Hamiltonicity and integrability of the Suslov problem

The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.     

Feedback Integrators for Nonholonomic Mechanical Systems

The theory of feedback integrators is extended to handle mechanical systems with nonholonomic constraints with or without symmetry, so as to produce numerical integrators that preserve the nonholonomic constraints as well as other conserved quantities. To extend the feedback integrators, we develop a suitable extension theory for nonholonomic systems and also a corresponding reduction theory for systems with symmetry. It is then applied to various nonholonomic systems such as the Suslov problem on \({\text {SO}}(3)\), the knife edge, the Chaplygin sleigh, the vertical rolling disk, the roller racer, the Heisenberg system, and the nonholonomic oscillator.

     

带一类第一积分的非完整系统的Szebehely问题

研究非完整系统动力学的一类逆问题·给出非完整系统的运动方程及其显式，考虑一类仅受齐次非完整约束的力学系统的Ｓｚｅｂｅｈｅｌｙ问题，研究已知一类第一积分的一般非完整系统的情形·最后举例说明其应用·     

A generalization of Chaplygin’s Reducibility Theorem

In this paper we study Chaplygin’s Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincaré-d’Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincaré-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems.     

Reduction of almost Poisson brackets for nonholonomic systems on Lie groups

We present a systematic geometric construction of reduced almost Poisson brackets for nonholonomic systems on Lie groups with invariant kinetic energy metric and constraints. Our construction is of geometric interest in itself and is useful in the hamiltonization of some classical examples of nonholonomic mechanical systems.      

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

Energy-Preserving Integrators Applied to Nonholonomic Systems

We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple \(({{\mathcal {D}}}^*, \varPi , \mathcal {H})\), where \({{\mathcal {D}}}^*\) is the dual of the vector bundle determined by the nonholonomic constraints, \(\varPi \) is an almost-Poisson bracket (the nonholonomic bracket) and \( \mathcal {H}: {{\mathcal {D}}}^*\rightarrow \mathbb {R}\) is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.

     

转动系统相对论性动力学方程的代数结构与Poisson积分

研究转动相对论系统动力学方程的代数结构,得到了完整保守转动相对论系统与特殊非完整转动相对论系统动力学方程具有Lie代数结构;一般完整转动相对论系统、一般非完整转动相对论系统动力学方程具有Lie容许代数结构。并给出转动相对论系统动力学方程的Poisson积分。     

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot). This work was partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, MTM 2006-10531, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM.     

非Chetaev型非完整系统的Lie对称性与Noether对称性

研究非Ｃｈｅｔａｅｖ型非完整系统的Ｌｉｅ对称性与Ｎｏｅｔｈｅｒ对称性,具体研究了非Ｃｈｅｔａｅｖ型常 质量非完整系统和非Ｃｈｅｔａｅｖ型变质量非完整系统的Ｌｉｅ对称性与Ｎｏｅｔｈｅｒ对称性．给出Ｌｉｅ对称 性导致Ｎｏｅｔｈｅｒ对称性以及Ｎｏｅｔｈｅｒ对称性导致Ｌｉｅ对称性的条件．     

The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions

We consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution ℜ°, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution ℜ°. Since the fibers of ℜ° contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given.      

Четаев型非完整力学系统的微分几何原理

本文应用现代微分几何的方法研究Четаев型非完整力学系统.通过恰当地定义Четаев型约束Pfaff系统,给出了非完整力学系统的微分几何结构,从而将带有非完整约束的Lagrange方程表达为一种与坐标无关的不变形式,并且采用这个新观点讨论了约束的嵌入和非完整力学系统的守恒定律等问题,得到了约束子流形上的Noether型定理.     

哈密顿-雅可比方法对非线性非完整系统的适用性

本文用庞加莱(Poincaré)形式将哈密顿-雅可比方法(Hamilton-Jacobi method)推广到具有非线性非完整约束的动力系统的运动情况.研究了本方法对该系统推广的必要条件和充分条件.并用非完整系统的某些具体实例进行了说明.     

Poincaré transformations in nonholonomic mechanics

We report on the application of the Poincaré transformation (from the theory of adaptive geometric integrators) to nonholonomic systems—mechanical systems with non-integrable velocity constraints. We prove that this transformation can be used to express the dynamics of certain nonholonomic systems at a fixed energy value in Hamiltonian form; examples and potential applications are also discussed.     

On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability

For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10. pp. 1411–1416, October, 1999.     

On instability of the equilibrium state of nonholonomic systems

We establish a criterion of instability for the equilibrium state of nonholonomic systems, in which gyroscopic forces may dominate over potential forces. We show that, similarly to the case of holonomic systems, the evident domination of gyroscopic forces over potential ones is not sufficient to ensure the equilibrium stability of nonholonomic systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 389–397, March, 1999.     

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S^n-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).     

积分非完整可控力学系统运动方程的场方法

本文将场方法推广应用于积分非完整可控力学系统的运动方程,并举例说明方法的应用.