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Euler–Lagrange equation

释义

History
Statement
Examples
Generalizations for several functions, several variables, and higher derivatives
Single function of single variable with higher derivatives Several functions of single variable with single derivative Single function of several variables with single derivative Several functions of several variables with single derivative Single function of two variables with higher derivatives Several functions of several variables with higher derivatives
Generalization to manifolds
See also
Notes
References

In the calculus of variations, the Euler–Lagrange equation, Euler's equation,^[1] or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential 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 to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative 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 the action 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 of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

History

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 space to at the point .

is a real-valued function with continuous first partial derivatives:

being the tangent bundle of defined by

;

The Euler–Lagrange equation, then, is given by

{{Equation box 1
|indent =:
|equation =

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|background colour = #ECFCF4}}

where and denote the partial derivatives of with respect to the second and third arguments, respectively.

If the dimension of the space is greater than 1, this is a system of differential equations, one for each component:

The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs in mathematics. It relies on the fundamental lemma of calculus of variations.

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.{{Cn|date=September 2013}}

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

When ε = 0 we have g_ε = f, F_ε = F(x, f(x), f'(x)) and J_ε has an extremum value, so that

The next step is to use integration by parts on the second term of the integrand, yielding

Using the boundary conditions ,

Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation

Derivation of 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.

Alternate derivation of one-dimensional Euler–Lagrange equation

Examples

A standard example is finding the real-valued function f on the interval [a, b], such that f(a) = c and f(b) = d, for which the path length along the curve traced by f is as short as possible.

the integrand function being {{nowrap|1=L(x, y, y′) = {{radic|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 constant first derivative, and thus its graph is a straight line.

Generalizations for several functions, several variables, and higher derivatives

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

is extremized only if f satisfies the partial differential equation

When n = 2 and functional is the energy functional, this leads to the soap-film minimal surface problem.

Several functions of several variables with single derivative

If there are several unknown functions to be determined and several variables such that

the system of Euler–Lagrange equations is^[4]

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

then the Euler–Lagrange equation is^[4]

which can be represented shortly as:

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 f_i to be determined that are dependent on m variables x₁ ... x_m and if the functional depends on higher derivatives of the f_i 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

Generalization to manifolds

Let be a smooth manifold, and let denote the space of smooth functions . Then, for functionals of the form

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:

Notes

1. ^{{cite book|first=Charles|last=Fox|title=An introduction to the calculus of variations|publisher=Courier Dover Publications|year=1987|isbn=978-0-486-65499-7}}
2. ^A short biography of Lagrange {{webarchive|url=https://web.archive.org/web/20070714022022/http://numericalmethods.eng.usf.edu/anecdotes/lagrange.pdf |date=2007-07-14 }}
3. ^{{harvnb|Courant|Hilbert|1953|p=184}}
4. ^¹²{{cite book | last1=Courant | first1=R | authorlink1=Richard Courant | last2=Hilbert | first2=D | authorlink2=David Hilbert | title = Methods of Mathematical Physics | volume = Vol. I | edition = First English | ref=harv | publisher = Interscience Publishers, Inc | year = 1953 | location = New York | isbn = 978-0471504474}}
5. ^{{cite book |last=Weinstock |first=R. |year=1952 |title=Calculus of Variations with Applications to Physics and Engineering |publisher=McGraw-Hill |location=New York }}

References

{{springer|title=Lagrange equations (in mechanics)|id=p/l057150}}
{{mathworld|urlname=Euler-LagrangeDifferentialEquation|title=Euler-Lagrange Differential Equation}}
{{planetmathref |id=1995|title=Calculus of Variations}}
{{cite book |last=Gelfand |first=Izrail Moiseevich |authorlink=Israel Gelfand |title=Calculus of Variations |publisher=Dover |year=1963 |isbn=0-486-41448-5}}
Roubicek, T.: Calculus of variations. Chap.17 in: Mathematical Tools for Physicists. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, {{ISBN|978-3-527-41188-7}}, pp.551-588.

5 : Ordinary differential equations|Partial differential equations|Calculus of variations|Articles containing proofs|Leonhard Euler

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