Particle on a torus knot: a Hamiltonian analysis

TL;DR

This paper analyzes the dynamics and symmetries of a particle constrained to a torus knot using Hamiltonian methods, revealing noncommutative structures and exploring motion both on and off specific tori.

Contribution

It provides a Hamiltonian framework for particle motion on torus knots, including the derivation of Dirac brackets and analysis of symmetries and fluctuations.

Findings

Hamiltonian system is second class in Dirac's formulation

Dirac brackets reveal noncommutative structures

Motion equations derived for general and specific torus knots

Abstract

We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac's formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific torus. The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed torus knot.