What is it about?
(Uploaded 2018-04-18). When the orientation of a free rotor is described by the Euler angles of three body fixed axes relative to three space fixed axes, the angular momentum has components along all six axes. The expressions for these in terms of the Euler angles and their conjugate momenta look fairly complicate. However: The components along the non-orthonormal axes formed by the line of nodes and the two z-axes are simply the momenta conjugate to the angles - so if we know the components of the vectors bi-orthogonal to these axes , we also know the components of the angular momentum looked for. A central point in the present little paper is that it turns out to be very easy to draw these bi-orthogonal vectors in the figure illustrating the Euler angles and then read off their components along the space fixed as well as the body fixed. Then the angular momentum problem is solved. As a "bonus" we also get the 6 components of the angular velocities and of the body-fixed axes along the spacefixed - the so-called direction cosines.
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Why is it important?
For simplicity the description above builds on the tacit assumption that the rotating axes are fixed to a rigid body and that their origin is fixed at the origin of the fixed axes. But the theory actually also works under much more useful conditions: Let the positions of N particles be described by the position of their centre of mass, by some coordinates which fix the positions relative to the rotating frame - and by the Euler angles for that frame relative to the space fixed. If we then increase one of these angles infinitesimally, while all other coordinates are kept fixed, the particles move rigidly. The reason why this observation is sufficient for the angular momentum results is explained in the brief appendix, which should be immediately understood as soon as one has learned how a conjugate momentum is defined.
Perspectives
The present article can be seen as an interesting example of bi-orthogonal sets of vectors and - more importantly - as a preparation for angular momentum defined as the generator of infinitesimal rotations. The last point of view is particularly important within molecular spectroscopy where we often need a quantum mechanical description of nuclei and electrons relative to molecule (body-) fixed axes as well as to laboratory (space-) axes. Here both sets of angular momentum components, expressed in the Euler angles, are essential. Furthermore: The quantum mechanical expressions for the various components of the angular momentum are formally the same as the classical.
Dr Flemming Jørgensen
Nygårdsvej 43, 4700 Næstved http://www.naestved-gym.dk/
Read the Original
This page is a summary of: Euler angles, direction cosines, and angular momentum, American Journal of Physics, August 1981, American Association of Physics Teachers (AAPT),
DOI: 10.1119/1.12421.
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