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Journal of Geometric Mechanics

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](96) +[PDF](600.1KB)
Abstract:
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](66) +[PDF](365.3KB)
Abstract:
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](103) +[PDF](343.6KB)
Abstract:
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](108) +[PDF](488.1KB)
Abstract:
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.

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