Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

Alber, Mark; Luther, G. G.; Marsden, Jerrold E.; Robbins, Jonathan M

doi:10.1016/s0167-2789(98)00127-4

Cited by 52 publications

(65 citation statements)

References 33 publications

(60 reference statements)

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“…In mechanics, the method was developed by Kummer, Cushman, Rod and coworkers in the 1980's. We will not attempt to give a literature survey here, other than to refer to Kummer [1990], Kirk, Marsden and Silber [1996], Alber, Luther, Marsden, and Robbins [1998] and the book of Cushman and for more details and references.…”

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

“…This sort of situation happens in interesting examples, such as Yang-Mills theory and general relativity. There are many other examples of singular reduction, such as those occurring in resonant phenomena (see, for example, Kummer [1981]; Cushman and Rod [1982] and Alber, Luther, Marsden, and Robbins [1998]. It was Sjamaar [1990] and Sjamaar and Lerman [1991] who began the systematic development of the corresponding singular reduction theory.…”

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confidence: 99%

“…In fact, quotients of manifolds by locally free and proper group actions are orbifolds, which follows by the use of the Palais slice theorem (see Palais [1957]). Orbifolds come up in a variety of interesting examples involving, for example, resonances; see, for instance, Cushman and Bates [1997] and Alber, Luther, Marsden, and Robbins [1998] for some specific examples.Symmetry and Singularities. If µ is a regular value of J then we claim that the action is automatically locally free at the elements of the corresponding level set J −1 (µ).…”

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confidence: 99%

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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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Section: This Takes Us Up To About 1972mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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“…17,18 Recently the geometric structure of the three-wave system was exploited to obtain these phase shifts. 10 In general these shifts have a geometric and a dynamic component. The geometric component is proportional to the symplectic area enclosed by an orbit on the three-wave surface.…”

Section: 8mentioning

confidence: 99%

“…10,11 This construction provides a means of visualizing the dynamics of three-wave interactions, analogous to the use of the Poincaré sphere for polarization dynamics, [12][13][14] and it enables existing analysis to be readily comprehended in geometrical terms. In this way, recent techniques of geometric mechanics provide fresh insight into the dynamics of second-harmonic generation and parametric frequency conversion in quadratic, or (2) , media.…”

Section: Introductionmentioning

confidence: 99%

Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

et al. 2000

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We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions.

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Wave Interactions

Vanneste

Nonlinear Waves in Fluids: Recent Advances and Modern Applications

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Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

Cited by 52 publications

References 33 publications

Hamiltonian Reduction by Stages

Hamiltonian Reduction by Stages

Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

Wave Interactions

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