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Beltrami Identity

An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is

(partialf)/(partialy)-d/(dx)((partialf)/(partialy_x))=0.

(1)

Now, examine the derivative of f with respect to x

(df)/(dx)=(partialf)/(partialy)y_x+(partialf)/(partialy_x)y_(xx)+(partialf)/(partialx).

(2)

Solving for the partialf/partialy term gives

(partialf)/(partialy)y_x=(df)/(dx)-(partialf)/(partialy_x)y_(xx)-(partialf)/(partialx).

(3)

Now, multiplying (1) by y_x gives

y_x(partialf)/(partialy)-y_xd/(dx)((partialf)/(partialy_x))=0.

(4)

Substituting (3) into (4) then gives

(df)/(dx)-(partialf)/(partialy_x)y_(xx)-(partialf)/(partialx)-y_xd/(dx)((partialf)/(partialy_x))=0

(5)

-(partialf)/(partialx)+d/(dx)(f-y_x(partialf)/(partialy_x))=0.

(6)

This form is especially useful if f_x=0, since in that case

d/(dx)(f-y_x(partialf)/(partialy_x))=0,

(7)

which immediately gives

f-y_x(partialf)/(partialy_x)=C,

(8)

where C 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.

See also

Brachistochrone Problem, Calculus of Variations, Euler-Lagrange Differential Equation, Surface of Revolution

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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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Beltrami Identity

Cite this as:

Weisstein, Eric W. "Beltrami Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BeltramiIdentity.html

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