# Transform a pde into rotating frame

by climbon
Tags: frame, rotating, transform
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 P: 18 Hi, I have an equation of the form; $$\frac{d}{dt}(W) = \omega \left(x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right) W + g \frac{\partial}{\partial y} W + k x \frac{\partial^2}{\partial y^2} W$$ I want to change it into the rotating frame using the transform; x = x' cos(wt) - y' sin(wt) y = x' sin(wt) + y' cos(wt) I have calculated the derivatives of these transforms to be; $$\frac{\partial}{\partial x} = -cos(\omega t) \frac{\partial}{\partial x'} - sin(\omega t) \frac{\partial}{\partial y'} \\ \frac{\partial}{\partial y} = -cos(\omega t) \frac{\partial}{\partial y'} + sin(\omega t) \frac{\partial}{\partial x'} \\ \frac{\partial^2}{\partial x^2} = -cos^(2)(\omega t) \frac{\partial^2}{\partial x'^2} - sin^(2)(\omega t) \frac{\partial^2}{\partial y'^2}$$ I am assuming I can just substitute these transforms for x, y and their derivatives into the original equation and this will give me the original equation in the rotating frame...but do I have to do something with the time derivative on the L.H.S of the original equation?? Thank you.

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