On symmetries and conserved quantities in Nambu mechanics

TL;DR

This paper explores the relationship between symmetries and conserved quantities in Nambu mechanics, highlighting differences from Hamiltonian mechanics and analyzing the role of differential forms and invariants.

Contribution

It clarifies how symmetries in Nambu mechanics lead to integral invariants rather than conserved functions, contrasting with Hamiltonian mechanics.

Findings

Symmetries in Nambu mechanics produce integral invariants.

The difference stems from the shift in degrees of forms in equations of motion.

The paper provides a conceptual framework for understanding conserved quantities in Nambu systems.

Abstract

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number. The origin of the difference may be traced back to a shift in degrees of relevant forms present in equations of motion, or, alternatively, to a corresponding shift in degrees of relevant objects in action integral for Nambu mechanics.