An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is
(1)
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Now, examine the derivative of with respect to
(2)
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Solving for the term gives
(3)
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Now, multiplying (1) by gives
(4)
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Substituting (3) into (4) then gives
(5)
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(6)
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This form is especially useful if , since in that case
(7)
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which immediately gives
(8)
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where is a constant of integration (Weinstock 1974, pp. 24-25; Arfken 1985, pp. 928-929; Fox 1988, pp. 8-9).
The Beltrami identity greatly simplifies the solution for the minimal area surface of revolution about a given axis between two specified points. It also allows straightforward solution of the brachistochrone problem.