Before break we derived the Euler-Lagrange equations from Newton II, but claimed that these equations could be derived from a deeper, unifying principle: Hamilton's principle, or the Principle of Least Action.
To understand this principle, we began an introduction to variational problems. We first reviewed ordinary calculus max/min problems, where the quantity to minimize was the value of some function f(x), which maps real numbers x to real numbers f(x). Here we wish to minimize f with respect to all possible arguments x.
Reviewed the definition of the derivative as the coefficient of the
first order variation in f due to infinitesimal change in its argument
x. Showed that nonvanishing 
implied that f could not be
extremal at a, since a Taylor expansion of f near a gives both higher
and lower values of f. Extremal points can occur when
vanishes (such points a are called "stationary points"). Reviewed
criteria for stationary points to be maximal, minimal, or inflection
points.
We then considered a generalized problem. We considered the case where we want to extremize something which depends, not just on a single point x, but on some entire path, or trajectory, f(x). That is, we tried to extremize a functional, I(f(x)), which maps a function f(x) to a single real number I(f(x)). So if we view f(x) as a curve y= f(x) in Cartesian space, I associates a single number with the entire curve. The number could be, for example, path length between endpoints a and b, or travel time between endpoints a and b, or other examples as given in class. Again, we wish to extremize the value of I with respect to all possible arguments f(x) -- all possible shapes the function could take.
Just like ordinary max/min problems, extrema occur when the variation
in I due to an infinitesimal change in its argument f(x)
vanishes. For I = I(f(x)), with no dependence on the x-derivatives
of f, we calculated the first order variation of I. We did this by
using a finite mesh to sample the function f(x), Taylor expanding
I about each sample point 
, then letting the mesh separation
between sampled points go to zero. We found the variation of I to be
the integral from a to b (the x-endpoints) of the partial
derivative of I with respect to f, weighted by the measure 
.
-- KB