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Because a differentiable functional is stationary at its local extrema, 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 to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.[2]
Statement
The Euler–Lagrange equation is an equation satisfied by a function q
of a real argument t, which is a stationary point of the functional
where:
is the function to be found:
such that is differentiable, , and .
is the derivative of :
denotes the tangent bundle to along the curve , (disjoint) union of all tangent spaces (see tangent space) to at the points of curve .
We wish to find a function which satisfies the boundary conditions , , and which extremizes the functional
We assume that is twice continuously differentiable.[3] A weaker assumption can be used, but the proof becomes more difficult.[citation needed]
If extremizes the functional subject to the boundary conditions, then any slight perturbation of that preserves the boundary values must either increase (if is a minimizer) or decrease (if is a maximizer).
Let be the result of such a perturbation of , where is small and is a differentiable function satisfying . Then define
where .
We now wish to calculate the total derivative of with respect to ε.
It follows from the total derivative that
So
When ε = 0 we have gε = f, Fε = F(x, f(x), f'(x)) and Jε has an extremum value, so that
The next step is to use integration by parts on the second term of the integrand, yielding
Alternate derivation of the one-dimensional Euler–Lagrange equation
Given a functional
on with the boundary conditions and , we proceed by approximating the extremal curve by a polygonal line with segments and passing to the limit as the number of segments grows arbitrarily large.
Divide the interval into equal segments with endpoints and let . Rather than a smooth function we consider the polygonal line with vertices , where and . Accordingly, our functional becomes a real function of variables given by
Extremals of this new functional defined on the discrete points correspond to points where
Evaluating this partial derivative gives
Dividing the above equation by gives
and taking the limit as of the right-hand side of this expression yields
The left hand side of the previous equation is the functional derivative of the functional . A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation.
Examples
A standard example is finding the real-valued function y(x) on the interval [a, b], such that y(a) = c and y(b) = d, for which the pathlength along the curve traced by y is as short as possible.
the integrand function being L(x, y, y′) = √1 + y′ ² .
The partial derivatives of L are:
By substituting these into the Euler–Lagrange equation, we obtain
that is, the function must have a constant first derivative, and thus its graph is a straight line.
Generalizations
Single function of single variable with higher derivatives
The stationary values of the functional
can be obtained from the Euler–Lagrange equation[4]
under fixed boundary conditions for the function itself as well as for the first derivatives (i.e. for all ). The endpoint values of the highest derivative remain flexible.
Several functions of single variable with single derivative
If the problem involves finding several functions () of a single independent variable () that define an extremum of the functional
then the corresponding Euler–Lagrange equations are[5]
Single function of several variables with single derivative
A multi-dimensional generalization comes from considering a function on n variables. If is some surface, then
Single function of two variables with higher derivatives
If there is a single unknown function f to be determined that is dependent on two variables x1 and x2 and if the functional depends on higher derivatives of f up to n-th order such that
wherein are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the indices is only over in order to avoid counting the same partial derivative multiple times, for example appears only once in the previous equation.
Several functions of several variables with higher derivatives
If there are p unknown functions fi to be determined that are dependent on m variables x1 ... xm and if the functional depends on higher derivatives of the fi up to n-th order such that
where are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is
where the summation over the is avoiding counting the same derivative several times, just as in the previous subsection. This can be expressed more compactly as
where is the Lagrangian, the statement is equivalent to the statement that, for all , each coordinate frame trivialization of a neighborhood of yields the following equations:
^ abcCourant, R; Hilbert, D (1953). Methods of Mathematical Physics. Vol. Vol. I (First English ed.). New York: Interscience Publishers, Inc. ISBN978-0471504474. {{cite book}}: |volume= has extra text (help); Invalid |ref=harv (help)