Re:[math/phys] Biggest area of ...

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Hi,
The problem that among figures (in R^{2}) with fixed length of
circumference circle has a biggest area,  can be solved by many different 
ways.   The simplest solution I know is based in find a minimum of
a functional with "bond".  (For physicists we use a "minimal principle of
actions".)

More precisely, we have
Area S={1/2} * \integral (ydx-xdy); and a "bond"
Lenghth L= \integral v dt = \integral [x'^{2}+y'^{2}]^{1/2} dt ;
(here x'=dx/dt and y'=dy/dt).
Next use Lagrange multipliers method (for finding extremums with bonds)
and provide a Euler-Lagrange equations for this functional.
(For mathematicians, this functional is convex and l.s.c, then the 
solution of Euler-Lagrange equations is a minimalizator of this
functional.  More delicate problem is a regurality problem, but let me
ignore it here ...) 
These equations have an unique solution which discribes a circle.
SN
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|                     Sonnet Nguyen                     |
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