This chapter discusses difference equations and how they arise from the temporal discretization of differential equations, emphasizing that temporal discretization is not a unique process. It shows how discrete dynamical systems can be described by difference equations. To do this the chapter first explains how discrete time mechanics action sums are related to continuous time action integrals, introducing the concept of system function. System functions are the discrete time analogues of Lagrangians in continuous time mechanics. The chapter discusses their relationship to Hamilton’s principal function. It uses a discrete time action principle to derive the discrete time equations of motion associated with a given system function. The chapter discusses Caldirola’s microverse model of the electron and show how Maxwell’s continuous time partial differential equations for electrodynamics can be discretized in terms of discrete-time difference equations.
Keywords: difference equations, temporal discretization, discrete dynamical systems, action sums, system functions, Hamilton’s principal function, Cadzow equations of motion, Caldirola’s microverse, discrete time electrodynamics
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