# Is there any Euler-Lagrange equation for functions dependent on only $x$ and $y$ and not on $y'$?

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 $$\omega$$?

• 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? Mar 28, 2021 at 18:05
• @Andrei I've added the entire problem Mar 28, 2021 at 18:10

## 1 Answer

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: $$\frac{\partial \mathscr{L}}{\partial y}=0$$

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