# Homework Help: Variational Calculus - Minimal Arc lenght for given surface

1. Dec 12, 2011

### Zaknife

1. The problem statement, all variables and given/known data
A curve is enclosing constant area P. By means of variational calculus show, that the curve with minimal arc length is a circle,

2. Relevant equations

3. The attempt at a solution

$$F(t)= \int_{x_{1}}^{x_{2}}\sqrt{1+(f^{'})^{2}}dt$$
$$G(t)= \int_{x_{1}}^{x_{2}}f(t)=const$$

If i use Lagrange theorem for functionals i will get

$$\int_{x_{1}}^{x_{2}}=\sqrt{1+(f^{'})^{2}} +\lambda f dx$$

Since, upper functional is not a function of T i can use Euler-Lagrange equation, after differentiation solution will be:
$$1+(f^{'})^{2}=\frac{1}{(C-\lambda f)^{2}}$$
What should i do now ?