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Stramigioli Stefano - One of the best experts on this subject based on the ideXlab platform.
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Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
'Elsevier BV', 2021Co-Authors: Rashad Ramy, Califano Federico, Schuller, Frederic P., Stramigioli StefanoAbstract:Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the Group of Diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved. In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities
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Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
2020Co-Authors: Rashad Ramy, Califano Federico, Schuller, Frederic P., Stramigioli StefanoAbstract:Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the Group of Diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved. In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.Comment: This is a prevprint submitted to the journal of Geometry and Physics. Please DO NOT CITE this version, but only the published manuscrip
Peter W Michor - One of the best experts on this subject based on the ideXlab platform.
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almost local metrics on shape space of hypersurfaces in n space
Siam Journal on Imaging Sciences, 2012Co-Authors: Martin Bauer, Philipp Harms, Peter W MichorAbstract:This paper extends parts of the results from [P. W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74-113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$ or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the Group of Diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $G_f(h,k) = \int_{M} \Phi(Vol(f),Tr(L))\bar{g}(h, k) vol(f^*\bar{g})$, where $\bar{g}$ is the Euclidean metric on $\mathbb R^n$, $f^*\bar{g}$ is the induced metric on $M$, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$ to the space of embeddings or immersions, where $\Phi:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $Vol(f) = \int_M vol(f^*\bar{g})$ is the total hypersurface volume of $f(M)$, and the trace $Tr(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $\Phi$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.
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almost local metrics on shape space of hypersurfaces in n space
arXiv: Differential Geometry, 2010Co-Authors: Martin Bauer, Philipp Harms, Peter W MichorAbstract:This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$, or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the Group of Diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $$ G_f(h,k) = \int_{M} \Phi(\on{Vol}(f),\operatorname{Tr}(L))\g(h, k) \operatorname{vol}(f^*\g),$$ where $\g$ is the Euclidean metric on $\mathbb R^n$, $f^*\g$ is the induced metric on $M$, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$ to the space of embeddings or immersions, where $\Phi:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $\operatorname{Vol}(f) = \int_M\operatorname{vol}(f^*\g)$ is the total hypersurface volume of $f(M)$, and the trace $\operatorname{Tr}(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $\Phi$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.
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cohomology for a Group of Diffeomorphisms of a manifold preserving an exact form
International Journal of Geometric Methods in Modern Physics, 2006Co-Authors: Mark Losik, Peter W MichorAbstract:Let M be a G-manifold and ω a G-invariant exact m-form on M. We indicate when these data allow us to construct a cocycle on a Group G with values in the trivial G-module ℝ, and when this cocycle is nontrivial.
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an overview of the riemannian metrics on spaces of curves using the hamiltonian approach
arXiv: Differential Geometry, 2006Co-Authors: Peter W Michor, David MumfordAbstract:Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the Group of Diffeomorphisms of $S^1$. We investige several Riemannian metrics on shape space: $L^2$-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order $n$ on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the Group of Diffeomorphisms on $\mathbb R^2$is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.
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extensions for a Group of Diffeomorphisms of a manifold preserving an exact 2 form
arXiv: Group Theory, 2004Co-Authors: Mark Losik, Peter W MichorAbstract:We construct a central extension by $\mathbb R$ of a Group of Diffeomorphisms of a manifold $M$ with an exact 2-form $\omega_M$ and give conditions of its triviality. When $H^1(M,\mathbb R)=0$ we prove that this extension is non-split for a manifold $\mathbb R^2\times M$ with a form $\omega=\omega_0+\omega_M$, where $\omega_0$ is the standard symplectic form on $\mathbb R^2$, of the Group of Diffeomorphisms of $\mathbb R^2\times M$ preserving the form $\omega$.
Rashad Ramy - One of the best experts on this subject based on the ideXlab platform.
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Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
'Elsevier BV', 2021Co-Authors: Rashad Ramy, Califano Federico, Schuller, Frederic P., Stramigioli StefanoAbstract:Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the Group of Diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved. In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities
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Port-Hamiltonian Modeling of Ideal Fluid Flow: Part II. Compressible and Incompressible Flow
2020Co-Authors: Rashad Ramy, Califano Federico, Schuller, Frederic P., Stramigioli StefanoAbstract:Part I of this paper presented a systematic derivation of the Stokes Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the Group of Diffeomorphisms as a configuration space for the fluid, the Stokes Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved. In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.Comment: This is a prevprint submitted to the journal of Geometry and Physics. Please DO NOT CITE this version, but only the published manuscrip
Yurii A Neretin - One of the best experts on this subject based on the ideXlab platform.
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some remarks on quasi invariant actions of loop Groups and the Group of Diffeomorphisms of the circle
Communications in Mathematical Physics, 1994Co-Authors: Yurii A NeretinAbstract:We construct the series of quasi-invariant actions of the Group Diff of Diffeomorphisms of the circle and loop Groups on the functional spaces provided by non-Wiener Gauss measures. We construct some measures which can be considered as analogues of Haar measure for loop Groups and the Group Diff. These constructions allow us to construct series of representations of these Groups including all known types of representations (highest weight representations, energy representations, almost invariant structures, etc.)
Hiroaki Shimomura - One of the best experts on this subject based on the ideXlab platform.
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quasi invariant measures on the Group of Diffeomorphisms and smooth vectors of unitary representations
Journal of Functional Analysis, 2001Co-Authors: Hiroaki ShimomuraAbstract:Abstract Let M be a smooth manifold and Diff 0 ( M ) the Group of all smooth Diffeomorphisms on M with compact support. Our main subject in this paper concerns the existence of certain quasi-invariant measures on Groups of Diffeomorphisms, and the denseness of C ∞ -vectors for a given unitary representation U of Diff* 0 ( M ), the connected component of the identity in Diff 0 ( M ). We first generalize some results of Shavgulidze on quasi-invariant measures on diffeomorphism Groups. Then we prove the following result: Suppose that M is compact and U has the property that the action extends continuously to Diff* k ( M ), the Group of C k Diffeomorphisms which are homotopic to the identity, for some finite k . Then U has a dense set of C ∞ -vectors. We also give an extension of our theorem to non-compact M .
