Can Anything from Noether’s Theorem Be Salvaged for Discrete Dynamical Systems?

  • Silvio Capobianco
  • Tommaso Toffoli
Conference paper

DOI: 10.1007/978-3-642-21341-0_13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6714)
Cite this paper as:
Capobianco S., Toffoli T. (2011) Can Anything from Noether’s Theorem Be Salvaged for Discrete Dynamical Systems?. In: Calude C.S., Kari J., Petre I., Rozenberg G. (eds) Unconventional Computation. UC 2011. Lecture Notes in Computer Science, vol 6714. Springer, Berlin, Heidelberg

Abstract

The dynamics of a physical system is linked to its phase-space geometry by Noether’s theorem, which holds under standard hypotheses including continuity. Does an analogous theorem hold for discrete systems? As a testbed, we take 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.

Keywords

analytical mechanics of cellular automata second-order dynamics energy conservation energy as generator of the dynamics Noether’s theorem in the discrete 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Silvio Capobianco
    • 1
  • Tommaso Toffoli
    • 2
  1. 1.Inst. of Cybernetics at TUTTallinnEstonia
  2. 2.ECE Dept.Boston Univ.BostonUSA

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