27.1: Definition of Euler Angles

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The rotational motion of a rigid body is completely defined by tracking the set of principal axes \(\left(x_{1}, x_{2}, x_{3}\right)\), with origin at the center of mass, as they turn relative to a set of fixed axes (X,Y,Z). The principal axes can be completely defined relative to the fixed set by three angles: the two angles \((\theta, \phi)\) fix the direction of \(x_{3}, \text { but that leaves the pair } x_{1}, x_{2}\) free to turn in the plane perpendicular to \(x_{3}\), the angle \(\psi\) fixes their orientation.

To see these angles, start with the fixed axes, draw a circle centered at the origin in the horizontal X,Y plane. Now draw a circle of the same size, also centered at the same origin, but in the principal axes \(x_{1}, x_{2}\) plane. Landau calls the line of intersection of these circles (or discs) the line of nodes. It goes through the common origin, and is a diameter of both circles.

The angle between these two planes, which is also the angle between Z, \(x_{3}\) (since they’re the perpendiculars to the planes) is labeled \(\theta\).

The angle between this line of nodes and the X axis is \(\phi\). It should be clear that \(\theta\), \(\phi\) together fix the direction of \(x_{3}\), then the other axes are fixed by giving \(\psi\), the angle between \(x_{1}\) and the line of nodes ON. The direction of measurement of \(\phi\), \(\psi\) around Z, \(x_{3}\) are given by the right-hand or corkscrew rule.