The twisted top

Thiffeault, Jean-Luc; Morrison, Philip

doi:10.1016/s0375-9601(01)00253-5

Cited by 12 publications

(19 citation statements)

References 19 publications

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“…The uniform magnetic rod gives a physical realisation to the abstract 'twisted top' equations introduced in [14]. The fourth member of the hierarchy also has a physical interpretation as a conducting rod in a non-uniform linearly-varying magnetic field.…”

Section: Resultsmentioning

confidence: 97%

“…We note that the Lagrange integrability condition K 1 = K 2 is unaltered by the magnetic field. The authors in [14] give numerical evidence (in the form of chaotic orbits) that the same is not true for the Kovalevskaya case: the magnetic rod with K 1 = K 3 = 2K 2 is not integrable. Of course, a B-perturbed condition on the stiffnesses may exist for which the system is integrable.…”

Section: The Conducting Rod In a Uniform Magnetic Field -Three-field mentioning

confidence: 99%

“…Eqs (3.23)-(3.25) give a physical realisation of the 'twisted top' introduced as an abstract mathematical system in [14]. There are three Casimirs, given by…”

Section: The Conducting Rod In a Uniform Magnetic Field -Three-field mentioning

confidence: 99%

“…In the isotropic case the system reduces further to a four-dimensional integrable canonical Hamiltonian system. We follow [14] but give details here to show the system is canonical. As it happens, the transformation is only canonical subject to a non-alignment condition.…”

Section: Reduction Of the Magnetic Rodmentioning

confidence: 99%

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Integrability of a conducting elastic rod in a magnetic field

Sinden¹,

Heijden²

2008

J. Phys. A: Math. Theor.

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Abstract. We consider the equilibrium equations for a conducting elastic rod placed in a uniform magnetic field, motivated by the problem of electrodynamic space tethers. When expressed in body coordinates the equations are found to sit in a hierarchy of noncanonical Hamiltonian systems involving an increasing number of vector fields. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable in the case of a transversely isotropic rod; they are in fact generated by a Lax pair. For the magnetic rod this gives a physical interpretation to a previously proposed abstract nine-dimensional integrable system. We use the conserved quantities to reduce the equations to a four-dimensional canonical Hamiltonian system, allowing the geometry of the phase space to be investigated through Poincaré sections. In the special case where the force in the rod is aligned with the magnetic field the system turns out to be superintegrable, meaning that the phase space breaks down completely into periodic orbits, corresponding to straight twisted rods.

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Section: Resultsmentioning

confidence: 97%

Section: The Conducting Rod In a Uniform Magnetic Field -Three-field mentioning

confidence: 99%

“…Eqs (3.23)-(3.25) give a physical realisation of the 'twisted top' introduced as an abstract mathematical system in [14]. There are three Casimirs, given by…”

Section: The Conducting Rod In a Uniform Magnetic Field -Three-field mentioning

confidence: 99%

Section: Reduction Of the Magnetic Rodmentioning

confidence: 99%

See 2 more Smart Citations

Integrability of a conducting elastic rod in a magnetic field

Sinden¹,

Heijden²

2008

J. Phys. A: Math. Theor.

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show abstract

“…As mentioned already these structures have applications in Hydrodynamics and Magnetohydrodynamics but there are examples of finite dimensional system having a Poisson-Lie structure resulting from one of the most simple extensions of that class [10]. Provided the above is a Lie bracket we obtain the algebra G W .…”

Section: Introductionmentioning

confidence: 98%