Dual pairs in fluid dynamics

Gay–Balmaz, François; Vizman, Cornelia

doi:10.1007/s10455-011-9267-z

Cited by 29 publications

(76 citation statements)

References 21 publications

(31 reference statements)

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“…In the case of classical mechanics, this fact led Marsden and Weinstein [47] to construct a dual pair of momentum maps underlying planar incompressible fluid flows. As reported also in the sections below, this construction has recently been developed further in [24,25], while the application to Liouville-type (Vlasov) equations was presented in [33]. The following section shows that an analogue construction also underlies quantum mixed states.…”

Section: Right Actions and Diffeomorphismsmentioning

confidence: 91%

“…to observe that the generalized mixture (14) identifies a momentum map for the natural right action ψ(r) → Uψ(r) of unitary operators U ∈ U(H ). We remark that the symplectic form (15) is strictly related to a class of symplectic forms previously appeared in [24,25]; let S be a compact orientable manifold with volume form µ S and let (M, ω) be an exact symplectic manifold. One can endow the manifold F (S, M) of smooth functions S → M with the symplectic formω…”

Section: Quantum Mixtures As Momentum Mapsmentioning

confidence: 99%

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Momentum maps for mixed states in quantum and classical mechanics

Tronci¹

2019

Journal of Geometric Mechanics

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This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann's density matrix [58] and Koopman wavefunctions [42] are shown to be special cases of this construction.

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Section: Right Actions and Diffeomorphismsmentioning

confidence: 91%

Section: Quantum Mixtures As Momentum Mapsmentioning

confidence: 99%

Momentum maps for mixed states in quantum and classical mechanics

Tronci¹

2019

Journal of Geometric Mechanics

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“…Thus the Euler-Arnold equation on the quantomorphism group of M is the quasigeostrophic equation in f -plane approximation on N , as in Holm and Zeitlin (1998) and Zeitlin and Pasmanter (1994); here α 2 is the Froude number. An alternative approach to the quantomorphism group is to view it as a central extension of the group D Ham (N ) of Hamiltonian diffeomorphisms of the symplectic manifold N ; this approach is used in Ratiu and Schmid (1981) and is also taken in the references Tronci (2009), Gay-Balmaz andVizman (2012) and GayBalmaz and Tronci (2012). Smolentsev (1994) computed the curvature tensor of the quantomorphism group under the same assumptions.…”

Section: Corollary 43mentioning

confidence: 99%

Riemannian Geometry of the Contactomorphism Group

Ebin

Preston

2014

Arnold Math J.

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Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L 2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an "Euler-Arnold" equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a "quasi-Lipschitz" estimate on the stream function, which leads to a Beale-Kato-Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler-Arnold equations are the Camassa-Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

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“…The space Gr(S 1 , R n ) is a special case of a non-linear Grassmannian [29], and it has a manifold structure under certain conditions on the space of embeddings and the space of diffeomorphisms [30]. When the parametrization is not removed, embedded curves and surfaces can be matched with the current dissimilarity measure [31,32].…”

Section: Curve and Surface Matchingmentioning

confidence: 99%

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

Sommer

Jacobs

2015

Symmetry

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Abstract:We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types.

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Dual pairs in fluid dynamics

Cited by 29 publications

References 21 publications

Momentum maps for mixed states in quantum and classical mechanics

Momentum maps for mixed states in quantum and classical mechanics

Riemannian Geometry of the Contactomorphism Group

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

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