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All content in this area was uploaded by Darryl Holm on Sep 05, 2014

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We show that Euler's equations for a free rigid body, and for a rigid body
with a controlled feedback torque each red lice to the classical simple pendulum
equation under an explicit cylindrical coordinate change of variables.
These examples illustrate several ideas in Hamiltonian mechanics: LiePoisson
reduction, cotangent bundle reduction, singular Lie-Poisson maps,
deformations of Lie algebras, brackets R3, simplifications obtained by
utilizing the representation-dependence of Lie-Poisson reduction, and controlling
instability by inducing global bifurcations among a set of equilibria
using a control parameter.

Content uploaded by Darryl Holm

Author content

All content in this area was uploaded by Darryl Holm on Sep 05, 2014

Content may be subject to copyright.

... where Σ 1 p and Σ 2 p are given by (8). The intersection of the level sets H(x, y, z) = h and C(x, y, z) = c suggests the existence of the periodic orbits (see Figure 3 (b),(d),(e)). ...

The dynamics of a three-dimensional Hamilton-Poisson system is closely related to its constants of motion, the energy or Hamiltonian function H and a Casimir C of the corresponding Lie algebra. The orbits of the system are included in the intersection of the level sets H = constant and C = constant. Furthermore, for some three-dimensional Hamilton-Poisson systems, connections between the associated energy-Casimir mapping (H, C) and some of their dynamic properties were reported. In order to detect new connections, we construct a Hamilton-Poisson system using two smooth functions as its constants of motion. The new system has infinitely many Hamilton-Poisson realizations. We study the stability of the equilibrium points and the existence of periodic orbits. Using numerical integration we point out four pairs of heteroclinic orbits.

... where x ∈ C 3 and F, G are differentiable functions on C 3 . The Lie bracket [·, ·] M and the associated Lie-Poisson bracket {·, ·} M are considered in [13] and used to relate the free rigid body and the simple pendulum. (It is further used in [16] to analyze the quantization problems.) ...

This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account on the geometric setting of the system, the structure of the Poisson commuting first integrals is discussed following the methods by Magri and Skrypnyk. Introducing supplementary coordinates, a geometric connection to Kummer surfaces, a typical class of K3 surfaces, is mentioned and also the system is linearized on the Jacobian of a hyperelliptic curve of genus two determined by the system. Further some special solutions contained in some vector subspace are discussed. Finally, an explicit computation of the action-angle coordinates is introduced.

... The relation among the parameters and integrals for the different types of dynamics is given. Note that in the study of the extended Euler system combinations of the integrals allow to do that analysis working with cylinders (see [10] and [3]). In this paper, as the reduced spaces are spheres and hyperboloids, thus we will keep that geometry all over. ...

Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in T∗R4 which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic ow, isotropic oscillator and free rigid body systems appear as particular cases

In this paper we design a linear time-invariant (LTI) two-input-two-output (TITO) controller for the Permanent Magnet Synchronous Motor (PMSM). First the nonlinear system of a PMSM is approximated using a linear system with structured uncertainties according to the geometric structures of PMSM. We then design a linear controller for the approximated linear system using a standard -synthesis robust control method. The main contribution of this paper is that we recognize that the nonlinear PMSM can be reduced to a linear system to apply the mature modern control theory, without referring to classical PID control, thus largely reducing the design effort. The newly designed robust controller is not only easy to calculate, it is also easy to implement and requires less sensors. By virtue of modern robust control theory, it responds fast to exogenous inputs, and robust against parameter uncertainties as well. The applicability of the designed robust controller is both numerically simulated and experimentally verified on a TMS320 F28335 based control board, with performance further compared with the widely used Field-oriented controller.

The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the Kähler metric are compared. It is discussed how a Kähler metric can arise from a complex structure induced by the Hilbert transform. Hybrid and metriplectic flows which combine Hamiltonian and gradient components are examined. A class of metriplectic systems that is generated by completely antisymmetric triple brackets (trilinear brackets) is described and for finite-dimensional systems given a Lie algebraic interpretation. A variety of explicit examples of the several types of flows are given. It is shown that this geometry describes a number of classical ordinary and partial differential equations of interest and that the different metrics give rise to different kinds of dissipation that occur in applications.

We study the extended Euler systems (EES) as an initial value problem. Particular realizations of it lead to several Lie-Poisson structures. We consider a 6-D Poisson structure that fit all of them together. The symplectic stratification of this non Lie-Poisson structure uses the first integrals which are elliptic and hyperbolic cylinders, although other quadrics may be used as well. A qualitative study of the solutions is carried out and the twelve Jacobi elliptic functions in the real domain are shown in an unified way as the solutions of the EES. As a consequence, Jacobi's transformation for the elliptic modulus is obtained. Likewise, introducing the square norm function we establish in a straightforward way the connection of the EES with the Weierstrass script P elliptic function, giving the relation of its invariants gi with the integrals and coefficients of the EES.

An optimal control problem on some particular Lie groups is defined and some of its properties are pointed out.

We describe some relevant dynamical and geometrical properties of the Rikitake dynamo system from the Poisson geometry point of view.

This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies, unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid plasmas are presented as examples. Starting with Lagrangian variables, our method explains in a direct way why semidirect products occur so frequently in examples. It also provides a framework for the systematic introduction of Clebsch, or canonical, variables.

Various holonomy phenomena are shown to be instances of the reconstruction procedure
for mechanical systems with symmetry. We systematically exploit this point of view for fixed
systems (for example with controls on the internal, or reduced, variables) and for slowly moving
systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a
connection which synthesizes the Cartan connection for moving mechanical systems with the
Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural
way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems.

Varietes de Poisson et applications. Decomposition. Structures de Poisson lineaires. Approximation lineaire. Systemes hamiltoniens. Le probleme de linearisation. Groupes de fonction, realisations et applications impulsion. Paires duales et groupes de jauge. Existence des realisations. Unicite des realisations. Les problemes des 3 corps restreints et autres exemples

Taking the Liouville theorem as a guiding principle, we propose a
possible generalization of classical Hamiltonian dynamics to a
three-dimensional phase space. The equation of motion involves two
Hamiltonians and three canonical variables. The fact that the Euler
equations for a rotator can be cast into this form suggests the
potential usefulness of the formalism. In this article we study its
general properties and the problem of quantization.

We show how the Energy-Casimir method can be used to prove stabilizability of the angular momentum equations of the rigid body about its intermediate axis of inertia, by a single torque applied about the major or minor axis. We also show how this system has associated with it, a Lie-Poisson bracket which is invariant under SO(3) for small feedback, but is invariant under SO(2, 1) for feedback large enough to achieve stability.