Optimal Reduction

Ortega, Juan‐Pablo

doi:10.1007/978-1-4757-3811-7_9

Cited by 3 publications

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“…In this case the coadjoint orbits are replaced by presymplectic homogeneous manifolds that admit a natural symplectic decomposition. The reader can check the details in 26 . Reduction by stages.…”

Section: Another Related Developmentsmentioning

confidence: 99%

Some Remarks About the Geometry of Hamiltonian Conservation Laws

Ortega

2003

Symmetry and Perturbation Theory

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Section: Another Related Developmentsmentioning

confidence: 99%

Some Remarks About the Geometry of Hamiltonian Conservation Laws

Ortega

2003

Symmetry and Perturbation Theory

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“…For instance, the convexity properties of its image will be presented in a future publication. Some natural questions about this object have been recently answered; for instance how to carry out orbit reduction and reduction by stages in the optimal framework is already well understood (Marsden, Misiolek, Ortega, Perlmutter, and Ratiu [2001]; Ortega [2001a]). Incidentally, this shows how to perform standard Hamiltonian singular reduction by stages without using the so called stages hypothesis.…”

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confidence: 99%

The Optimal Momentum Map

Ortega¹,

Raţiu²

Geometry, Mechanics, and Dynamics

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The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden-Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian system with fewer degrees of freedom. However, in a large number of situations, this standard approach does not work or is not efficient enough, in the sense that it does not use all the information encoded in the symmetry of the system. In this work, a new momentum map will be defined that is capable of overcoming most of the problems encountered in the traditional approach.Let (M, ω, h) be a Hamiltonian system and G be a Lie group with Lie algebra g, acting canonically on M ; ω denotes the symplectic two-form on the phase space M and h : M → R is the Hamiltonian function. The triplet (M, ω, h) is called a G-Hamiltonian system or one says that (M, ω, h) has symmetry G , if h is a G-invariant function. The G-action on M is said to be globally Hamiltonian if there exists a G-equivariant map J : M → g * with respect to the G-action on M and the coadjoint action on the dual g * of the Lie algebra g, such that, for each ξ ∈ g the vector field associated to the infinitesimal generator ξ M is Hamiltonian with Hamiltonian function J ξ := J, ξ (the symbol ·, · denotes the natural pairing of g with its dual g * ). The map J is called the momentum map associated to the canonical G-action on M . The main interest in finding the symmetries of a given system lies in the conservation laws associated to them provided by the following classical result due to E. Noether (see Noether [1918]).1.1 Theorem (Noether). Let (M, ω, h) be a G-Hamiltonian system. If the G-action on M is globally Hamiltonian with associated momentum map J : M → g * , then J is a constant of the motion for h, that is:where F t is the flow of X h , the Hamiltonian vector field associated to h.In other words, for each µ ∈ g * J := J(M ), the (connected components of the) level set J −1 (µ) is preserved by the dynamics induced by any Ginvariant Hamiltonian. Notice that this allows us to look at g * J as a set of labels that index a family of sets that are invariant under the flows associated to G-invariant Hamiltonian functions. The problem with this classical approach to the interplay between symmetries and conservation laws resides in the fact that in a number of important situations it cannot be implemented or, even if it can be implemented, it is grossly inefficient in the sense that the sets labeled by g * J are not the smallest subsets of M preserved by G-invariant dynamics. The following situations exemplify these problems:(i) The simplest situation in which the labeling by g * J is not optimal is when the level sets J −1 (µ) are not connected. Notice that Ginvariant dynamics preserves not only J −1 (µ), that is, the sets labeled by g * J , but also their connected components. Even though in several importa...

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Singular dual pairs

Ortega¹

2003

Differential Geometry and its Applications

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We generalize the notions of dual pair and polarity introduced by S. Lie [Lie90] and A. Weinstein [W83] in order to accommodate very relevant situations where the application of these ideas is desirable. The new notion of polarity is designed to deal with the loss of smoothness caused by the presence of singularities that are encountered in many problems in Poisson and symplectic geometry. We study in detail the relation between the newly introduced dual pairs, the quantum notion of Howe pair, and the symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs arising in the context of symmetric Poisson manifolds are treated with special attention. We show that in this case and under very reasonable hypotheses we obtain a particularly well behaved kind of dual pairs that we call von Neumann pairs. Some of the ideas that we present in this paper shed some light on the optimal momentum maps introduced in [OR02a]. ContentsAbstract 1 are symplectically orthogonal distributions. This remark motivates the following definition: the diagramis called a Lie-Weinstein dual pair when ker T π 1 and ker T π 2 are symplectically orthogonal distributions. Every Lie-Weinstein dual pair where the submersions π 1 and π 2 have connected fibers is a Howe pair (see Corollary 5.4).The geometries underlying two Poisson manifolds forming a Lie-Weinstein pair are very closely related. For instance, if the fibers of the submersions π 1 and π 2 are connected, the symplectic leaves of P 1 and P 2 are in bijection [KKS78, W83, Bl01] and, for any m ∈ M , the transverse Poisson structures on P 1 and P 2 at π 1 (m) and π 2 (m), respectively, are anti-isomorphic [W83].Apart from the already mentioned studies on representation theory, dual pairs occur profusely in finite and infinite dimensional classical mechanics (see for instance [MRW84,MW83], and references therein). Another intimately related concept that we will not treat in our study is that of the Morita equivalence of two Poisson manifolds [Mor58, Xu91]. Nice presentations of the classical theory of dual pairs can be found in [CaWe99] and in [Land98].In this paper we will pay special attention to the dual pairs that appear in the reduction of Poisson symmetric systems. We introduce this situation with a very simple example: let (M, ω) be a symplectic manifold and G be a Lie group acting freely, canonically, and properly on M . Suppose that this action has a momentum map J : M → g * associated. If we denote by g * J the image of M by J, it is easy to check that the diagramis a Lie-Weinstein dual pair, and consequently, if the group is connected and J has connected fibers, a Howe pair. However, in most cases of practical interest, the freeness assumption on the G-action is not satisfied hence it is worth studying the impact of dropping this condition in the duality between M/G and g * J . When the action is not free M/G and g * J still form a Lie-Weinstein dual pair in a generalized sense since, even though M/G is not a smooth manifold anymore, the tangent space to the G...

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Optimal Reduction

Cited by 3 publications

References 16 publications

Some Remarks About the Geometry of Hamiltonian Conservation Laws

Some Remarks About the Geometry of Hamiltonian Conservation Laws

The Optimal Momentum Map

Singular dual pairs

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