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  • 1
    Electronic Resource
    Electronic Resource
    Nonholonomic mechanical systems with symmetry (1996)
    Springer
    Archive for rational mechanics and analysis 136 (1996), S. 21-99 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d'Alembert formulation, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration-space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a “body reference frame” relates part of the momentum equation to the components of the Euler-Poincaré equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02199365
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  • 2
    Electronic Resource
    Electronic Resource
    Nonholonomic Mechanical Systems with Symmetry (1996)
    Springer
    Archive for rational mechanics and analysis 136 (1996), S. 21-99 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d'Alembert formulation, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration-space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a "body reference frame" relates part of the momentum equation to the components of the Euler-Poincaré equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02199365
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Completely integrable gradient flows (1992)
    Springer
    Communications in mathematical physics 147 (1992), S. 57-74 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099528
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  • 4
    Electronic Resource
    Electronic Resource
    Double Bracket Equations and Geodesic Flows on Symmetric Spaces (1997)
    Springer
    Communications in mathematical physics 187 (1997), S. 357-373 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: In this paper we consider the geometry of Hamiltonian flows on the cotangent bundle of coadjoint orbits of compact Lie groups and on symmetric spaces. A key idea here is the use of the normal metric to define the kinetic energy. This leads to Hamiltonian flows of the double bracket type. We analyze the integrability of geodesic flows according to the method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity. We demonstrate for example integrability of the geodesic flow on the real and complex Grassmannians. We also consider right invariant systems and the generalized rigid body equations in this setting.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050140
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  • 5
    Electronic Resource
    Electronic Resource
    Generalized Serret-Andoyer Transformation and Applications for the Controlled Rigid Body (1999)
    Springer
    Dynamics and control 9 (1999), S. 39-66 
    Details
    ISSN: 1573-8450
    Keywords: Hamiltonian system ; canonical transformation ; group symmetry ; symplectic form ; symplectic reduction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008342708491
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  • 6
    Electronic Resource
    Electronic Resource
    Stability analysis of a rotating flexible system (1989)
    Springer
    Acta applicandae mathematicae 15 (1989), S. 211-234 
    Details
    ISSN: 1572-9036
    Keywords: 70F99 ; 70K20 ; 73C02 ; 35P30 ; Flexible body ; Lagrangian and Hamiltonian dynamics ; stability ; partial differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyse here the equations of motion of a planar body consisting of a rigid body with attached flexible rod. These equations take the form of coupled ordinary and partial differential equations. We analyse the equations both with and without centrifugal stiffening effects. Using the ‘energy-momentum’ method, we analyse nonlinear stability of the equilibria in each case. We also analyse the Hamiltonian and Poisson bracket structure of the system as well as the energy-momentum map and associated relative equilibria.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00047531
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  • 7
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    Hamel’s Formalism for Infinite-Dimensional Mechanical Systems (2016)
    Springer
    Details
    Publication Date: 2016-09-01
    Print ISSN: 0938-8974
    Electronic ISSN: 1432-1467
    Topics: Mathematics , Physics
    Published by Springer
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  • 8
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    Quantization of a Nonholonomic System (2008)
    American Physical Society
    Details
    Publication Date: 2008-07-15
    Print ISSN: 0031-9007
    Electronic ISSN: 1079-7114
    Topics: Physics
    Published by American Physical Society
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  • 9
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    Asymptotic Hamiltonian dynamics: the Toda lattice, the three-wave interaction and the non-holonomic Chaplygin sleigh (2000)
    Elsevier
    Details
    Publication Date: 2000-07-01
    Print ISSN: 0167-2789
    Electronic ISSN: 1872-8022
    Topics: Physics
    Published by Elsevier
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  • 10
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    Qualitative behavior of non-Abelian Toda-like flows (2004)
    Elsevier
    Details
    Publication Date: 2004-12-01
    Print ISSN: 0167-2789
    Electronic ISSN: 1872-8022
    Topics: Physics
    Published by Elsevier
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