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 and are constants and.
If, then the Euler–Lagrange equation reduces to the Beltrami identity,
where is a constant.

Derivation

The following derivation of the Beltrami identity starts with the Euler–Lagrange equation,
Multiplying both sides by,
According to the chain rule,
where.
Rearranging this yields
Thus, substituting this expression for into the second equation of this derivation,
By the product rule, the last term is re-expressed as
and rearranging,
For the case of, this reduces to
so that taking the antiderivative results in the Beltrami identity,
where is a constant.

Applications

Solution to the brachistochrone problem

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.