ISSN:
 1941-4889

eISSN:
 1941-4897

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Volume 10, 2018

Volume 9, 2017

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Volume 5, 2013

Volume 4, 2012

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Volume 2, 2010

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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A note on time-optimal paths on perturbed spheroid
Piotr Kopacz
2018, 10(2) : 139-172 doi: 10.3934/jgm.2018005 +[Abstract](310) +[HTML](169) +[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.

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](187) +[HTML](118) +[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.

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](197) +[HTML](163) +[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.

A note on the normalization of generating functions
Simone Vazzoler
2018, 10(2) : 209-215 doi: 10.3934/jgm.2018008 +[Abstract](175) +[HTML](132) +[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.

Double groupoids and the symplectic category
Santiago Cañez
2018, 10(2) : 217-250 doi: 10.3934/jgm.2018009 +[Abstract](162) +[HTML](123) +[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.

Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics
Manuel de León, Juan Carlos Marrero and David Martín de Diego
2010, 2(2) : 159-198 doi: 10.3934/jgm.2010.2.159 +[Abstract](721) +[PDF](475.8KB) Cited By(30)
Nonholonomic Hamilton-Jacobi equation and integrability
Tomoki Ohsawa and Anthony M. Bloch
2009, 1(4) : 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](602) +[PDF](789.5KB) Cited By(17)
On Euler's equation and 'EPDiff'
David Mumford and Peter W. Michor
2013, 5(3) : 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](560) +[PDF](661.6KB) Cited By(16)
Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
Ioan Bucataru and Matias F. Dahl
2009, 1(2) : 159-180 doi: 10.3934/jgm.2009.1.159 +[Abstract](658) +[PDF](318.8KB) Cited By(16)
Integrable Euler top and nonholonomic Chaplygin ball
Andrey Tsiganov
2011, 3(3) : 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](533) +[PDF](488.1KB) Cited By(15)
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
Joachim Escher and Boris Kolev
2014, 6(3) : 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](604) +[PDF](587.5KB) Cited By(15)
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras
Andrew N. W. Hone and Matteo Petrera
2009, 1(1) : 55-85 doi: 10.3934/jgm.2009.1.55 +[Abstract](592) +[PDF](494.1KB) Cited By(14)
Clebsch optimal control formulation in mechanics
François Gay-Balmaz and Tudor S. Ratiu
2011, 3(1) : 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](580) +[PDF](597.2KB) Cited By(14)
$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball
Simon Hochgerner and Luis García-Naranjo
2009, 1(1) : 35-53 doi: 10.3934/jgm.2009.1.35 +[Abstract](621) +[PDF](272.6KB) Cited By(13)
Geodesic Vlasov equations and their integrable moment closures
Darryl D. Holm and Cesare Tronci
2009, 1(2) : 181-208 doi: 10.3934/jgm.2009.1.181 +[Abstract](583) +[PDF](360.8KB) Cited By(12)
Lagrange-d'alembert-poincaré equations by several stages
Hernán Cendra and Viviana A. Díaz
2018, 10(1) : 1-41 doi: 10.3934/jgm.2018001 +[Abstract](643) +[HTML](350) +[PDF](622.49KB) PDF Downloads(116)
Classical field theory on Lie algebroids: Multisymplectic formalism
Eduardo Martínez
2018, 10(1) : 93-138 doi: 10.3934/jgm.2018004 +[Abstract](587) +[HTML](422) +[PDF](708.21KB) PDF Downloads(112)
The projective Cartan-Klein geometry of the Helmholtz conditions
Carlos Durán and Diego Otero
2018, 10(1) : 69-92 doi: 10.3934/jgm.2018003 +[Abstract](537) +[HTML](313) +[PDF](437.43KB) PDF Downloads(86)
On some aspects of the discretization of the suslov problem
Fernando Jiménez and Jürgen Scheurle
2018, 10(1) : 43-68 doi: 10.3934/jgm.2018002 +[Abstract](498) +[HTML](296) +[PDF](1325.47KB) PDF Downloads(74)
A note on time-optimal paths on perturbed spheroid
Piotr Kopacz
2018, 10(2) : 139-172 doi: 10.3934/jgm.2018005 +[Abstract](310) +[HTML](169) +[PDF](3595.86KB) PDF Downloads(54)
The physical foundations of geometric mechanics
Andrew D. Lewis
2017, 9(4) : 487-574 doi: 10.3934/jgm.2017019 +[Abstract](1089) +[HTML](248) +[PDF](2331.7KB) PDF Downloads(45)
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](197) +[HTML](163) +[PDF](1421.47KB) PDF Downloads(44)
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](187) +[HTML](118) +[PDF](440.5KB) PDF Downloads(43)
A note on the normalization of generating functions
Simone Vazzoler
2018, 10(2) : 209-215 doi: 10.3934/jgm.2018008 +[Abstract](175) +[HTML](132) +[PDF](297.83KB) PDF Downloads(36)
Double groupoids and the symplectic category
Santiago Cañez
2018, 10(2) : 217-250 doi: 10.3934/jgm.2018009 +[Abstract](162) +[HTML](123) +[PDF](485.27KB) PDF Downloads(31)

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