Application
An example of an application of the Beltrami identity is the Brachistochrone problem, which involves finding the curve y = y(x) that minimizes the integral
The integrand
does not depend explicitly on the variable of integration x, so the Beltrami identity applies,
Substituting for L and simplifying,
which can be solved with the result put in the form of parametric equations
where A is a constant and φ is a variable. These are the parametric equations for a cycloid.
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