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Journal of Geometric Mechanics

September 2011 , Volume 3 , Issue 3

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Infinitesimal gauge symmetries of closed forms
Olivier Brahic
2011, 3(3): 277-312 doi: 10.3934/jgm.2011.3.277 +[Abstract](2875) +[PDF](600.1KB)
Motivated by the relationship between symplectic fibrations and classical Yang-Mills theories, we study the closedness of a $n$-form ($n$=2,3) defined on the total space of a fibration as a simple model for an abstract field theory. We introduce $2$-plectic fibrations and interpret geometrically the corresponding equations for coupling in terms of higher analogues of connections.
Euler equations on a semi-direct product of the diffeomorphisms group by itself
Joachim Escher, Rossen Ivanov and Boris Kolev
2011, 3(3): 313-322 doi: 10.3934/jgm.2011.3.313 +[Abstract](3026) +[PDF](365.3KB)
The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
Killing's equations for invariant metrics on Lie groups
Firas Hindeleh and Gerard Thompson
2011, 3(3): 323-335 doi: 10.3934/jgm.2011.3.323 +[Abstract](3252) +[PDF](343.6KB)
This article is the first in a series that will investigate symmetry and curvature properties of a right-invariant metric on a Lie group. This paper will consider Lie groups in dimension two and three and will focus on the solutions of Killing's equations. A striking result is that several of the three-dimensional Lie groups turn out to be spaces of constant curvature.
Integrable Euler top and nonholonomic Chaplygin ball
Andrey Tsiganov
2011, 3(3): 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](4561) +[PDF](488.1KB)
We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.

2021 Impact Factor: 0.737
5 Year Impact Factor: 0.713
2021 CiteScore: 1.3



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