Dirac structures in nonholonomic mechanics
Katarzyna Grabowska, Michalina Borczyńska, Joanna Majsak, Tomasz Sobczak
TL;DR
This work addresses how to obtain a universal, geometry-based phase description for nonholonomic systems with linear constraints, including cases with magnetic-like terms. It develops Dirac algebroids as linear almost Dirac structures on $E^*$ and integrates them into the Tulczyjew triple to generate dynamics that do not depend on a particular Hamiltonian or Lagrangian. The key contribution is a constraint-driven, metric-free framework that can describe both mechanical and gyroscopic potentials using the same Dirac structure, providing explicit relations via $D_L=\varepsilon(dL(E))$ and $D_H=\Lambda^#(dH(E^*))$ and extending to affine (magnetic) settings through affine morphisms. This approach offers a unifying perspective for nonholonomic mechanics and has potential to simplify and generalize phase descriptions for a broad class of constrained systems.
Abstract
The concept of a Dirac algebroid, which is a linear almost Dirac structure on a vector bundle, was designed to generate phase equations for mechanical systems with linear nonholonomic constraints. We apply it to systems with magnetic-like or gyroscopic potentials, that were previously described by means of almost Poisson structures. The almost Poisson structures present in the literature in this context were constructed using constraints, metrics and information about magnetic or gyroscopic potential present in the Hamiltonian function of the system. The Dirac algebroid we use is constructed out of constraints and canonical geometric structures of the underlying bundles and is universal in the sense that it is independent on the particular Hamiltonian or Lagrangian. We provide examples showing that using the same Dirac structure we can describe systems with different potentials, magnetic or mechanical, added freely to a function generating the dynamics.