# 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.

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