Calculus of Variations

     I am currently taking a class in the calculus of variations.  I find it an odd course, since it is taught in French, each class is almost entirely lecture, and we have neither a textbook nor any graded homework.  Regardless of my personal confusion as to the pedagogic methods (well, except the French- that's because I'm at a French-speaking university), I have learned quite a bit about the calculus of variations. My first introduction was in my mechanics class, through Taylor's Classical Mechanics, and while I found the Euler-Lagrange equation as covered in my class useful, the generalizations I have learned this term could have been and could continue to be useful.

     Alright, so in case you have never heard of the calculus of variations, let me explain a bit of what it is. In calculus, generally you look at a function and try to find its maximum and minimum values; at least, that is one thing you can do. The calculus of variations looks at functionals rather than functions. A functional is a type of function that takes a function as its input, rather than a number or vector. Generally, a functional looks like an integral of a function of the independent variable(s), the function, and its derivative(s).  Wolfram has an example and a more precise definition, of course. Functionals do not have derivatives, per se, but they have variations, which are similar.  The Euler-Lagrange equation, which in standard form looks like:
, is a differential equation whose solutions are the functions for which the functional is stationary (about the same as a critical point in calculus).
     Standard versions of the equation can be found on Wikipedia. The one variant I have not seen outside of my course is the "first integral" (translation from French, I'm not certain if that is the correct term), often represented by H. This version can be used when the function within the integral of the functional does not depend explicitly on the independent variable(s). In physics, using the first integral is essentially using conservation of energy rather than the principle of least time. is the form of the integral, although I know the equation as the negative of this (C being any real number, a negative is unimportant).
     In physics, the Euler-Lagrange equation is used with f, which is generally the kinetic energy minus the potential, and would be inside the integral of a functional. The Euler-Lagrange equation is used under the principle of least time in optics, which states that light will travel the fastest way possible. The explanation of its use in other realms of physics is currently beyond me, but Classical Mechanics by Taylor which I mentioned before, explains it well, as I remember.
     I just wanted to get that first integral a bit more space out there among any physicists or engineers that could use it. In general, it is a much easier differential equation to solve, as it is first-order, rather than second, which means using it can free up time for the rest of your work from solving a math problem, perhaps without calling a mathematician. A really short, cheap book that covers the calculus of variations is Calculus of Variations by Gelfand and Fomin, which I have been using in the absence of a textbook for my class.