prolate symmetric rotors asymmetric rotors This classification depends on the relative magnitudes of the principal moments of inertia. Coordinates of the rigid rotor Different branches of physics and engineering use different coordinates for the description of the kinematics of a rigid rotor. In molecular physics Euler angles are used almost exclusively. In quantum mechanical applications it is advantageous to use Euler angles in a convention that is a simple extension of the physical convention of spherical polar coordinates. The first step is the attachment of a right-handed orthonormal frame (3-dimensional system of orthogonal axes) to the rotor (a body-fixed frame) . This frame can be attached arbitrarily to the body, but often one uses the principal axes frame—the normalized eigenvectors of the inertia tensor, which always can be chosen orthonormal, since the tensor is symmetric. When the rotor possesses a symmetry-axis, it usually coincides with one of the principal axes. It is convenient to choose as body-fixed z-axis the highest-order symmetry axis. One starts by aligning the body-fixed frame with a space-fixed frame (laboratory axes), so that the body-fixed x, y, and z axes coincide with the space-fixed X, Y, and Z axis. Secondly, the body and its frame are rotated actively over a positive angle around the z-axis (by the right-hand rule), which moves the - to the -axis. Thirdly, one rotates the body and its frame over a positive angle around the -axis. The z-axis of the body-fixed frame has after these two rotations the longitudinal angle (commonly designated by ) and the colatitude angle (commonly designated by ), both with respect to the space-fixed frame. If the rotor were cylindrical symmetric around its z-axis, like the linear rigid rotor, its orientation in space would be unambiguously specified at this point. If the body lacks cylinder (axial) symmetry, a last rotation around its z-axis (which has polar coordinates and ) is necessary to specify its orientation completely. Traditionally the last rotation angle is called . The convention for Euler angles described here is known as the convention; it can be shown (in the same manner as in this article) that it is equivalent to the convention in which the order of rotations is reversed. The total matrix of the three consecutive rotations is the product Let be the coordinate vector of an arbitrary point in the body with respect to the body-fixed frame. The elements of are the