- #1
Zaknife
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Homework Statement
A curve is enclosing constant area P. By means of variational calculus show, that the curve with minimal arc length is a circle,
Homework Equations
The Attempt at a Solution
[tex]F(t)= \int_{x_{1}}^{x_{2}}\sqrt{1+(f^{'})^{2}}dt[/tex]
[tex]G(t)= \int_{x_{1}}^{x_{2}}f(t)=const[/tex]
If i use Lagrange theorem for functionals i will get
[tex]\int_{x_{1}}^{x_{2}}=\sqrt{1+(f^{'})^{2}} +\lambda f dx [/tex]
Since, upper functional is not a function of T i can use Euler-Lagrange equation, after differentiation solution will be:
[tex]1+(f^{'})^{2}=\frac{1}{(C-\lambda f)^{2}}[/tex]
What should i do now ?