On novel Hamiltonian description of the nonholonomic Suslov problem

TL;DR

This work addresses reformulating the nonholonomic Suslov problem within a Hamiltonian framework by constructing invariant Poisson bivectors. It identifies two rank-four invariant bivectors, $P_a$ and $P_b$, that produce cubic Poisson brackets with globally defined Casimir functions, enabling Hamiltonian descriptions $X=P_a H_a$ and $X=P_b H_b$ on regular leaves. For gyrostat variations and the Suslov system in a potential field, it also derives rank-two invariant bivectors (e.g., $P_c$, $P_d$, and $P_e$ in special cases) that yield formal Hamiltonian representations with two Casimirs, though the full leaf structure and global coordinates remain open problems. Overall, the results extend Hamiltonian-style analysis to a class of nonholonomic systems, offering a path to Darboux-coordinate descriptions on leaves in the rank-four case while outlining current limitations for rank-two structures.

Abstract

We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem. Two rank four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets on the five dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.