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Discrete mechanics on unitary octonions

Janusz Grabowski, Zohreh Ravanpak

Abstract

In this article we generalize the discrete Lagrangian and Hamiltonian mechanics on Lie groups to non-associative objects generalizing Lie groups (smooth loops). This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. As a working example we obtain the discrete Lagrangian and Hamiltonian mechanics on unitary octonions.

Discrete mechanics on unitary octonions

Abstract

In this article we generalize the discrete Lagrangian and Hamiltonian mechanics on Lie groups to non-associative objects generalizing Lie groups (smooth loops). This shows that the associativity assumption is not crucial for mechanics and opens new perspectives. As a working example we obtain the discrete Lagrangian and Hamiltonian mechanics on unitary octonions.

Paper Structure

This paper contains 8 sections, 8 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Smooth loops
  3. Discrete mechanics on smooth loops
  4. Lagrangian formalism
  5. Hamiltonian formalism
  6. The smooth loop of octonions
  7. Mechanics on octonions
  8. Concluding remarks

Key Result

Theorem 1

Invertible octonions ${\mathbb O}^\times$ form a smooth inverse Moufang loop under the octonion multiplication..

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Definition 1
  • Proposition 1
  • Theorem 3
  • proof
  • Definition 2
  • Theorem 4
  • proof
  • ...and 7 more