The Euler-Lagrange equation is a fundamental equation of the calculus of variations that is used to determine if a function has a stationary value. In physical terms a stationary value corresponds to a minimum or maximum of a function. The Euler-Lagrange equation is usually written as: (∂f/∂y) - (d/dt)(∂f/∂ &ydot;) = 0. One of the fundamental equations of calculus of variations states that for a function P defined by an integral of the form P = f(x,y, &ydot;)dx, where &ydot; = dy/dt, there is a stationary value if the corresponding Euler-Lagrange equation for that function is satisfied. When the time derivative notation is replaced by spacevariable notation often times, in physical problems, the partial derivative of f with respect to x is equal to 0. When this is the case the Euler-Lagrange equation is reduced to a simplified form known as the Beltrami identity: f - yx(∂f/∂yx) = c, where c is a constant. The Euler-Lagrange equation plays a pivotal role in the calculus of variations since this generalization of calculus focuses on finding the path, curve, surface, etc., for which a given function has a stationary value. Solving an appropriate Euler-Lagrange equation often lends itself to solutions of problems in the calculus of variations.
The Euler-Lagrange equation arose out of studies conducted by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the mid 1700s. During the early 1750s Lagrange began studying the tautochrone, the curve on which a weighted particle always arrives at a fixed point in the same time regardless of its initial position.
His discoveries concerning this curve were substantial to the calculus of variations. In 1755 Lagrange sent Euler his results on the tautochrone which contained his ideas for methods determining maxima and minima. A year later Lagrange sent Euler the results of applying the calculus of variations to mechanics. These results were a generalized form of results Euler had already obtained himself. From these communications rose the Euler-Lagrange equation. Lagrange, using this equation, went on to publish works in the foundations of dynamics which were based on the principle of least action and on kinetic energy. In 1766 Euler officially named these studies calculus of variations.
The Euler-Lagrange equation and calculus of variations are well known in classical mechanics since they are often used to determine minimum and/or maximum points of functions. Studying critical behavior such as in surface tension phenomena often employs Euler-Lagrange equations and calculus of variations. A generalized form of calculus of variations, called Morse theory, employs nonlinear techniques in studying variational problems. This particular theory relies on the thought that there is a relationship between the stationary points of a smooth, real-valued function on a manifold and the global topology of that manifold. Morse theory was related to quantum field theory in 1982 by a paper published by Edward Witten.
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