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

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## 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## Notes

### Acknowledgments

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.

## References

- Arnold V (2010) Mathematical methods of classical mechanics, 2nd edn, corr. 4th printing. Springer, HeidelbergGoogle Scholar
- Bach T (2007) Methodology and implementation of a software architecture for cellular and lattice-gas automata programming. PhD thesis, Boston University. Also in http://www.ioc.ee/~silvio/nrg/
- Bennett CH, Margolus N, Toffoli T (1988) Bond-energy variables for Ising spin-glass dynamics. Phys Rev B 37:2254CrossRefGoogle Scholar
- Boykett T, Kari J, Taati S (2008) Conservation laws in rectangular CA. J Cell Automata 3:115–122MathSciNetMATHGoogle Scholar
- Complements to this paper (2012) http://www.ioc.ee/~silvio/nrg/
- Creutz M (1983) Microcanonical Monte Carlo simulation. Phys Rev Lett 50:1411–1414MathSciNetCrossRefGoogle Scholar
- Doolen G et al (eds) (1988) Lattice gas methods for partial differential equations. Addison–Wesley, BostonGoogle Scholar
- Feynman R, Leighton R, Sands M (1963) Conservation of energy. The Feynman Lectures on Physics, vol 1, Sections 4-1–4-8. Addison–Wesley, Boston (“Feynman’s blocks”) can be read in their entirety in http://www.ioc.ee/~silvio/nrg/
- Gibbs J (1902) elementary principles in statistical mechanics. Developed with especial reference to the rational foundation of thermodynamics. Scribners, LondonGoogle Scholar
- Goldstein H, Poole C, Safko J (2001) Classical mechanics, 3rd edn. Addison–Wesley, BostonGoogle Scholar
- Grant B (2011) News in a nutshell. The Scientist, 31 MarchGoogle Scholar
- Hardy J, de Pazzis O, Pomeau Y (1976) Molecular dynamics of a classical lattice gas: transport properties and time correlation functions. Phys Rev A13:1949–1960CrossRefGoogle Scholar
- Kari J (1996) Representation of reversible cellular automata with block permutations. Math Syst Theory 29:47–61MathSciNetMATHGoogle Scholar
- Kari J (2005) Theory of cellular automata: a survey. Theor Comput Sci 334:3–33MathSciNetMATHCrossRefGoogle Scholar
- Lanczos C (1986) The variational principles of mechanics, 4th edn. Dover, New YorkGoogle Scholar
- Landau L, Lifshitz E (1976) Mechanics, 3rd edn. Oxford: Pergamon.Google Scholar
- Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, CambridgeGoogle Scholar
- Lind D (2004) Multi-dimensional symbolic dynamics. Symbolic dynamics and its applications. American Mathematical Society, Providence, pp 61–79; also in Proceedings of Symposium of Applied Mathematics, 60Google Scholar
- Mackey G (1963) Mathematical foundations of quantum mechanics. Benjamin, New YorkGoogle Scholar
- Margolus N, Toffoli T, Vichniac G (1986) Cellular-automata supercomputers for fluid dynamics modeling. Phys Rev Lett 56:1694–1696CrossRefGoogle Scholar
- Noether E (1918) Invariante Variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse 235–257 (English translation at arxiv.org/abs/physics/0503066v1)Google Scholar
- Pomeau Y (1984) Invariant in cellular automata. J Phys A 17:L415–L418MathSciNetCrossRefGoogle Scholar
- Toffoli T (1984) Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics. Physica D 10:117–127MathSciNetCrossRefGoogle Scholar
- Toffoli T (1999) Action, or the fungibility of computation. In Hey A (ed) Feynman and computation. Perseus, Reading, pp 349–392Google Scholar
- Toffoli T, Capobianco S, Mentrasti P (2004) A new inversion scheme, or how to turn second-order cellular automata into lattice gases. Theor Comput Sci 325:329–344MathSciNetMATHCrossRefGoogle Scholar
- Tyagi A (1994) A principle of least computational action. In: Workshop on physics and computation. IEEE Computer Society Press, Los Alamitos, pp 262–266Google Scholar
- Vichniac G (1984) Simulating physics with cellular automata. Physica D 10:96–115MathSciNetCrossRefGoogle Scholar