Natural Computing

, Volume 11, Issue 4, pp 565–577

Conserved quantities in discrete dynamics: what can be recovered from Noether’s theorem, how, and why?


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


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.


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

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institute of CyberneticsTallinn University of TechnologyTallinnEstonia
  2. 2.ECE DepartmentBoston UniversityBostonUSA

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