Beltrami identity

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

{\displaystyle u'(x)={\frac {du}{dx}}}

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

[2][note 1] 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

{\displaystyle {\frac {dL}{dx}}}

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, 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,

These are the parametric equations for a cycloid.

[3] Consider a string with uniform density

suspended from two points of equal height and at distance

By the formula for arc length,

are the boundary conditions.

The curve has to minimize its potential energy

and is subject to the constraint

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

is the Lagrange multiplier.

It is possible to simplify the differential equation as such:

g ρ y − λ

Solving this equation gives the hyperbolic cosine, where

is a second constant obtained from integration

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.