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Journal of Geometric Mechanics

2009 , Volume 1 , Issue 1

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The ubiquity of the symplectic Hamiltonian equations in mechanics
P. Balseiro , M. de León , Juan Carlos Marrero and  D. Martín de Diego
2009, 1(1): 1-34 doi: 10.3934/jgm.2009.1.1 +[Abstract](38) +[PDF](391.2KB)
In this paper, we derive a "Hamiltonian formalism" for a wide class of mechanical systems, that includes, as particular cases, classical Hamiltonian systems, nonholonomic systems, some classes of servomechanisms... This construction strongly relies on the geometry characterizing the different systems. The main result of this paper is to show how the general construction of the Hamiltonian symplectic formalism in classical mechanics remains essentially unchanged starting from the more general framework of algebroids. Algebroids are, roughly speaking, vector bundles equipped with a bilinear bracket of sections and two vector bundle morphisms (the anchors maps) satisfying a Leibniz-type property. The bilinear bracket is not, in general, skew-symmetric and it does not satisfy, in general, the Jacobi identity. Since skew-symmetry is related with preservation of the Hamiltonian, our Hamiltonian framework also covers some examples of dissipative systems. On the other hand, since the Jacobi identity is related with the preservation of the associated linear Poisson structure, then our formalism also admits a Hamiltonian description for systems which do not preserve this Poisson structure, like nonholonomic systems.
   Some examples of interest are considered: gradient extension of dynamical systems, nonholonomic mechanics and generalized nonholonomic mechanics, showing the applicability of our theory and constructing the corresponding Hamiltonian formalism.
$G$-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](62) +[PDF](272.6KB)
Via compression ([18, 8]) we write the $n$-dimensional Chaplygin sphere system as an almost Hamiltonian system on T*$\SO(n)$ with internal symmetry group $\SO(n-1)$. We show how this symmetry group can be factored out, and pass to the fully reduced system on (a fiber bundle over) T*$S^{n-1}$. This approach yields an explicit description of the reduced system in terms of the geometric data involved. Due to this description we can study Hamiltonizability of the system. It turns out that the homogeneous Chaplygin ball, which is not Hamiltonian at the T*$\SO(n)$-level, is Hamiltonian at the T*$S^{n-1}$-level. Moreover, the $3$-dimensional ball becomes Hamiltonian at the T*$S^{2}$-level after time reparametrization, whereby we re-prove a result of [4, 5] in symplecto-geometric terms. We also study compression followed by reduction of generalized Chaplygin systems.
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](62) +[PDF](494.1KB)
Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system.
   The Euler top is naturally written in terms of the $\mathfrak{so}(3)$ Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.
Dirac cotangent bundle reduction
Hiroaki Yoshimura and  Jerrold E. Marsden
2009, 1(1): 87-158 doi: 10.3934/jgm.2009.1.87 +[Abstract](47) +[PDF](1379.4KB)
The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie--Dirac reduction . This procedure simultaneously includes Lagrangian, Hamiltonian, and a variational view of reduction. The goal of the present paper is to generalize Lie--Dirac reduction to the case of a general configuration manifold; we refer to this as Dirac cotangent bundle reduction . This reduction procedure encompasses, in particular, a reduction theory for Hamiltonian as well as implicit Lagrangian systems, including the case of degenerate Lagrangians.
     First of all, we establish a reduction theory starting with the Hamilton-Pontryagin variational principle, which enables one to formulate an implicit analogue of the Lagrange-Poincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure . Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called Lagrange-Poincaré-Dirac reduction . This procedure naturally induces the horizontal and vertical implicit Lagrange-Poincaré equations , which are consistent with those derived from the reduced Hamilton-Pontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, Hamilton-Poincaré-Dirac reduction for the horizontal and vertical Hamilton-Poincaré equations . We illustrate the reduction procedures by an example of a satellite with a rotor.
     The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages . This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.

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