# Cylindrical coordinate of Galilean transformation

• Pring
In summary, a cylindrical coordinate system is a three-dimensional coordinate system that uses two perpendicular axes and a vertical axis to represent a point in space. The Galilean transformation is applied to cylindrical coordinates by converting them between reference frames, taking into account relative motion. Using cylindrical coordinates in the Galilean transformation has advantages for problems with cylindrical symmetry and certain types of motion. The Galilean transformation does not affect the coordinate axes in cylindrical coordinates, and it is appropriate to use with cylindrical coordinates when the relative motion between frames is linear and uniform. This is commonly used in classical mechanics to describe the motion of objects.

#### Pring

$r\rightarrow r-2qz$ and $\psi\rightarrow\psi+q\cdot(r-qz)$, I don't know how to derive it, anybody know?

This question results from the book "Optical Solitons: From Fibers to Photonic Crystals [1 ed.]" section 6.5

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Does the expression or theorem have a name? Perhaps you can search on it to see what pops up.

## 1. What is a cylindrical coordinate system?

A cylindrical coordinate system is a three-dimensional coordinate system that uses two perpendicular axes (usually labeled as r and z) and a vertical axis (labeled as θ) to represent a point in space.

## 2. How is the Galilean transformation applied to cylindrical coordinates?

The Galilean transformation is applied to cylindrical coordinates by converting the coordinates from one reference frame to another, taking into account the relative motion between the two frames. This involves a translation in the r and z directions and a rotation in the θ direction.

## 3. What are the advantages of using cylindrical coordinates in the Galilean transformation?

One advantage of using cylindrical coordinates in the Galilean transformation is that they are well-suited for problems with cylindrical symmetry, such as those involving circular or cylindrical objects. Additionally, they can simplify calculations for certain types of motion, such as rotational motion.

## 4. How does the Galilean transformation affect the coordinate axes in cylindrical coordinates?

The Galilean transformation does not affect the coordinate axes in cylindrical coordinates. They remain perpendicular to each other and maintain their orientation in relation to the point in space being described.

## 5. When is it appropriate to use the Galilean transformation with cylindrical coordinates?

The Galilean transformation with cylindrical coordinates is appropriate in situations where the relative motion between two reference frames is linear and uniform. This is often the case in classical mechanics, where the Galilean transformation is commonly used to describe the motion of objects in different frames of reference.