INTEGRABILITY AND STABILITY OF NONHOLONOMIC SYSTEMS
Author
ZENKOV, DMITRY V.
Year
1998
Advisor
BLOCH, ANTHONY M.
Abstract
In this thesis, methods of geometric mechanics have been used to study the integrability and stability of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. Unlike Hamiltonian dynamics, symmetries do not always lead to conservation laws as in the classical Noether theorem, but rather to an interesting momentum equation. This momentum equation has the structure of a parallel transport equation for momentum corrected by additional terms and plays an important role in both integrability and stability analyses. For a number of systems, the momentum equation is pure transport. For such systems we show that the relative equilibria cannot be asymptotically stable and find the energy-momentum conditions for stability of these equilibria. Often in this case of a pure transport momentum equation, the nonholonomic system is integrable. We show that the invariant manifolds of such integrable systems such as the Routh problem and the rolling disk are tori filled out with quasi-periodic motions. However, the way these tori are embedded in the phase space is quite different from that of the Liouville tori of integrable Hamiltonian systems. To carry out the stability analysis in the general non-pure transport case, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. We develop a new energy-based approach, which allows finding asymptotically stable equilibria. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer, and the rattleback top.
ZENKOV, DMITRY V.
Year
1998
Advisor
BLOCH, ANTHONY M.
Abstract
In this thesis, methods of geometric mechanics have been used to study the integrability and stability of nonholonomic systems (that is, mechanical systems with nonintegrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. Unlike Hamiltonian dynamics, symmetries do not always lead to conservation laws as in the classical Noether theorem, but rather to an interesting momentum equation. This momentum equation has the structure of a parallel transport equation for momentum corrected by additional terms and plays an important role in both integrability and stability analyses. For a number of systems, the momentum equation is pure transport. For such systems we show that the relative equilibria cannot be asymptotically stable and find the energy-momentum conditions for stability of these equilibria. Often in this case of a pure transport momentum equation, the nonholonomic system is integrable. We show that the invariant manifolds of such integrable systems such as the Routh problem and the rolling disk are tori filled out with quasi-periodic motions. However, the way these tori are embedded in the phase space is quite different from that of the Liouville tori of integrable Hamiltonian systems. To carry out the stability analysis in the general non-pure transport case, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. We develop a new energy-based approach, which allows finding asymptotically stable equilibria. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer, and the rattleback top.