Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space

Patrick, George W.

doi:10.1016/0393-0440(92)90015-s

Cited by 68 publications

(103 citation statements)

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“…An extension to energy methods, such as the energy-Casimir method, that can be used to prove this kind of stronger stability conclusion is presented in Leonard and Marsden (1996). This work makes use of reduction by stages and a result for compact symmetry groups due to Patrick (1992). We note that Lamb (1932) does not extend his classical stability analysis of a submerged rigid body to the full twelve-dimensional phase space T*SE(3), i.e.…”

Section: Stabilitymentioning

confidence: 99%

“…here G = SE(3) or G = SE(2) X R). Stability modulo G is just the usual notion of Lyapunov stability, except that one allows arbitrary drift along the orbits of G (Patrick, 1992). For example, stability in T*SE(3) modulo SE(3) means that there is the possibility that there will be drift in the rotational and translational parameters but not in the linear and angular momentum parameters.…”

Section: Stabilitymentioning

confidence: 99%

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Stability of a bottom-heavy underwater vehicle

Leonard¹

1997

Automatica

176

109

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Section: Stabilitymentioning

confidence: 99%

Section: Stabilitymentioning

confidence: 99%

Stability of a bottom-heavy underwater vehicle

Leonard¹

1997

Automatica

176

109

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“…The third argument is commonly used in the literature on finite dimensional Hamiltonian systems [Pat92], and appears also in [Stu08] in the infinite dimensional case. It is not universally useable, since it depends on the existence of a G µ -invariant Euclidean structure on the dual of the Lie-algebra of G, as we will see in Section 8.…”

Section: First Argumentmentioning

confidence: 99%

Orbital Stability: Analysis Meets Geometry

Bièvre

Genoud

Nodari

2015

Nonlinear Optical and Atomic Systems

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ABSTRACT. We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

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“…For example, if G acts on itself on the left by group multiplication and if we lift this to an action on T * G by the cotangent lift, then the action is free and so all µ are regular values, but such values (for instance, the zero element in so(3)) need not be regular. On the other hand, in many important stability considerations, a regularity assumption on the point µ is required; see, for instance, Patrick [1992], Ortega and Ratiu [1999b] and Patrick, Roberts, and Wulff [2004].…”

Section: And Only If J(z) = Admentioning

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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Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space

Cited by 68 publications

References 1 publication

Stability of a bottom-heavy underwater vehicle

Stability of a bottom-heavy underwater vehicle

Orbital Stability: Analysis Meets Geometry

Hamiltonian Reduction by Stages

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