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I am currently taking a class in which we are working on calculus of variations, but I was given the following problem:

A circular column of water of radius $l$ is rotated about its vertical axis atconstant angular velocity, $ω$, so that the surface is a radially symmetric function whose low point is at the center of the circular cross-section. The (upper) free surface assumes a shape which preserves the volume $$V=2\pi \int_{0}^{l} xy(x)dx$$

and minimizes the potential energy $$\rho \pi\int_0^l [gy^2(x)-\omega^2 x^2y(x)]xdx$$.

$ρ$ is the density of the water, g is the standard gravitational constant and $y(x)$ is the height of the water at a radial distance $x$ from the center.

Now, I've looked at the 3 separate cases for applying the E-L equation, but as far as I know, they're only for $F(y'), F(x,y')$ and $F(y,y')$. Is there something I'm missing? Or any way to work out how to find $y(x)$, or minimize $\omega$?

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    $\begingroup$ The second formula is not an equation. And the first formula seems like a definition for $V$. Could you write exactly the problem that you were given? $\endgroup$
    – Andrei
    Mar 28, 2021 at 18:05
  • $\begingroup$ @Andrei I've added the entire problem $\endgroup$ Mar 28, 2021 at 18:10

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The Euler-Lagrange equation is still valid. In your case, the Lagrangian $\mathscr{L}$ is given by $$\mathscr{L}(x, y, y')=\varrho \pi (gy^2-\omega^2x^2 y)+\lambda 2\pi x y$$ So the Euler-Lagrange equation is: $$\frac{\partial \mathscr{L}}{\partial y}=0$$

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  • $\begingroup$ OHH yes alright, this helped a lot, thank you! $\endgroup$ Mar 28, 2021 at 19:24

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