The geometric structure of nonholonomic mechanics

Many important problems in multibody dynamics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory. Despite considerable advances on both Hamiltonian and Lagrangian sides of the theory, there remains much to do. We report on progress on two of these fronts. The first is a Poisson description of the equations that is equivalent to those given by Lagrangian reduction, and second, a deeper understanding of holonomy for such systems. These results promise to lead to further progress on the stability issues and on locomotion generatio

The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems

This paper compares the Hamiltonian approach to systems with nonholonomic constraints (see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke [1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996] and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the approaches and each has its own advantages; some structures have been discovered on one side and their analogues on the other side are interesting to clarify. For example, the momentum equation and the reconstruction equation were first found on the Lagrangian side and are useful for the control theory of these systems, while the failure of the reduced two form to be closed (i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the Hamiltonian side. Clarifying the relation between these approaches is important for the future development of the control theory and stability and bifurcation theory for such systems. In addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum phase) control system

Poisson reduction for nonholonomic mechanical systems with symmetry

This paper continues the work of Koon and Marsden [1997b] that began the comparison of the Hamiltonian and Lagrangian formulations of nonholonomic systems. Because of the necessary replacement of conservation laws with the momentum equation, it is natural to let the value of momentum be a variable and for this reason it is natural to take a Poisson viewpoint. Some of this theory has been started in van der Schaft and Maschke [1994]. We build on their work, further develop the theory of nonholonomic Poisson reduction, and tie this theory to other work in the area. We use this reduction procedure to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic momentum equation and the reduced Lagrange d’Alembert equations in Hamiltonian form. We also show that these equations are equivalent to those given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden and Murray [1996]. Because of the results of Koon and Marsden [1997b], this is also equivalent to the results of Bates and Sniatycki [1993], obtained by nonholonomic symplectic reduction. Two interesting complications make this effort especially interesting. First of all, as we have mentioned, symmetry need not lead to conservation laws but rather to a momentum equation. Second, the natural Poisson bracket fails to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum that vanishes when the Jacobi identity holds), or equivalently, the Schouten bracket, is an interesting expression involving the curvature of the underlying distribution describing the nonholonomic constraints. The Poisson reduction results in this paper are important for the future development of the stability theory for nonholonomic mechanical systems with symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they should be useful for the development of the powerful block diagonalization properties of the energy-momentum method developed by Simo, Lewis and Marsden [1991]

Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden and Murray [1995]. The snakeboard is used to illustrate the method

Quartically hyponormal weighted shifts need not be 3-hyponormal

We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal

A Gravity Dual of RHIC Collisions

In the context of the AdS/CFT correspondence we discuss the gravity dual of a heavy-ion-like collision in a variant of ${\cal N}=4$ SYM. We provide a gravity dual picture of the entire process using a model where the scattering process creates initially a holographic shower in bulk AdS. The subsequent gravitational fall leads to a moving black hole that is gravity dual to the expanding and cooling heavy-ion fireball. The front of the fireball cools at the rate of $1/\tau$, while the core cools as $1/\sqrt{\tau}$ from a cosmological-like argument. The cooling is faster than Bjorken cooling. The fireball freezes when the dual black hole background is replaced by a confining background through the Hawking-Page transition.Comment: 25 pages, 8 figures, Added references, Falling picture elucidate

Intramolecular energy transfer and the driving mechanisms for large-amplitude collective motions of clusters

This paper uncovers novel and specific dynamical mechanisms that initiate large-amplitude collective motions in polyatomic molecules. These mechanisms are understood in terms of intramolecular energy transfer between modes and driving forces. Structural transition dynamics of a six-atom cluster between a symmetric and an elongated isomer is highlighted as an illustrative example of what is a general message. First, we introduce a general method of hyperspherical mode analysis to analyze the energy transfer among internal modes of polyatomic molecules. In this method, the (3n−6) internal modes of an n-atom molecule are classified generally into three coarse level gyration-radius modes, three fine level twisting modes, and (3n−12) fine level shearing modes. We show that a large amount of kinetic energy flows into the gyration-radius modes when the cluster undergoes structural transitions by changing its mass distribution. Based on this fact, we construct a reactive mode as a linear combination of the three gyration-radius modes. It is shown that before the reactive mode acquires a large amount of kinetic energy, activation or inactivation of the twisting modes, depending on the geometry of the isomer, plays crucial roles for the onset of a structural transition. Specifically, in a symmetric isomer with a spherical mass distribution, activation of specific twisting modes drives the structural transition into an elongated isomer by inducing a strong internal centrifugal force, which has the effect of elongating the mass distribution of the system. On the other hand, in an elongated isomer, inactivation of specific twisting modes initiates the structural transition into a symmetric isomer with lower potential energy by suppressing the elongation effect of the internal centrifugal force and making the effects of the potential force dominant. This driving mechanism for reactions as well as the present method of hyperspherical mode analysis should be widely applicable to molecular reactions in which a system changes its overall mass distribution in a significant way

A new approach to the 2-variable subnormal completion problem

We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion