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Journal of Geometric Mechanics

2016 , Volume 8 , Issue 1

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Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group
James Benn
2016, 8(1): 1-12 doi: 10.3934/jgm.2016.8.1 +[Abstract](223) +[PDF](435.8KB)
Abstract:
Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.
Symplectic reduction at zero angular momentum
Joshua Cape, Hans-Christian Herbig and Christopher Seaton
2016, 8(1): 13-34 doi: 10.3934/jgm.2016.8.13 +[Abstract](261) +[PDF](528.4KB)
Abstract:
We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $O_n$ on $k$ copies of $T^\ast\mathbb{R}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $O_n$- and $SO_n$-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when $n \leq k$, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of $k$, the ring of regular functions on the symplectic quotient is graded Gorenstein.
Lagrangian reduction of discrete mechanical systems by stages
Javier Fernández, Cora Tori and Marcela Zuccalli
2016, 8(1): 35-70 doi: 10.3934/jgm.2016.8.35 +[Abstract](242) +[PDF](686.3KB)
Abstract:
In this work we introduce a category of discrete Lagrange--Poincaré systems $\mathfrak{L}\mathfrak{P}_d$ and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete dynamical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in $\mathfrak{L}\mathfrak{P}_d$. We introduce a notion of symmetry group for objects of $\mathfrak{L}\mathfrak{P}_d$ as well as a reduction procedure that is closed in the category $\mathfrak{L}\mathfrak{P}_d$. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in $\mathfrak{L}\mathfrak{P}_d$ to the reduction by the full symmetry group.
Free Courant and derived Leibniz pseudoalgebras
Benoît Jubin, Norbert Poncin and Kyosuke Uchino
2016, 8(1): 71-97 doi: 10.3934/jgm.2016.8.71 +[Abstract](250) +[PDF](529.9KB)
Abstract:
We introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over `functions'. The free generalized Courant pseudoalgebra is built from two components: the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra and the free symmetric Leibniz pseudoalgebra on an anchored module. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids, which were introduced in [12] as geometric replacements of standard Leibniz algebroids. Eventually, we apply our algebro-categorical machinery to associate a differential graded Lie algebra to any symmetric Leibniz pseudoalgebra, such that the Leibniz bracket of the latter coincides with the derived bracket of the former.
Linearisation of tautological control systems
Andrew D. Lewis
2016, 8(1): 99-138 doi: 10.3934/jgm.2016.8.99 +[Abstract](289) +[PDF](657.4KB)
Abstract:
The framework of tautological control systems is one where ``control'' in the usual sense has been eliminated, with the intention of overcoming the issue of feedback-invariance. Here, the linearisation of tautological control systems is described. This linearisation retains the feedback-invariant character of the tautological control systems framework and so permits, for example, a well-defined notion of linearisation of a system about an equilibrium point, something which has surprisingly been missing up to now. The linearisations described are of systems, first, and then about reference trajectories and reference flows.

2016  Impact Factor: 0.857

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