We continued with variational problems, considering a special form for
the functional which occurs frequently -- e.g. for time of flight or
path length problems. In this form, the functional I is an integral
between fixed x-endpoints a and b of an integrand F(f(x),
f'(x), x) which depends on f and its first derivative f'
only. (The integration measure is dx.) We calculated the first order
variation of I explicitly, by Taylor expanding I about f(x) and
integrating by parts. (The parts integration was possible because we
recognized that f and f' could not be varied independently, that
instead 
.) We found that the first
order variation 
was given by the integral from a to b,
with measure 
, of the integrand 
, along with a surface term
. Thus, if is to
vanish for all variations 
, the quantity must vanish at
all interior points. This gives the Euler-Lagrange equation. (Special
conditions may hold at the endpoints if the variations of f are not
required to vanish there.)
So we see that the Euler-Lagrange equation can come from a variational
principle. Particularly, if we take 
-- a
quantity known as the "action" S -- we obtain the Euler-Lagrange
equation of mechanics from the following principle, which generates
equations of motion equivalent to Newton II:
Hamilton's Principle:
A particle always follows a path x(t) between two fixed endpoints
, 
that extremizes the action, 
.
Hamilton's principle thus gives us a much easier, more elegant way of finding Newton II for a given physical system. It tells us that the classical motion x(t) is the one function x(t) that extremizes the action. It thus obeys the Euler-Lagrange equation, a differential equation for the motion x(t).
We then began considering how we might solve such an equation. We
first set out to integrate the Euler-Lagrange equation to find a
constant of the motion. We defined the "Jacobi integral" J by 
. We then calculated the total
derivative dJ/dx, using the chain rule to account for the implicit
dependences of F on x through its dependences on f(x) and
f'(x). We showed, invoking the Euler-Lagrange equation, that 
. That is, the absolute, total, physical
rate of change of J with respect to x (taking account of all
implicit, or indirect, variations that propagate out through f and
f' due to a change in x) equals minus the apparent rate of change
of F with respect to x, considering only F's explicit dependence
on x.
Thus, if F does not explicitly depend on x -- that is, if you can write F in terms of f and f' without the variable x appearing -- J must remain constant for all x. We have integrated our differential equation (Euler-Lagrange equation) to find a constant of the motion. Next time we will apply this method to solve a specific variational problem: on a sphere, find the shortest path between any two points.
-- KB