For a function we speak of a stationary point when the function doesn’t change under a small change, i.e., if we take , and thus , the change in , , to first order in . This leads to the obvious relation .
As for functions, we are extremely interested in stationary points of functionals:
A functional has a stationary point for any function such that a small change leads to no change in to first order in .
A key difference with the stationary points of functions is that smallness of only implies that as a function it is everywhere (mathematician would say “uniformly”) close to zero, but can still vary arbitrarily.
An important class of functionals is given by
(5.1) |
where , are fixed, and is specified at the boundaries, i.e., the values of and are specified as boundary conditions. Thus under a (small) change the preservation of the boundary conditions implies
(5.2) |
Now substitute and expand to first order in ,
where we have integrated by parts to obtain the penultimate line, using the boundary conditions on .
Since is allowed to vary arbitrarily, this is only zero if the quantity multiplying is zero at every point . For example, you can choose a set of ’s that are all peaked around a particular value of . We thus see that the term proportional to vanishes at each point , and we get the Euler-Lagrange equation
(5.3) |
Remarks
for the term in the functional proportional to ,
we can turn it into an integral by adding a delta function. In the case above,
For a general functional, the equation
is called the Euler-Lagrange equation.
Solutions to this equation define stationary points of the functional.