Natural Computing

, Volume 11, Issue 4, pp 565–577 | Cite as

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

  • Silvio Capobianco
  • Tommaso Toffoli


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 



This research was supported by the European Regional Development Fund (ERDF) through the Estonian Center of Excellence in Computer Science (EXCS), by the Estonian Science Foundation under grant no. 7520, and by the Estonian Ministry of Education and Research target-financed research theme no. 0140007s12.


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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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