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It's actually quite astonishing that this term is nowadays only understood as some synonym for numerical approximations of differential equations stemming from the treatment of mechanical systems. Whereas traditionally this topic has been in the hand of continous mathematics the discrete (finite) incarnation lives in its very own right!
Some more concrete aspects now about our abstract physical systems: A Chess piece movement could be seen as a kinematic system, having velocity, energy. There simply must be a due treatment possible regarding relativity and dynamics (s.th. like principle of least action).
Let's treat a 2-body system. One can create the following system of two coupled linear difference equations, which can (since they are linear this is for sure) be rewritten to make them explicit in highest order and then be used to run some (Runge-Kutta and friends) forward simulation (simultaneously or better taking turns). In Equation 17.1, “Simple Dynamical 2-Body System” the bold font indicates that we have s.th. like vectors, and t denotes the time coordinate of course.
Let's think in the context of our holonomy groups now. Integrating the acceleration gives the velocity, and summing up the latter gives to current space(-time) 'vector'. The interesting case is of course to keep the velocity to length one, so there is some delay (the more the longer the velocity is already) for acceleration to take effect: is this s.th. like mass (inertia)? Otherwise those pieces are moving around in a way not in accordance with the Chess rules. Of course this can be generalized to varying acceleration.
(results, pics or whatever, a link to a Java class for animation, await next release)
Another goal should be to solve this system 'in analytically closed form' (z-transformation or tricks from recursions theory). And those linear 'greedy' forces are only a first shot. Yet another question afterwards is what sort of forces (potential, whatever) is actually appropriate to enforce an optimal Chess play. It's a framework where deterministic algorithmic strategies can be expressed and investigated (integrated) in the context of an integrable system. And anyway we have the transition from ordinary to partial difference equations as another tool to apply.
Symmetry theory, finite analogs to continous (Lie) transformations, invariants and conserved quantities, Noether theorem and perhaps even gauge theory might help to get into the matter. Anyway instead of using forces (Newton) one could try more elaborate things (Lagrange, Hamilton), but let's always keep to the ground and focus on real working examples :-)
Another point of view would be to think about what sort of knots and links arise from the piece movements.