Converted Cartesian coordinates to polar coordinates

[touqra]

I don't know where have I gone wrong...
I converted Cartesian coordinates to polar coordinates:

$$\frac{\partial^2\Psi}{\partial x^2} +\frac{\partial^2\Psi}{\partial y^2}= \frac{1}{2}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2 }{\partial y^2})\Psi^2 - \Psi(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\Psi =\frac{\partial^2\Psi}{\partial r^2}+ \frac{1}{r^2}\frac{\partial^2\Psi}{\partial \Phi^2}$$

But on the left hand side (the Cartesian components) is just the Laplacian in 2D, but the final answer I got for the polar components is not equivalent to the Laplacian for polar coordinate system.
I'm missing the term $$\frac{1}{r}\frac{\partial\Psi}{\partial r}$$

 

[AlephZero]

You are misunderstanding something here.

$${\frac{\partial^2}{\partial x^2}\Psi^2 = \frac{\partial}{\partial x}\frac{\partial}{\partial x}\Psi^2 = \frac{\partial}{\partial x}( 2\Psi \frac{\partial \Psi}{\partial x}) = 2 ({\frac{\partial \Psi}{\partial x}})^2 + 2\Psi \frac{\partial^2 \Psi}{\partial^2 x}$$

That doesn't seem consistent with your (wrong) equation.

 

[touqra]

$$\frac{\partial^2\Psi}{\partial x^2} +\frac{\partial^2\Psi}{\partial y^2}= \frac{1}{2}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2 }{\partial y^2})\Psi^2 - \Psi(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\Psi=\frac{\partial^2\Psi}{\partial r^2}+ \frac{1}{r^2}\frac{\partial^2\Psi}{\partial \Phi^2}$$

Ooops, I typed the wrong stuffs. I'm sorry.
It should read:

$$(\frac{\partial\Psi}{\partial x})^2 +(\frac{\partial\Psi}{\partial y})^2= \frac{1}{2}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2 }{\partial y^2})\Psi^2 - \Psi(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\Psi$$

Next, I plug in the Laplacian for the polar coordinates, essentially cylindrical coordinate, with z constant, and I end up with:
$$(\frac{\partial\Psi}{\partial x})^2 +(\frac{\partial\Psi}{\partial y})^2= (\frac{\partial\Psi}{\partial r})^2+ \frac{1}{r^2}(\frac{\partial\Psi}{\partial \Phi})^2$$

Next, I am required to get the Euler Lagrange equation for a system. The above is just the potential part. The time derivative kinetic is just $$\frac{1}{2}m\dot{\Psi}^2$$
Taking the Euler Lagrange for the Cartesian, I end up with an expression from the potential part:
$$\frac{\partial^2\Psi}{\partial x^2} +\frac{\partial^2\Psi}{\partial y^2}$$
And this is just a Laplacian.

But when taking the Euler Lagrange for the polar coordinates, I end up with an expression $$\frac{\partial^2\Psi}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2\Psi}{\partial \Phi^2}$$
and this is not equal to the Laplacian for the polar coordinates.
I am missing $$\frac{1}{r}\frac{\partial\Psi}{\partial r}$$