A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs?

Question

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one?

I was looking for the tag "circumference", supplied the newer synonym (in this field) perimeter. Apparently this tag is not related, since it was removed.

I will link the Wikipedia article on arc length from suggestions, even though I don't know if it fits; it certainly doesn't answer my question.

https://en.wikipedia.org/wiki/Arc_length

The proof is likely related to isoperimetry as per suggestion, but again I'm asking for proofs.

https://en.wikipedia.org/wiki/Isoperimetric_inequality

I heavily suspect any intuitive proof to not be analytic, since mine isn't. If someone made a simpler intuitive proof, I want to know it and who to credit; if someone made an analytic proof, I will use it as advanced course reference.

Answer 1

The historical information your request can be found here :

http://mathworld.wolfram.com/IsoperimetricProblem.html

A proof using the calculus of variation can be found here :

https://mathematicalgarden.wordpress.com/2008/12/21/the-problem-of-dido/

Answer 2

OK. Since you persist I give a short abbreviated well known Dido's isoperimetric problem. The object and constraint functions are

$$ \int y dx -R \sqrt{1+y^{'2}} dx$$ where $R$ is a constant Lagrangian multiplier. The Lagrangian $F(x,y,y^{'})$ is given in square brackets

Euler-Lagrange equation of Variational calculus is:

$$ F-y^{' }\frac{\partial F}{\partial y^{'}}= const. c$$

$$[y-R\sqrt{1+y^{'2}}]-y^{'}\frac{-R y^{'}}{\sqrt{1+y^{'2}}}= c $$

$$y- \frac{R}{1+y^{'2}}= c$$ since $ \sec \phi=\sqrt{1+y^{'2}} $ the DE required is

$$ \frac{y-c}{ \cos \phi}=R $$

which represents all circles with center shifted up/down on x-axis by an amount $c.$ Note that the Lagrange multiplier $R$ has an arbitrary magnitude which is nothing but the radius of a circle thus defined.

Historical

Newton's work on Variational approach in fact predated Euler and Lagrange. But the definition of variation and its formalism ( Lagrange included proof with freedom from direct geometric representations) and rigorous structuring of Euler-Lagrange of variational definitions gained acceptance and adoption. Newton first used the method to find optimal shape of a projectile (travelling at supersonic speed ?)that offers minimum aerodynamic resistance to flight. The rocket shape has cusps at $\pm 30^{\circ}$ to flight path.

Aerodynamic Newton Prob Ref

Trivia about Aerodynamic Newton's Problem .. a edges of pages written/published in Newton's own handwriting after Principia's publication got burnt after his cat tipped a lighted candle, in Motte's Principia translation iirc..