Rotating frame transformation

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This tutorial needs review

1 Introduction
2 Rotating vector in a fixed frame
3 Fixed vector, rotating frame
4 Full rotating frame transformation

Introduction

Rotating frame transformation is often used in NMR spectroscopy to simplify description of magnetization vector in the magnetic field.

Behavior of a vector can be described mathematically if vector's derivative is known. Therefore the goal of this transformation is to establish relationship of vector's derivatives in both frames.

Perhaps it is easier to first calculate derivative of a vector rotating in a fixed laboratory frame and after that switch to the case of static vector and rotating frame.

In the end we can add motion of vector in the laboratory frame and get the final formula.

Rotating vector in a fixed frame

In fixed frame, imagine vector A rotating with a positive angular frequency ω. Direction of such rotation of A will be by convention - counter-clockwise.

Let's determine derivative $\frac{\vec{\delta A}}{\delta t}$ .

First calculate $\vec{\delta A}$ - how much $\vec{A}$ changes within time $\displaystyle \delta t$ :

$\vec{\delta A} = \delta\phi \cdot |\vec{a}| \cdot \vec{n}$

In the equation above $\displaystyle \delta \phi$ - rotation angle, $\vec{a}$ - projection of $\vec{A}$ on the plane normal to $\displaystyle \vec{\omega}$ , and $\vec{n}$ - unit vector collinear with $\vec{\delta A}$

Notice that $\vec{\delta A}$ is normal to the plane formed by $\vec{\omega}$ and $\vec{A}$ .

Rotation angle can be expressed as

$\delta \phi = |\vec{\omega}| \cdot \delta t$

So, we have

$\vec{\delta A} = |\vec{\omega}| \cdot |\vec{a}| \cdot \vec{n} \cdot \delta t = (\vec{\omega}\times\vec{A}) \cdot \delta t$

In the previous step the definition of vector cross product was used. Therefore:

$\frac{\vec{\delta A}}{\delta t} = (\vec{\omega}\times\vec{A})$

Fixed vector, rotating frame

If instead $\vec{A}$ is fixed in the laboratory frame, but the reference frame is rotating in the same direction and rate as above, our derivative of vector within the rotating frame will simply equal above with the minus sign:

$\frac{\vec{\delta A'}}{\delta t} = -(\vec{\omega}\times\vec{A'}) = (\vec{A'}\times\vec{\omega})$

Full rotating frame transformation

Now we have $\vec{A}$ also changing in the static laboratory frame with a derivative $\frac{\vec{d A}}{d t}$ .

Then full transformation will look like:

$\frac{\vec{\delta A'}}{\delta t} = \left(\frac{\vec{d A}}{d t}\right)' + (\vec{A'}\times\vec{\omega})$

where $\displaystyle \vec{A'}$ is coordinate of the vector in the rotating frame,

$\vec{A}$ is coordinate of the vector in the laboratory frame, $\left(\frac{\vec{d A}}{d t}\right)'$ is $\frac{\vec{d A}}{d t}$ , recalculated in the new rotating coordinates,