If one can define such a thing as a discrete dynamical system, can a Lagrangian and/or Hamiltonian for this also be defined and if so how, i.e. what do these correspond to?
Apologies if this is "dynamical systems 101", or little more. If so, or perhaps in any event, I guess the most helpful answer might just be to cite a good reference where this is explained.
Geometry and Hamiltonian mechanics on discrete spaces (2004). This paper$^\ast$ also contains an overview of earlier publications on this topic:
The goal of this paper is to provide a discrete analogue of differential geometry, and to define on these discrete models a formal discrete Hamiltonian structure in space and time. While the discretization techniques themselves have been the subject of a great deal of research, not much is known about the formal mathematical/geometrical structure of the final discrete model in relation to the structure of the smooth model. What we mean by this is, for example, suppose we have a smooth model defined in the Hamiltonian framework, and we have all the associated structure on the cotangent bundle. Let us now discretize this model; what is then the associated discrete Hamiltonian structure?
$^\ast$Behind a paywall, regrettably, I can email it to you if you ask me, I guess this is "fair use".
