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Journal of Geometric Mechanics

2014 , Volume 6 , Issue 3

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Warped Poisson brackets on warped products
Yacine Aït Amrane, Rafik Nasri and Ahmed Zeglaoui
2014, 6(3): 279-296 doi: 10.3934/jgm.2014.6.279 +[Abstract](255) +[PDF](452.6KB)
Abstract:
In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure ([10]) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Poisson structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-Civita connection and the curvatures associated with.
Reduction of cluster iteration maps
Inês Cruz and M. Esmeralda Sousa-Dias
2014, 6(3): 297-318 doi: 10.3934/jgm.2014.6.297 +[Abstract](289) +[PDF](510.0KB)
Abstract:
We study iteration maps of difference equations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and presymplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several periodic quivers using either the presymplectic reduction or a Poisson reduction via log-canonical Poisson structures.
Discriminantly separable polynomials and quad-equations
Vladimir Dragović and Katarina Kukić
2014, 6(3): 319-333 doi: 10.3934/jgm.2014.6.319 +[Abstract](210) +[PDF](356.6KB)
Abstract:
We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler, Bobenko, and Suris.
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
Joachim Escher and Boris Kolev
2014, 6(3): 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](422) +[PDF](587.5KB)
Abstract:
In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies
Lyudmila Grigoryeva, Juan-Pablo Ortega and Stanislav S. Zub
2014, 6(3): 373-415 doi: 10.3934/jgm.2014.6.373 +[Abstract](265) +[PDF](883.9KB)
Abstract:
This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron". We study the nonlinear stability of a branch of equatorial relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that allow us to conclude the sharpness of some of the nonlinear stability conditions obtained.
Poisson structures for two nonholonomic systems with partially reduced symmetries
Andrey Tsiganov
2014, 6(3): 417-440 doi: 10.3934/jgm.2014.6.417 +[Abstract](250) +[PDF](468.7KB)
Abstract:
We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.

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