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Journal of Geometric Mechanics

2016 , Volume 8 , Issue 4

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No monodromy in the champagne bottle, or singularities of a superintegrable system
Larry M. Bates and Francesco Fassò
2016, 8(4): 375-389 doi: 10.3934/jgm.2016012 +[Abstract](329) +[PDF](1294.2KB)
The three-dimensional champagne bottle system contains no mondromy, despite being entirely composed of invariant two-dimensional champagne bottle systems, each of which posesses nontrivial monodromy. We explain where the monodromy went in the three-dimensional system, or perhaps, where it did come from in the two-dimensional system, by regarding the three-dimensional system not as completely integrable, but as superintegrable (or non-commutatively integrable), and explaining the role of the singularities of its isotropic-coisotropic pair of foliations.
The Frank tensor as a boundary condition in intrinsic linearized elasticity
Nicolas Van Goethem
2016, 8(4): 391-411 doi: 10.3934/jgm.2016013 +[Abstract](345) +[PDF](499.8KB)
The Frank tensor plays a crucial role in linear elasticity, and in particular in the presence of dislocation lines, since its curl is exactly the elastic strain incompatibility. Furthermore, the Frank tensor also appears in Cesaro decomposition, and in Volterra theory of dislocations and disclinations, since its jump is the Frank vector around the defect line. The purpose of this paper is to show to which functional space the compatible strain $e$ belongs in order to imply a homogeneous boundary conditions for the induced displacement field on a portion $\Gamma_0$ of the boundary. This will allow one to define the homogeneous, or even the mixed problem of linearized elasticity in a variational setting involving the strain $e$ in place of displacement $u$. With other purposes, this problem was originaly treated by Ph. Ciarlet and C. Mardare, and termed the intrinsic formulation. In this paper we propose alternative conditions on $e$ expressed in terms of $e$ and the Frank tensor Curl$^t$ $e$ only, yielding a clear physical understanding and showing as equivalent to Ciarlet-Mardare boundary condition.
The Tulczyjew triple in mechanics on a Lie group
Katarzyna Grabowska and Marcin Zając
2016, 8(4): 413-435 doi: 10.3934/jgm.2016014 +[Abstract](473) +[PDF](657.1KB)
Tulczyjew triple for physical systems with configuration manifold equipped with a Lie group structure is constructed and discussed. Systems invariant with respect to group and subgroup actions are considered together with appropriate reductions of the Tulczyjew triple. The theory is applied to free and constrained rigid-body dynamics.
On strain measures and the geodesic distance to $SO_n$ in the general linear group
Raz Kupferman and Asaf Shachar
2016, 8(4): 437-460 doi: 10.3934/jgm.2016015 +[Abstract](272) +[PDF](456.7KB)
We consider various notions of strains---quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of [12], is to select a Riemannian metric on $GL_n$, and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-$GL_n$-invariant and right-$O_n$-invariant. We proceed to investigate alternative distance functions on $GL_n$, and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on $GL_n$. Lastly, we investigate strain measures induced by inverse-invariant distances.
Kirchhoff's equations of motion via a constrained Zakharov system
Banavara N. Shashikanth
2016, 8(4): 461-485 doi: 10.3934/jgm.2016016 +[Abstract](282) +[PDF](1119.0KB)
The Kirchhoff problem for a neutrally buoyant rigid body dynamically interacting with an ideal fluid is considered. Instead of the standard Kirchhoff equations, equations of motion in which the pressure terms appear explicitly are considered. These equations are shown to satisfy a Hamiltonian constraint formalism, with the pressure playing the role of the Lagrange multiplier. The constraint is imposed on the shape of a compact fluid surface whose dynamics is governed by the canonical variables introduced by Zakharov for a free-surface. It is also shown that the assumption of neutral buoyancy can be relaxed.

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