Angular momentum central force problem

Using the area velocity D, see Eq. (1.16), which is a constant of motion, D = A in the classical case of the central force problem, one derives straightforwardly the following relations in standard polar coordinates r,

particle motion is in a plane perpendicular to the angular momentum vector L)... 

Generalization of quantum mechanics to D spatial dimensions is illustrated explicitly for a few elementary examples. For central force problems the sole effect is to augment the orbital angular momentum by I I + — 3). As shown by Rost, this relation holds even for... 

Therefore the radial equation for the probability amplitude of any >-dimensional central force problem is the same as that for D = 3, but with the orbital angular momentum given by Eq.(lO). This is the key result, establishing that D is isomorphic with the orbital angular momentum [5], such that D D 2 is equivalent to / + 1. 

At this point we are in a position to understand why physicists resolutely adhere to Newton s three laws of mechanics while engineers always adopt some form of Euler s two laws. The physicist, with an overriding Interest in the motion of particles, finds it convenient to tacitly accept the central force law in the discussion of non-relativlstic mechanics since this idea is easily altered when relativistic problems are encountered. If the physicist were to adopt Euler s two axioms of mechanics, the second axiom would require alteration when relativistic problems arise. Engineers, on the other hand, are immersed in the study of continua and Euler s laws for linear and angular momentum are perfectly suited to their purposes which rarely include relativistic effects. 

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