So, this is a problem actually motivated by a physics problem that I was facing but is again a math question. So, in polar coordinates, we can write $x=r\cos(\theta)$ and $y=r\sin(\theta)$ and using that we can derive the expression for the laplacian- $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}=\frac{1}{r}\frac{\partial}{\partial r}+\frac{\partial^2}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}$$
But now, if I have a problem where suppose the body rotates in a circle of some constant radius R, then without any logical explanation, many authors write that the Laplacian transforms as follows- $$\nabla^2=\frac{1}{R^2}\frac{\partial^2}{\partial \theta^2}$$ For some reason, the derivatives with respect to r vanish and become 0. I understand that probably if I try to derive the Laplacian with $x=R\cos(\theta)$ and $y=R\sin(\theta)$ where $R$ is a constant, I would get the above expression. But suppose, I know the expression for the Laplacian in polar coordinates, is there any way to transform this so that I directly get the result for the transformed expression when $r$ is constant (which ultimately begs for the explanation of the expression above)? Thanks a lot.