Conserved quantities in discrete dynamics: what can be recovered from Noether’s theorem, how, and why? Article First Online: 08 August 2012 DOI :
10.1007/s11047-012-9336-7

Cite this article as: Capobianco, S. & Toffoli, T. Nat Comput (2012) 11: 565. doi:10.1007/s11047-012-9336-7
Abstract The connections between symmetries and conserved quantities of a dynamical system brought to light by Noether’s theorem depend in an essential way on the symplectic nature of the underlying kinematics. In the discrete dynamics realm, a rather suggestive analogy for this structure is offered by second-order cellular automata. We ask to what extent the latter systems may enjoy properties analogous to those conferred, for continuous systems, by Noether’s theorem. For definiteness, as a second-order cellular automaton we use the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that—and why—energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant. We then begin to explore the issue of whether, in these systems, it may hold as well that translation invariance entails momentum conservation.

Keywords Analytical mechanics of cellular automata Second-order dynamics Energy conservation Energy as generator of the dynamics Noether’s theorem in discrete systems

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Authors and Affiliations 1. Institute of Cybernetics Tallinn University of Technology Tallinn Estonia 2. ECE Department Boston University Boston USA