A geometric setting for Hamiltonian perturbation theory

Blaom, Anthony D.

doi:10.1090/memo/0727

Cited by 20 publications

(38 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the defining condition of a dual pair. As as corollary of a result by Weinstein [1983a]; Blaom [2001] in the context of the theory of dual pairs (the so called Symplectic Leaves Correspondence Theorem) we see that if J has connected fibers then there is a bijective correspondence between the symplectic leaves of M/G, namely the symplectic orbit reduced spaces, and those of g * J , that is, the coadjoint orbits inside J(M ). The notions of duality and dual pair have been generalized in Ortega [2003a] in such way that in many situations the optimal momentum map provides an example of these newly introduced dual pairs.…”

Section: Momentum Level Sets and Associated Isotropiesmentioning

confidence: 77%

“…The proof of this theorem when O µ is an embedded submanifold of g * can be found in Marle [1976], Kazhdan, Kostant and Sternberg [1978], with useful additions given in Marsden [1981] and Blaom [2001]. For nonfree actions and when O µ is not an embedded submanifold of g * see [HRed].…”

Section: Theorem (Symplectic Orbit Reduction Theorem)mentioning

confidence: 99%

“…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%

“…* forms a dual pair and it has been shown (see Weinstein [1983a]; Blaom [2001]) that whenever we have two Poisson manifolds in the legs of a dual pair (…”

Section: Optimal Point Reductionmentioning

confidence: 99%

“…In this paper the authors use a combination of distribution theory with Sikorski differential spaces to show that the orbit reduced space is a symplectic manifold. Nevertheless, the first reference where the standard formula (13.5.1) appears at this level of generality is Blaom [2001]. In that paper the author only deals with the free case.…”

Section: Orbit Reduction: Beyond Compact Groupsmentioning

confidence: 99%

See 4 more Smart Citations

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

View full text Add to dashboard Cite

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...

show abstract

Section: Momentum Level Sets and Associated Isotropiesmentioning

confidence: 77%

Section: Theorem (Symplectic Orbit Reduction Theorem)mentioning

confidence: 99%

Section: This Takes Us Up To About 1972mentioning

confidence: 99%

“…* forms a dual pair and it has been shown (see Weinstein [1983a]; Blaom [2001]) that whenever we have two Poisson manifolds in the legs of a dual pair (…”

Section: Optimal Point Reductionmentioning

confidence: 99%

Section: Orbit Reduction: Beyond Compact Groupsmentioning

confidence: 99%

See 3 more Smart Citations

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

View full text Add to dashboard Cite

show abstract

Nekhoroshev Theory

Niederman

2009

Perturbation Theory

View full text Add to dashboard Cite

Optimal Reduction

Ortega¹

2004

Momentum Maps and Hamiltonian Reduction

View full text Add to dashboard Cite

We generalize various symplectic reduction techniques of Marsden, Weinstein, Sjamaar, Bates, Lerman, Marle, Kazhdan, Kostant, and Sternberg to the context of the optimal momentum map. We see that, even though all those reduction procedures had been designed to deal with canonical actions on symplectic manifolds in the presence of a momentum map, our construction allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map.We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of the theory of singular dual pairs. More specifically, this condition ensures that the orbits in the momentum space of the optimal momentum map (we call them polar reduced spaces) admit a presymplectic structure that generalizes the Kostant-Kirillov-Souriau symplectic structure of the coadjoint orbits in the dual of a Lie algebra. Using this presymplectic structure, the optimal orbit reduced spaces are symplectic with a form that satisfies a relation identical to the classical one obtained by Marle, Kazhdan, Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold. In the Poisson case we provide a sufficient condition for the polar reduced spaces to be symplectic. In the symplectic case the polar reduced spaces are symplectic if and only if certain relation between the tangent space to the orbit and its symplectic orthogonal with the tangent space to the isotropy type submanifolds is satisfied. In general, the presymplectic polar reduced spaces are foliated by symplectic submanifolds that are obtained through a generalization to the optimal context of the so called Sjamaar Principle, already existing in the theory of Hamiltonian singular reduction. We call these subspaces the regularized polar reduced spaces.We use these ideas to shed some light in the problem of orbit reduction of globally Hamiltonian actions when the symmetry group is not compact and in the construction of a family of presymplectic homogeneous manifolds and of its symplectic foliation.We also show that these reduction techniques can be implemented in stages whenever we are in the presence of certain hypotheses that generalize those already existing for free globally Hamiltonian actions. ContentsAbstract 1 Bibliography 41reduction [MW74, SL91, ACG91, BL97, O98, CS01, OR02b]. For the last thirty years, Marsden-Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. More recently, a new momentum map, we call it optimal momentum map, has been introduced [OR02a]. This object is partially inspired by the distribution theoretical approach to the conservation laws induced by symmetry that one can find in the works of Cartan [C22]. The use of this tool allows the construction of symplectically reduced space...

show abstract

A geometric setting for Hamiltonian perturbation theory

Cited by 20 publications

References 2 publications

Hamiltonian Reduction by Stages

Hamiltonian Reduction by Stages

Nekhoroshev Theory

Optimal Reduction

Contact Info

Product

Resources

About