Euler–Lagrange equation

转自：http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation

In calculus of variations, the Euler–Lagrange equation, Euler's equation,^[1] orLagrange's equation, is adifferential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph Louis Lagrange in the 1750s.

Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing (or maximizing) it. This is analogous toFermat's theorem in calculus, stating that where a differentiable function attains its local extrema, itsderivative is zero.

In Lagrangian mechanics, because of Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for theaction of the system. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system ofgeneralized coordinates, and it is better suited to generalizations.

Statement

Examples

A standard example is finding the real-valued function on the interval [a,b], such thatf(a) = c and f(b) =d, the length of whose graph is as short as possible. The length of the graph of f is: