# American Institute of Mathematical Sciences

ISSN:
1941-4889

eISSN:
1941-4897

All Issues

## Journal of Geometric Mechanics

June 2017 , Volume 9 , Issue 2

Special issue on infinite-dimensional Riemannian geometry, Part I

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2017, 9(2): i-iii doi: 10.3934/jgm.201702i +[Abstract](1531) +[HTML](1124) +[PDF](31.6KB)
Abstract:
2017, 9(2): 131-156 doi: 10.3934/jgm.2017005 +[Abstract](2361) +[HTML](67) +[PDF](950.9KB)
Abstract:

This paper focuses on the study of open curves in a Riemannian manifold \begin{document} $M$ \end{document}, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [29] to define a Riemannian metric on the space of immersions \begin{document} $\mathcal{M}=\text{Imm}([0,1],M)$ \end{document} by pullback of a natural metric on the tangent bundle \begin{document} $\text{T}\mathcal{M}$ \end{document}. This induces a first-order Sobolev metric on \begin{document} $\mathcal{M}$ \end{document} and leads to a distance which takes into account the distance between the origins in \begin{document} $M$ \end{document} and the \begin{document} $L^2$ \end{document}-distance between the SRV representations of the curves. The geodesic equations for this metric are given and exploited to define an exponential map on \begin{document} $\mathcal M$ \end{document}. The optimal deformation of one curve into another can then be constructed using geodesic shooting, which requires to characterize the Jacobi fields of \begin{document} $\mathcal M$ \end{document}. The particular case of curves lying in the hyperbolic half-plane \begin{document} $\mathbb H$ \end{document} is considered as an example, in the setting of radar signal processing.

2017, 9(2): 157-165 doi: 10.3934/jgm.2017006 +[Abstract](2567) +[HTML](79) +[PDF](353.5KB)
Abstract:

The Madelung transform relates the non-linear Schrödinger equation and a compressible Euler equation known as the quantum hydrodynamical system. We prove that the Madelung transform is a momentum map associated with an action of the semidirect product group \begin{document} $\mathrm{Diff}(\mathbb{R}^{n}) \ltimes H^∞(\mathbb{R}^n; \mathbb{R})$ \end{document}, which is the configuration space of compressible fluids, on the space \begin{document} $Ψ = H^∞(\mathbb{R}^{n}; \mathbb{C})$ \end{document} of wave functions. In particular, this implies that the Madelung transform is a Poisson map taking the natural Poisson bracket on \begin{document} $Ψ$ \end{document} to the compressible fluid Poisson bracket. Moreover, the Madelung transform provides an example of "Clebsch variables" for the hydrodynamical system.

2017, 9(2): 167-189 doi: 10.3934/jgm.2017007 +[Abstract](2853) +[HTML](75) +[PDF](498.0KB)
Abstract:

This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on \begin{document}$\text{Dif}{{\text{f}}^{\infty }}\left( {{\mathbb{T}}^{d}} \right)$\end{document} and \begin{document}$\text{Dif}{{\text{f}}^{\infty }}\left( {{\mathbb{R}}^{d}} \right)$\end{document} when the inertia operator is a non-local Fourier multiplier.

2017, 9(2): 191-206 doi: 10.3934/jgm.2017008 +[Abstract](2633) +[HTML](63) +[PDF](366.9KB)
Abstract:

In this paper rational differential invariants are used to classify various plane shapes as well as plane domains equipped with an additional geometrical object.

2017, 9(2): 207-225 doi: 10.3934/jgm.2017009 +[Abstract](2115) +[HTML](71) +[PDF](485.4KB)
Abstract:

The group \begin{document} $\text{Diff}\left( {{S}^{1}} \right)$ \end{document} of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups \begin{document} $\text{U}(p,q)$ \end{document}, \begin{document} $\text{Sp}(2n,\mathbb{R})$ \end{document}, \begin{document} $\text{SO}^*(2n)$ \end{document}; the space \begin{document} $Ξ$ \end{document} of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of \begin{document} $\text{Diff}\left( {{S}^{1}} \right)$ \end{document} in the space of holomorphic functionals on \begin{document} $Ξ$ \end{document}, reproducing kernels on \begin{document} $Ξ$ \end{document} determining inner products, and expressions ('canonical cocycles') replacing spherical functions.

2017, 9(2): 227-256 doi: 10.3934/jgm.2017010 +[Abstract](2348) +[HTML](62) +[PDF](1716.5KB)
Abstract:

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in [13] on the space of arc-length parameterized curves. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of un-parameterized curves is concerned, pushing aside the tricky quotient procedure detailed in [12] of the preshape space of parameterized curves by the reparameterization (semi-)group. In order to study the problem of finding geodesics between two given arc-length parameterized curves under these quotient elastic metrics, we give a precise computation of the gradient of the energy functional in the smooth case as well as a discretization of it, and implement a path-straightening method. This allows us to have a better understanding of how the landscape of the energy functional varies with respect to the parameters.

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