Angular momentum first-order derivatives

Unfortunately, the simple aufbau principle, which only involves angular momentum together with a fixed scheme for the ordering of the energy levels derived from observations of the alkali spectra (see chapter 2) fails beyond the first three periods, and straightforward ordering of the subshells breaks down. The reasons for this failure are interesting and important, and will be taken up in detail in chapter 5. 

The description of rotational motion is naturally performed in spherical coordinates. The two angular variables of rotational motion are generalized coordinates, free of additional constraints, as introduced in chapter 2. The transformation to spherical coordinates affects the definition of the angular momentum (operator) and subsequently the squared angular momentum (operator) which enters the kinetic energy (operator) expression. In order to avoid lengthy coordinate transformations of the latter containing second derivatives with respect to Cartesian coordinates, we may consider the situation in classical mechanics first and subsequently apply the correspondence principle. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

---