The Euler-Lagrange differential equation is the fundamental equation of calculus of variations. It states that if is defined by an integral of the form
(1)
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where
(2)
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then has a stationary value if the Euler-Lagrange differential equation
(3)
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is satisfied.
If time-derivative notation is replaced instead by space-derivative notation , the equation becomes
(4)
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The Euler-Lagrange differential equation is implemented as EulerEquations[f, u[x], x] in the Wolfram Language package VariationalMethods` .
In many physical problems, (the partial derivative of with respect to ) turns out to be 0, in which case a manipulation of the Euler-Lagrange differential equation reduces to the greatly simplified and partially integrated form known as the Beltrami identity,
(5)
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For three independent variables (Arfken 1985, pp. 924-944), the equation generalizes to
(6)
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Problems in the calculus of variations often can be solved by solution of the appropriate Euler-Lagrange equation.
To derive the Euler-Lagrange differential equation, examine
(7)
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(8)
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(9)
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since . Now, integrate the second term by parts using
(10)
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(11)
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(12)
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so
(13)
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Combining (◇) and (◇) then gives
(14)
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But we are varying the path only, not the endpoints, so and (14) becomes
(15)
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We are finding the stationary values such that . These must vanish for any small change , which gives from (15),
(16)
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This is the Euler-Lagrange differential equation.
The variation in can also be written in terms of the parameter as
(17)
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(18)
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where
(19)
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(20)
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and the first, second, etc., variations are
(21)
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(22)
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(23)
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(24)
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The second variation can be re-expressed using
(25)
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so
(26)
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But
(27)
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Now choose such that
(28)
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and such that
(29)
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so that satisfies
(30)
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It then follows that
(31)
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(32)
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