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The Rotor and the Pendulum

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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.
... 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)). ...
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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.) ...
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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. ...
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