American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

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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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2019, 11(1) : 1-22 doi: 10.3934/jgm.2019001 +[Abstract](157) +[HTML](99) +[PDF](1329.23KB)
Abstract:

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

2019, 11(1) : 23-44 doi: 10.3934/jgm.2019002 +[Abstract](160) +[HTML](81) +[PDF](504.76KB)
Abstract:

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

2019, 11(1) : 45-58 doi: 10.3934/jgm.2019003 +[Abstract](136) +[HTML](59) +[PDF](403.46KB)
Abstract:

Motivated by the work of Leznov-Mostovoy [17], we classify the linear deformations of standard \begin{document}$2n$\end{document}-dimensional phase space that preserve the obvious symplectic \begin{document}$\mathfrak{o}(n)$\end{document}-symmetry. As a consequence, we describe standard phase space, as well as \begin{document}$T^{*}S^{n}$\end{document} and \begin{document}$T^{*}\mathbb{H}^{n}$\end{document} with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in \begin{document}${\mathbb{R}}^{n+2}$\end{document}.

2019, 11(1) : 59-76 doi: 10.3934/jgm.2019004 +[Abstract](144) +[HTML](65) +[PDF](408.79KB)
Abstract:

The Piola identity \begin{document}$\operatorname{div}\; \operatorname{cof} \;\nabla f = 0$\end{document} is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

2019, 11(1) : 77-122 doi: 10.3934/jgm.2019005 +[Abstract](112) +[HTML](75) +[PDF](737.83KB)
Abstract:

We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of generalized variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective. In particular, it allows to derive, interpret and generalize many known results on non-Abelian Chaplygin systems. We apply it also to a class of systems on Lie groups with a left-invariant constraints distribution. Concrete examples of the unicycle in a potential field, the two-wheeled carriage and the generalized Heisenberg system are discussed.

2010, 2(2) : 159-198 doi: 10.3934/jgm.2010.2.159 +[Abstract](1470) +[PDF](475.8KB) Cited By(30)
2009, 1(4) : 461-481 doi: 10.3934/jgm.2009.1.461 +[Abstract](1257) +[PDF](789.5KB) Cited By(17)
2013, 5(3) : 319-344 doi: 10.3934/jgm.2013.5.319 +[Abstract](1291) +[PDF](661.6KB) Cited By(16)
2009, 1(2) : 159-180 doi: 10.3934/jgm.2009.1.159 +[Abstract](1518) +[PDF](318.8KB) Cited By(16)
2011, 3(3) : 337-362 doi: 10.3934/jgm.2011.3.337 +[Abstract](1160) +[PDF](488.1KB) Cited By(15)
2014, 6(3) : 335-372 doi: 10.3934/jgm.2014.6.335 +[Abstract](1209) +[PDF](587.5KB) Cited By(15)
2009, 1(1) : 55-85 doi: 10.3934/jgm.2009.1.55 +[Abstract](1252) +[PDF](494.1KB) Cited By(14)
2011, 3(1) : 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](1189) +[PDF](597.2KB) Cited By(14)
2009, 1(1) : 35-53 doi: 10.3934/jgm.2009.1.35 +[Abstract](1242) +[PDF](272.6KB) Cited By(13)
2009, 1(2) : 181-208 doi: 10.3934/jgm.2009.1.181 +[Abstract](1266) +[PDF](360.8KB) Cited By(12)

2017  Impact Factor: 0.561