# American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

## Journal Home

All Issues

### Volume 1, 2009

The Journal of Geometric Mechanics (JGM) aims to publish research articles devoted to geometric methods (in a broad sense) in mechanics and control theory, and intends to facilitate interaction between theory and applications. Advances in the following topics are welcomed by the journal:

1. Lagrangian and Hamiltonian mechanics
2. Symplectic and Poisson geometry and their applications to mechanics
3. Geometric and optimal control theory
4. Geometric and variational integration
5. Geometry of stochastic systems
6. Geometric methods in dynamical systems
7. Continuum mechanics
8. Classical field theory
9. Fluid mechanics
10. Infinite-dimensional dynamical systems
11. Quantum mechanics and quantum information theory
12. Applications in physics, technology, engineering and the biological sciences

More detailed information on the subjects covered by the journal can be found by viewing the fields of research of the members of the editorial board.

Contributions to this journal are published free of charge.

• AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
• Publishes 4 issues a year in March, June, September and December.
• Publishes online only.
• Indexed in Science Citation Index-Expanded, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
• Archived in Portico and CLOCKSS.
• JGM is a publication of the American Institute of Mathematical Sciences with the support of the Consejo Superior de Investigaciones Científicas (CSIC). All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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2018, 10(2) : 139-172 doi: 10.3934/jgm.2018005 +[Abstract](300) +[HTML](165) +[PDF](3595.86KB)
Abstract:

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.

2018, 10(2) : 173-187 doi: 10.3934/jgm.2018006 +[Abstract](178) +[HTML](115) +[PDF](440.5KB)
Abstract:

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.

2018, 10(2) : 189-208 doi: 10.3934/jgm.2018007 +[Abstract](193) +[HTML](158) +[PDF](1421.47KB)
Abstract:

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.

2018, 10(2) : 209-215 doi: 10.3934/jgm.2018008 +[Abstract](172) +[HTML](127) +[PDF](297.83KB)
Abstract:

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.

2018, 10(2) : 217-250 doi: 10.3934/jgm.2018009 +[Abstract](157) +[HTML](121) +[PDF](485.27KB)
Abstract:

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.

2010, 2(2) : 159-198 doi: 10.3934/jgm.2010.2.159 +[Abstract](714) +[PDF](475.8KB) Cited By(30)
2009, 1(4) : 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](593) +[PDF](789.5KB) Cited By(17)
2013, 5(3) : 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](555) +[PDF](661.6KB) Cited By(16)
2009, 1(2) : 159-180 doi: 10.3934/jgm.2009.1.159 +[Abstract](651) +[PDF](318.8KB) Cited By(16)
2011, 3(3) : 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](528) +[PDF](488.1KB) Cited By(15)
2014, 6(3) : 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](599) +[PDF](587.5KB) Cited By(15)
2009, 1(1) : 55-85 doi: 10.3934/jgm.2009.1.55 +[Abstract](584) +[PDF](494.1KB) Cited By(14)
2011, 3(1) : 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](573) +[PDF](597.2KB) Cited By(14)
2009, 1(1) : 35-53 doi: 10.3934/jgm.2009.1.35 +[Abstract](615) +[PDF](272.6KB) Cited By(13)
2009, 1(2) : 181-208 doi: 10.3934/jgm.2009.1.181 +[Abstract](572) +[PDF](360.8KB) Cited By(12)
2018, 10(1) : 1-41 doi: 10.3934/jgm.2018001 +[Abstract](639) +[HTML](350) +[PDF](622.49KB) PDF Downloads(116)
2018, 10(1) : 93-138 doi: 10.3934/jgm.2018004 +[Abstract](580) +[HTML](422) +[PDF](708.21KB) PDF Downloads(112)
2018, 10(1) : 69-92 doi: 10.3934/jgm.2018003 +[Abstract](531) +[HTML](313) +[PDF](437.43KB) PDF Downloads(86)
2018, 10(1) : 43-68 doi: 10.3934/jgm.2018002 +[Abstract](493) +[HTML](296) +[PDF](1325.47KB) PDF Downloads(74)
2018, 10(2) : 139-172 doi: 10.3934/jgm.2018005 +[Abstract](300) +[HTML](165) +[PDF](3595.86KB) PDF Downloads(53)
2017, 9(4) : 487-574 doi: 10.3934/jgm.2017019 +[Abstract](1083) +[HTML](248) +[PDF](2331.7KB) PDF Downloads(45)
2018, 10(2) : 189-208 doi: 10.3934/jgm.2018007 +[Abstract](193) +[HTML](158) +[PDF](1421.47KB) PDF Downloads(44)
2018, 10(2) : 173-187 doi: 10.3934/jgm.2018006 +[Abstract](178) +[HTML](115) +[PDF](440.5KB) PDF Downloads(43)
2018, 10(2) : 209-215 doi: 10.3934/jgm.2018008 +[Abstract](172) +[HTML](127) +[PDF](297.83KB) PDF Downloads(36)
2018, 10(2) : 217-250 doi: 10.3934/jgm.2018009 +[Abstract](157) +[HTML](121) +[PDF](485.27KB) PDF Downloads(31)

2017  Impact Factor: 0.561