Discrete mechanics on unitary octonions

TL;DR

This paper extends discrete Lagrangian and Hamiltonian mechanics from Lie groups to non-associative structures like unitary octonions, demonstrating that associativity is not essential for formulating mechanics and opening new research avenues.

Contribution

It introduces a framework for mechanics on non-associative smooth loops, specifically applying it to unitary octonions, which broadens the scope of geometric mechanics.

Findings

Mechanics can be formulated on non-associative algebraic structures.

Discrete Lagrangian and Hamiltonian mechanics are developed for unitary octonions.

The associativity assumption is shown to be non-essential for mechanics.

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.