# Rotation conventions

Rotation conventions are a veritable minefield. It is truly remarkable how much confusion the subject has amassed over the years. *Spinach* uses the conventions specified by Brink and Satchler in their book on angular momentum. The book is included with the kernel (see kernel/conventions). The relevant pages are reproduced below.

## Sense and sign

*Spinach* uses the active convention - it always rotates the spin system, not the coordinate axes. Positive rotation is counter-clockwise, as shown in the figure below.

The function that builds rotation matrices is euler2dcm.m; it uses the Euler angle convention described below.

## Euler angles and Wigner functions

*Spinach* uses active ZYZ convention described in Page 20 of Brink and Satchler (reproduced below). Note particularly the text immediately after Equation 2.12 - the rotation specification in which the axes do not drift is easy to visualize.

Equation 2.13 is used by wigner.m to build Wigner D matrices.

*Version 2.2, authors: Ilya Kuprov*