The mechanical and the wave-theoretical aspects of momentum considering discrete action

The mechanical aspect of momentum, basically its role as a tangent vector of the trajectory of the particle, is related to properties of the momentum found in the contexts of Hamilton's optico-mechanical analogy, de Broglie's matter waves, and quantum mechanics. These properties are treated in a systematic way by considering an approximation of the particle mechanical action of the particle by a step function. A special method of discretizing partial differential equations is shown to be required. Using this method, a discrete dynamics is developed. It is shown that particle dynamics can be regarded as the limit case of the discrete dynamics as the step functions tend to the continuous ones. The equation of motion of a free particle in an arbitrary reference system is deduced in two ways: (i) in continuous dynamics by making use of the invariance of action within changes of reference systems, and (ii) by taking the mentioned limit in discrete dynamics of an equation which expresses that the mechanical and wave-theoretical aspects of the momentum are interrelated in specific way.