Maximizing the Isoperimetric Problem with Integral Constraints | MathWorld

[HACR]

Homework Statement

The isoperimetric problem is of the finding the object that has the largest area with the equal amount of perimeters; however, how does the integral constrained by the arc length get maximized? http://mathworld.wolfram.com/IsoperimetricProblem.html

Homework Equations

The Attempt at a Solution

...finding a point at which the integral is like finding the max and min of a function of two variables...

 

[LCKurtz]

Homework Statement

The isoperimetric problem is of the finding the object that has the largest area with the equal amount of perimeters; however, how does the integral constrained by the arc length get maximized? http://mathworld.wolfram.com/IsoperimetricProblem.html

The Attempt at a Solution

...finding a point at which the integral is like finding the max and min of a function of two variables...

The general subject is Calculus of Variations. One place to read about it is here:

http://www.google.com/url?sa=t&rct=...sg=AFQjCNGZoq3YfZweM8ZFKBWuB062RSvkZQ&cad=rja

 

[HACR]

It says the shortest path is the straight line; however, the brachistochrone problem proves that it is actually a curved line on which a stone could accelerate more. OK, brachistochrone problem is discussed. But why is on page 1163, the Euler Lagrangian equal to $$-\frac{u"}{(1+(u')^2)^{\frac{3}{2}}}$$? I got -u"+(u')^{2}u" for numerator.