they give
which is just Newton's equation of motion, subject to the conditions that x(0)=x, x(t)=x'. Thus, the classical path and those near it contribute the most to the path integral.
The classical path condition was derived by requiring that 
to first
order. This is known as an action stationarity principle. However, it
turns out that there is also a principle of least action, which states
that the classical path minimizes the action as well. This is an important
consideration when deriving the dominant paths for the density matrix, which
takes the form
The action appearing in this expression is
which is known as the Euclidean action and is just the integral over a
path of the total energy or Euclidean Lagrangian 
.
Here, we see that a minimum action principle is needed, since the smallest
values of 
will contribute most to the integral. Again, we require
that to first order 
. Applying the same logic
as before, we obtain the condition
which is just Newton's equation of motion on the inverted potential surface
-U(x), subject to the conditions x(0)=x, 
.
For the partition function 
, the same equation of motion must
be solved, but subject to the conditions that 
, i.e.,
periodic paths.