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Journal of Geometric Mechanics

June 2018 , Volume 10 , Issue 2

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A note on time-optimal paths on perturbed spheroid
Piotr Kopacz
2018, 10(2): 139-172 doi: 10.3934/jgm.2018005 +[Abstract](146) +[HTML](72) +[PDF](3595.86KB)

We consider Zermelo's problem of navigation on a spheroid in the presence of space-dependent perturbation \begin{document}$W$\end{document} determined by a weak velocity vector field, \begin{document}$|W|_h<1$\end{document}. The approach is purely geometric with application of Finsler metric of Randers type making use of the corresponding optimal control represented by a time-minimal ship's heading \begin{document}$\varphi(t)$\end{document} (a steering direction). A detailed exposition including investigation of the navigational quantities is provided under a rotational vector field. This demonstrates, in particular, a preservation of the optimal control \begin{document}$\varphi(t)$\end{document} of the time-efficient trajectories in the presence and absence of acting perturbation. Such navigational treatment of the problem leads to some simple relations between the background Riemannian and the resulting Finsler geodesics, thought of the deformed Riemannian paths. Also, we show some connections with Clairaut's relation and a collision problem. The study is illustrated with an example considered on an oblate ellipsoid.

Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems
Božzidar Jovanović
2018, 10(2): 173-187 doi: 10.3934/jgm.2018006 +[Abstract](100) +[HTML](55) +[PDF](440.5KB)

We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transparent formulation of the momentum equation and the Noether theorem in their general forms.

Vortex pairs on a triaxial ellipsoid and Kimura's conjecture
Adriano Regis Rodrigues, César Castilho and Jair Koiller
2018, 10(2): 189-208 doi: 10.3934/jgm.2018007 +[Abstract](109) +[HTML](87) +[PDF](1421.47KB)

We consider the problem of point vortices moving on the surface of a triaxial ellipsoid. Following Hally's approach, we obtain the equations of motion by constructing a conformal map from the ellipsoid into the sphere and composing with stereographic projection. We focus on the case of a pair of opposite vortices. Our approach is validated by testing a prediction by Kimura that a (infinitesimally close) vortex dipole follows the geodesic flow. Poincaré sections suggest that the global flow is non-integrable.

A note on the normalization of generating functions
Simone Vazzoler
2018, 10(2): 209-215 doi: 10.3934/jgm.2018008 +[Abstract](87) +[HTML](58) +[PDF](297.83KB)

In the present note, I will propose some insights on the normalization of generating functions for Lagrangian submanifolds. From the literature (see, for example [4], [6], [7], [3] and [1]), it is clear that a problem exists concerning the nonuniqueness of generating functions and, in particular, of the generating functions quadratic at infinity (GFQI). This problem can be avoided introducing a normalization on the whole set of generating functions that will allow us to

(ⅰ) choose an unique GFQI for Lagrangian submanifolds of the form \begin{document}$\varphi(L)$\end{document}, where \begin{document}$L$\end{document} is a Lagrangian submanifold and \begin{document}$\varphi$\end{document} is an Hamiltonian isotopy;

(ⅱ) compare the critical values \begin{document}$c(α, S_1)$\end{document} and \begin{document}$c(α, S_2)$\end{document} of two GFQI generating the submanifolds, \begin{document}$\varphi_1(L)$\end{document} and \begin{document}$\varphi_2(L)$\end{document}, where \begin{document}$\varphi_1$\end{document} and \begin{document}$\varphi_2$\end{document} are Hamiltonian isotopies relative to two Hamiltonians \begin{document}$H_1$\end{document} and \begin{document}$H_2$\end{document}, respectively.

Double groupoids and the symplectic category
Santiago Cañez
2018, 10(2): 217-250 doi: 10.3934/jgm.2018009 +[Abstract](82) +[HTML](58) +[PDF](485.27KB)

We introduce the notion of a symplectic hopfoid, a "groupoid-like" object in the category of symplectic manifolds whose morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on a new realization of the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.

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