The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations.

The Euler–Lagrange equation serves to extremize action functionals of the form

where and are constants and .[1]

If , then the Euler–Lagrange equation reduces to the Beltrami identity,

where C is a constant.[2][note 1]

Derivation

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By the chain rule, the derivative of L is

 

Because  , we write

 

We have an expression for   from the Euler–Lagrange equation,

 

that we can substitute in the above expression for   to obtain

 

By the product rule, the right side is equivalent to

 

By integrating both sides and putting both terms on one side, we get the Beltrami identity,

 

Applications

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Solution to the brachistochrone problem

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The solution to the brachistochrone problem is the cycloid.

An example of an application of the Beltrami identity is the brachistochrone problem, which involves finding the curve   that minimizes the integral

 

The integrand

 

does not depend explicitly on the variable of integration  , so the Beltrami identity applies,

 

Substituting for   and simplifying,

 

which can be solved with the result put in the form of parametric equations

 
 

with   being half the above constant,  , and   being a variable. These are the parametric equations for a cycloid.[3]

Solution to the catenary problem

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A chain hanging from points forms a catenary.

Consider a string with uniform density   of length   suspended from two points of equal height and at distance  . By the formula for arc length,   where   is the path of the string, and   and   are the boundary conditions.

The curve has to minimize its potential energy   and is subject to the constraint   where   is the force of gravity.

Because the independent variable   does not appear in the integrand, the Beltrami identity may be used to express the path of the string as a separable first order differential equation

  where   is the Lagrange multiplier.

It is possible to simplify the differential equation as such:  

Solving this equation gives the hyperbolic cosine, where   is a second constant obtained from integration

 

The three unknowns  ,  , and   can be solved for using the constraints for the string's endpoints and arc length  , though a closed-form solution is often very difficult to obtain.

Notes

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  1. ^ Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant along the dynamical path.

References

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  1. ^ Courant R, Hilbert D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. p. 184. ISBN 978-0471504474.
  2. ^ Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. See Eq. (5).
  3. ^ This solution of the Brachistochrone problem corresponds to the one in — Mathews, Jon; Walker, RL (1965). Mathematical Methods of Physics. New York: W. A. Benjamin, Inc. pp. 307–9.