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Date: 29-9-2018
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Date: 13-6-2019
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Date: 19-5-2019
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An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is
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(1) |
Now, examine the derivative of with respect to
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(2) |
Solving for the term gives
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(3) |
Now, multiplying (1) by gives
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(4) |
Substituting (3) into (4) then gives
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(5) |
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(6) |
This form is especially useful if , since in that case
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(7) |
which immediately gives
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(8) |
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.
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
Fox, C. An Introduction to the Calculus of Variations. New York: Dover, 1988.
Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, 1974.
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