et
be a smooth function. We introduce the
action
functional
taking vector functions of
and producing a number. We consider a problem of minimization of
across a variety of smooth functions
with fixed values of
at
the ends of the interval
.
If a function
minimizes
then we must have for any perturbation function
and number
:
We integrate the first term by parts and use the fact that the perturbation
must be zero at the ends of the
interval:
The above is true for any
.
Hence


(Euler Lagrange equation)

when evaluated at the minimizing function
.
The last equation is called the EulerLagrange equation.
