Yong-Xin Guo

Birkhoffian formulations of nonholonomic constrained systems

Abstract Only for some special nonholonomic constrained systems can a canonical Hamiltonian structure be realized. Based on a reduction of a nonholonomic system to a conditional holonomic system, a universal symplectic structure for a constrained system can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics, which preserves symbiotic character among derivability from a variational principle, Lie algebra and symplectic geometry. Two examples are presented.

Nonholonomic versus vakonomic dynamics on a Riemann–Cartan manifold

For the Chaplygin’s nonholonomic constrained systems, the constraint manifold can be endowed with Riemann–Cartan geometric structure by nonholonomic mapping into a Riemann manifold. The two kinds of existing dynamics, nonholonomic dynamics and vakonomic dynamics, are compared in the framework of Riemann–Cartan geometry. It is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the Riemann–Cartan constraint manifold, respectively. If the metricity condition of Riemann–Cartan connection is satisfied, the torsion (contorsion) of the Riemann–Cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations.

Influence of nonholonomic constraints on variations, symplectic structure, and dynamics of mechanical systems

Based on a serious analysis of the Frobenius integrability condition for affine differential constraints that mechanical systems are subject to the necessary and sufficient conditions for coincidence of three kinds of unfree variations, the existence of simple symplectic structure of the constraint submanifold and equivalence of nonholonomic and vakonomic dynamics for the constrained systems are, respectively, obtained, which are all related with the Frobenius integrability condition in their special forms. Two illustrative examples are presented to verify the results.

Poincare-Cartan Integral Invariants of Nonconservative Dynamical Systems

Traditionally there do not exist integralinvariants for a nonconservative system in the phasespace of the system. For weak nonconservative systems,whose dynamical equations admit adjoint symmetries, there exist Poincare and Poincare-Cartanintegral invariants on an extended phase space, wherethe set of dynamical equations and their adjointequations are canonical. Moreover, integral invariantsalso exist for pseudoconservative dynamical systemsin the original phase space if the adjoint symmetriessatisfy certain condtions.

Lie symmetries and conserved quantities of the constraint mechanical systems on time scales

We introduce a new method to study Lie symmetries and conserved quantities of constraint mechanical systems which include Lagrangian systems, nonconservative systems and nonholonomic systems on time scales T . For the constraint mechanical systems on time scales, based on the transformation Lie group, we get a series of significant results including the variational principle of systems on time scales, the equations of motion, the determining equations, the structure equations, the restriction equations as well as the Lie theorems of the Lie symmetries of the systems on time scales. Furthermore, a set of new conserved quantities of the constraint mechanical systems on time scales are given. More significant is that this work unifies the theories of Lie symmetries of the two cases for the continuous and the discrete constraint mechanical systems by applying the time scales. And then taking the discrete ( T = ℤ ) nonholonomic system for example, we derive the corresponding discrete Lie symmetry theory. Finally, two examples are designed to illustrate these results.