# 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(3) : 251-291 doi: 10.3934/jgm.2018010 +[Abstract](127) +[HTML](34) +[PDF](674.71KB)
Abstract:

Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.

2018, 10(3) : 293-329 doi: 10.3934/jgm.2018011 +[Abstract](132) +[HTML](69) +[PDF](564.97KB)
Abstract:

We consider the Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point with parameters in a complex domain. We suppose that these equations admit a first integral functionally independent of the three already known integrals which does not depend on all the variables. We prove that this may happen only in the already known three integrable cases or in the trivial case of kinetic symmetry. We provide a method for finding such a fourth integral, when it exists.

2018, 10(3) : 331-357 doi: 10.3934/jgm.2018012 +[Abstract](72) +[HTML](37) +[PDF](5451.01KB)
Abstract:

About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngọc by means of five invariants. Standard examples are the coupled spin oscillator on \begin{document}$\mathbb{S}^2 \times \mathbb{R}^2$\end{document} and coupled angular momenta on \begin{document}$\mathbb{S}^2 \times \mathbb{S}^2$\end{document}, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a \begin{document}$6$\end{document}-parameter family of integrable systems on \begin{document}$\mathbb{S}^2 \times \mathbb{S}^2$\end{document} and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.

2018, 10(3) : 359-372 doi: 10.3934/jgm.2018013 +[Abstract](83) +[HTML](37) +[PDF](379.97KB)
Abstract:

The \begin{document}$\mathcal{KS}$\end{document} map is revisited in terms of an \begin{document}$S^1$\end{document}-action in \begin{document}$T^*\mathbb{H}_0$\end{document} with the bilinear function as the associated momentum map. Indeed, the \begin{document}$\mathcal{KS}$\end{document} transformation maps the \begin{document}$S^1$\end{document}-fibers related to the mentioned action to single points. By means of this perspective a second twin-bilinear function is obtained with an analogous \begin{document}$S^1$\end{document}-action. We also show that the connection between the 4-D isotropic harmonic oscillator and the spatial Kepler systems can be done in a straightforward way after regularization and through the extension to 4 degrees of freedom of the Euler angles, when the bilinear relation is imposed. This connection incorporates both bilinear functions among the variables. We will show that an alternative regularization separates the oscillator expressed in Projective Euler variables. This setting takes advantage of the two bilinear functions and another integral of the system including them among a new set of variables that allows to connect the 4-D isotropic harmonic oscillator and the planar Kepler system. In addition, our approach makes transparent that only when we refer to rectilinear solutions, both bilinear relations defining the \begin{document}$\mathcal{KS}$\end{document} transformations are needed.

2010, 2(2) : 159-198 doi: 10.3934/jgm.2010.2.159 +[Abstract](829) +[PDF](475.8KB) Cited By(30)
2009, 1(4) : 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](704) +[PDF](789.5KB) Cited By(17)
2013, 5(3) : 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](667) +[PDF](661.6KB) Cited By(16)
2009, 1(2) : 159-180 doi: 10.3934/jgm.2009.1.159 +[Abstract](757) +[PDF](318.8KB) Cited By(16)
2011, 3(3) : 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](620) +[PDF](488.1KB) Cited By(15)
2014, 6(3) : 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](679) +[PDF](587.5KB) Cited By(15)
2009, 1(1) : 55-85 doi: 10.3934/jgm.2009.1.55 +[Abstract](711) +[PDF](494.1KB) Cited By(14)
2011, 3(1) : 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](671) +[PDF](597.2KB) Cited By(14)
2009, 1(1) : 35-53 doi: 10.3934/jgm.2009.1.35 +[Abstract](709) +[PDF](272.6KB) Cited By(13)
2009, 1(2) : 181-208 doi: 10.3934/jgm.2009.1.181 +[Abstract](677) +[PDF](360.8KB) Cited By(12)

2017  Impact Factor: 0.561