Nonholonomic Hamiltonian Systems Research Articles

Variational reduction of Hamiltonian systems with general constraints

In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a variational reduction procedure has already been developed for Hamiltonian systems without constraints. In this paper we present a procedure of the same kind, but for the entire class of the higher order constrained systems (HOCS), described in the Hamiltonian formalism. Last systems include the standard and generalized nonholonomic Hamiltonian systems as particular cases. When restricted to Hamiltonian systems without constraints, our procedure gives rise exactly to the so-called Hamilton-Poincaré equations, as expected. In order to illustrate the procedure, we study in detail the case in which both the configuration space of the system and the involved symmetry define a trivial principal bundle.

Nonholonomic dynamical reduction of open-chain multi-body systems: A geometric approach

This paper studies the geometry behind nonholonomic Hamilton's equation to present a two-stage reduction procedure for the dynamical equations of nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. In this process, we use the Chaplygin reduction and an almost symplectic reduction theorem. We first restate the Chaplygin reduction theorem on cotangent bundle for nonholonomic Hamiltonian mechanical systems with symmetry. Then, under some conditions we extend this theorem to include a second reduction stage using an extended version of the symplectic reduction theorem for almost symplectic manifolds. We briefly introduce the displacement subgroups and accordingly open-chain multi-body systems consisting of such joints. For a holonomic open-chain multi-body system, the relative configuration manifold corresponding to the first joint is a symmetry group. Hence, we focus on a class of nonholonomic distributions on the configuration manifold of an open-chain multi-body system that is invariant under the action of this group. As the first stage of reduction procedure, we perform the Chaplygin reduction for such systems. We then introduce a number of sufficient conditions for a reduced system to admit more symmetry due to the action of the relative configuration manifolds of other joints. Under these conditions, we present the second stage of the reduction process for nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a two degree-of-freedom crane mounted on a four-wheel car to illustrate the results of this paper.

Noether conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices

The Noether conserved quantities and the Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie transformations acting on the lattice, as well as the equations and the determining equations of the Lie symmetries are obtained for the nonholonomic Hamiltonian systems. The discrete analogue of the Noether conserved quantity is constructed by using the Lie point symmetries. An example is discussed to illustrate the results.

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.

Modeling and passivity analysis of nonholonomic Hamiltonian systems with rheonomous affine constraints

This paper is devoted to modeling and theoretical analysis of Hamiltonian systems subject to nonholonomic rheonomous affine constraints. We first define rheonomous affine constraints and explain geometric representation of them. Next, a complete nonholonomicity condition for the rheonomous affine constraints is developed in terms of the rheonomous bracket. Then, the nonholonomic Hamiltonian system with rheonomous affine constraints (NHSRAC) is derived via a transformation and model reduction for the expanded Hamiltonian system defined on the expanded phase space. After that, we investigate passivity of the NHSRAC with the control input term and the output equation. Finally, in order to confirm the application potentiality of our new results, we show an example, a radius-variable ball on rotating table with a time-varying angular velocity.

Noether Symmetry and First Integral of Discrete Nonconservative and Nonholonimic Hamiltoinian System

The letter focuses on studying Noether symmetry and conserved quantity of discrete nonconservative and nonholonomic Hamiltonian system. Firstly, the discrete Hamiltonian canonical equations and discrete energy equations of nonconservative and nonholonomic Hamiltonian systems are derived with discrete Hamiltonian action. Secondly, based on the quasi-invariance of discrete Hamiltonian action and equation of lattice under the infinitesimal transformation with respect to time, generalized coordinates and generalized momentums, the discrete analogue of Noether’s identity and determining equation of lattice are obtained for the systems. Thirdly, the discrete analogues of Noether’s theorems and conserved quantities of the systems are presented. Finally, one example is discussed to illustrate the application of the results.

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

Modeling and Passivity Analysis of Nonholonomic Hamiltonian Systems with Time-varying Affine Constraints

AbstractThis paper is devoted to modeling and analysis of Hamiltonian systems subject to nonholonomic time-varying affine constraints. First, time-varying affine constraints are defined and an integrability condition of them is shown. We next derive nonholonomic Hamiltonian systems with time-varying affine constraints (NHSTAC) by using a transformation and reduction on the expanded phase space. Then, we investigate passivity of the NHSTAC with the control input term and the output equation. Finally, in order to confirm the results, a boat on a running river is illustrated as a physical example.

Hamiltonian systems, information transmittal and special relativity

Linear transformations of special relativity considered in [Albert Einstein, Zur Elektrodynamik der bewegte Körper. Annalen der Physik. 17 (1905) 891–921] are applied to Hamiltonian systems and Dirac equations, as presented in [Paul A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, New York, 1964]. To account for Weber’s electro-dynamic law of particle attraction and for the relativistic increase of the mass in particle accelerators, an extension of the second Newton’s law for motion subject to external forces that may depend on accelerations and higher order derivatives of velocities is considered and the Buquoy–Mestschersky generalization for motion of bodies with variable masses is included. The causality of systems driven by such forces is assured by consideration of left higher order derivatives in the right-hand sides of the equations of motion. The consistency condition is presented, and existence of solutions for the equations of motion driven by forces with left higher order derivatives is proved, leading to the generalized Lagrange and Hamilton equations that incorporate those extensions. Then, non-holonomic Hamiltonian systems with time-invariant constraints introduced by P.G. Bergmann and P.A.M. Dirac are considered, as suggested by Dirac for atomic models. Since one and the same process or particle may be observed as different images, the procedure is developed for identification of processes in moving systems by inverse relativistic transformations applied over small intervals of time to discrete experimental measurements obtained as the images of those processes in the observed (relativistic) coordinates. It is demonstrated that motions and processes evolving in still and/or moving systems can be described in the proper (of a still system) and/or relativistic (of different moving systems) coordinates in one common system of equations under the condition that all components of that system are referred to one and the same time of a still or moving observer. The results open new avenues for further research in relativistic systems theory and provide a basis for development of computing software for process identification in the images transmitted from distant or fast moving systems.

アフィン拘束を受ける非ホロノミックハミルトニアンシステムに対する一般化正準変換と受動性に基づく制御

アフィン拘束を受ける非ホロノミックハミルトニアンシステムに対する一般化正準変換と受動性に基づく制御

アファイン拘束を受ける非ホロノミックハミルトニアンシステムのモデリングと受動性解析

In this paper, we introduce and analyze Hamiltonian systems subject to nonholonomic affine constraints. We first sum up some concepts of Hamiltonian systems defined on both the phase space and the expanded phase space. Next, we derive nonholonomic Hamiltonian systems with affine constraints (NHSAC) by using a transformation and reduction on the expanded phase space. Then, passivity of the NHSAC with the control input term and the output equation is investigated. Finally, a coin on a rotating table is illustrated to confirm the results as a physical example.

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

This paper focuses on studying a new energy-work relationship numerical integration scheme of nonholonomic Hamiltonian systems. The signal-stage numerical, multi-stage and parallel composition numerical integration schemes are presented. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multi-stage schemes of order 2, its order of accuracy is 2n. The connection, which is discrete analogue of usual case, between the change of energy and work of nonholonomic constraint forces is obtained for nonholonomic Hamiltonian systems. This paper also gives that there is smaller error of the scheme when taking a large number of stages than a less one. Finally, an applied example is discussed to illustrate these results.

Construction of exact invariants of time-dependent linear nonholonomic dynamical systems

In this work, we build exact dynamical invariants for time-dependent, linear, nonholonomic Hamiltonian systems in two dimensions. Our aim is to obtain an additional insight into the theoretical understanding of generalized Hamilton canonical equations. In particular, we investigate systems represented by a quadratic Hamiltonian subject to linear nonholonomic constraints. We use a Lie algebraic method on the systems to build the invariants. The role and scope of these invariants is pointed out.

Trajectory Tracking Control of Nonholonomic Hamiltonian Systems via Generalized Canonical Transformations

This paper is devoted to a unified approach to trajectory tracking control of nonholonomic portcontrolled Hamiltonian systems via generalized canonical transformations. The basic strategy of this approach is to construct an error system, which describes the dynamics of the tracking error, by a passive port-controlled Hamiltonian system. This technique works for both holonomic and nonholonomic port-controlled Hamiltonian systems. A practical design procedure to derive global tracking controllers for those systems is proposed. This method is a natural extension of the conventional passivity based control.

Form invariance, Noether symmetry and Lie symmetry of Hamiltonian systems in phase space

For the holonomic and non-holonomic Hamiltonian systems in phase space, the definitions and criterions of the form invariance of both Hamilton and generalized Hamilton canonical equations are given. The relations among the form invariance, Noether symmetry and Lie symmetry are studied. Two examples are given to illustrate these results.

Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations

This paper is concerned with the stabilization of nonholonomic systems in port-controlled Hamiltonian formulae based on time-varying generalized canonical transformations. A special class of time-varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity. Utilizing these transformations, time-varying asymptotically stabilizing controllers for the nonholonomic Hamiltonian systems are derived. Since the proposed method is a natural generalization of passivity based control for conventional holonomic systems, it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method.