Reconstruction phases via Poisson reduction

Blaom, Anthony D.

doi:10.1016/s0926-2245(00)00014-0

Cited by 14 publications

(32 citation statements)

References 12 publications

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“…The corresponding connection one-form is then obtained in terms of the momentum map, symplectic and complex structures. These expressions enable us to further simplify the computation of the reconstruction phases as described in [3] for the case of Abelian groups. The requirement of the group being Abelian is essential for the geometric phase part (see Section 4.2), but it is not used in the construction of the map L involving the abstract inertia tensor and in the expression for the dynamic phase.…”

Section: Overview Of the Resultsmentioning

confidence: 99%

“…The relation between the abstract mechanical connection and the well-known mechanical connection is considered in [3], where the reconstruction phases for the cotangent bundles were analyzed, though not from the point of view of almost Kähler manifolds and corresponding abstract mechanical connections.…”

Section: Overview Of the Resultsmentioning

confidence: 99%

“…In this section, we briefly overview the results on reconstruction phases obtained in [3]. We refer the reader to the original paper for detailed and comprehensive treatment of the subject.…”

Section: Reconstruction Connection and The Associated Phasesmentioning

confidence: 99%

“…In [3], a general formula is derived which expresses a reconstruction phase in terms of the associated reduced solution, viewed as a curve in the Poissonreduced phase space P/G, and certain derivatives transverse to the symplectic leaf in P/G containing the curve. Specifically, the dynamic part of the phase depends on transverse derivative in the Poisson-reduced Hamiltonian, while the geometric part is determined by transverse derivatives in the leaf symplectic structures.…”

Section: Reconstruction Connection and The Associated Phasesmentioning

confidence: 99%

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Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Pekarsky

Marsden

2001

Journal of Applied Mathematics

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When the phase space P of a Hamiltonian G-system (P, ω, G, J, H) has an almost Kähler structure, a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (due to the work of A. Blaom), thus significantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.

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Section: Overview Of the Resultsmentioning

confidence: 99%

Section: Overview Of the Resultsmentioning

confidence: 99%

Section: Reconstruction Connection and The Associated Phasesmentioning

confidence: 99%

Section: Reconstruction Connection and The Associated Phasesmentioning

confidence: 99%

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Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Pekarsky

Marsden

2001

Journal of Applied Mathematics

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“…For a general introduction to some of these ideas and for further references, see Marsden, Montgomery and Ratiu [1990]; Simo, Lewis and Marsden [1991]; Marsden and Ostrowski [1996]; Marsden and Ratiu [1999]; Montgomery [1988Montgomery [ , 1990Montgomery [ , 1991aMontgomery [ ,b, 1993; Blaom [2000Blaom [ , 2001, and Kanso, Marsden, Rowley, and Melli-Huber [2005].…”

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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PrefaceThis book is about Symplectic Reduction by Stages for Hamiltonian systems with symmetry. Reduction by stages means, roughly speaking, that we have two symmetry groups and we want to carry out symplectic reduction by both of these groups, either sequentially or all at once. More precisely, we shall start with a "large group" M that acts on a phase space P and assume that M has a normal subgroup N . The goal is to carry out reduction of the phase space P by the action of M in two stages; first by N and then by the quotient group M/N . For example, M might be the Euclidean group of R 3 , with N the translation subgroup so that M/N is the rotation group. In the Poisson context such a reduction by stages is easily carried out and we shall show exactly how this goes in the text. However, in the context of symplectic reduction, things are not nearly as simple because one must also introduce momentum maps and keep track of the level set of the momentum map at which one is reducing. But this gives an initial flavor of the type of problem with which the book is concerned.As we shall see in this book, carrying out reduction by stages, first by N and then by M/N , rather than all in "one-shot" by the "large group" M is often not only a much simpler procedure, but it also can give nontrivial additional information about the reduced space. Thus, reduction by stages can provide an essential and useful tool for computing reduced spaces, including coadjoint orbits, which is useful to researchers in symplectic geometry and geometric mechanics.Reduction theory is an old and time-honored subject, going back to the early roots of mechanics through the works of Euler, Lagrange, Poisson, vi Preface Liouville, Jacobi, Hamilton, Riemann, Routh, Noether, Poincaré, and others. These founding masters regarded reduction theory as a useful tool for simplifying and studying concrete mechanical systems, such as the use of Jacobi's elimination of the node in the study of the n-body problem to deal with the overall rotational symmetry of the problem. Likewise, Liouville and Routh used the elimination of cyclic variables (what we would call today an Abelian symmetry group) to simplify problems and it was in this setting that the Routh stability method was developed.The modern form of symplectic reduction theory begins with the works of Arnold [1966a], Smale [1970], Meyer [1973], and Marsden and Weinstein [1974]. A more detailed survey of the history of reduction theory can be found in the first Chapter of the present book. As was the case with Routh, this theory has close connections with the stability theory of relative equilibria, as in Arnold [1969] and Simo, Lewis and Marsden [1991]. The symplectic reduction method is, in fact, by now so well known that it is used as a standard tool, often without much mention. It has also entered many textbooks on geometric mechanics and symplectic geometry, such as Abraham and Marsden [1978], Arnold [1989], Guillemin and Sternberg [1984], Libermann and Marle [1987], and McDuff and Salamon [1995]. De...

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Mechanical Systems: Symmetries and Reduction

Marsden¹,

Raţiu²

2009

Encyclopedia of Complexity and Systems Science

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Reconstruction phases via Poisson reduction

Cited by 14 publications

References 12 publications

Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Abstract mechanical connection and abelian reconstruction foralmost Kähler manifolds

Hamiltonian Reduction by Stages

Mechanical Systems: Symmetries and Reduction

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