The Euler-Poincaré equations and double bracket dissipation

Bloch, Anthony M.; Krishnaprasad, P. S.; Marsden, Jerrold E.; Raţiu, Tudor S.

doi:10.1007/bf02101622

Cited by 179 publications

(182 citation statements)

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“…In Marsden and Scheurle [1993b] it was shown that, for matrix groups, one could view the Euler-Poincaré equations via the reduction of Hamilton's variational principle from T G to g. The work of Bloch, Krishnaprasad, Marsden and Ratiu [1996] established the EulerPoincaré variational structure for general Lie groups.…”

Section: Reduction Theory: Historical Overview 31mentioning

confidence: 99%

“…Of course, in general, one of the great successes of reduction theory has been the advancement of stability theory, starting with Poincaré [1901aPoincaré [ ,b, 1910 and Arnold [1966a], and continuing with the energy-momentum method developed in Simo, Lewis and Marsden [1991]; see also Bloch, Krishnaprasad, Marsden and Ratiu [1996]. Apart from the many applications of standard semidirect product theory, we have already mentioned the significant application to underwater vehicle dynamics and stability by Leonard and Marsden [1997].…”

Section: Applications and Infinite Dimensional Problemsmentioning

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: Reduction Theory: Historical Overview 31mentioning

confidence: 99%

Section: Applications and Infinite Dimensional Problemsmentioning

confidence: 99%

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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“…(For a discussion with the links with the Lie-Poisson equations, see for example, Marsden and Ratiu [1994]; also see this reference and Bloch, Krishnaprasad, Marsden and Ratiu [1996] for the proof.) Theorem 2.4 Let G be a Lie group and L : T G → R a left invariant Lagrangian.…”

Section: The Euler-poincaré Equations and Variational Principlesmentioning

confidence: 99%

“…Symmetry plays a special role in variational principles. Not only does it lead to conservation laws of Noether, but the reduced variational principle for the Euler-Poincaré equations on a general Lie algebra induced by Hamilton's principle on the corresponding Lie group was only recently found (Marsden and Scheurle [1993b] and Bloch, Krishnaprasad, Marsden and Ratiu [1996]). In fluid mechanics, such variational principles were associated with "Lin constraints", but even here it was only with work such as Seliger and Whitham [1968] and Bretherton [1970] that the situation was clarified.…”

Section: Introductionmentioning

confidence: 99%

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

Marsden

Wendlandt

1997

Current and Future Directions in Applied Mathematics

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This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov.The general idea for these variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators.

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“…More recently, several scientists working in the field called "Geometric Mechanics", used the equations obtained by Poincaré (which they called "Euler-Poincaré equations") to solve various problems. Following a remark made by Poincaré at the end of his note, several authors observed that these equations become very simple when the Lagrangian L is such that its value L(v) at a vector v tangent to the configuration space at a point x, only depends on the element of the Lie algebra of vector fields which, at the point x, takes the value v. Modern authors sometimes call "Lagrangian reduction" [2,3,4,5,6,7,8,9,10,18] the use of that property to make easier the determination of motions of the system. Assumptions made in these recent papers and books seem to us very often more restrictive than those made by Poincaré himself; for example, several modern authors assume that the mechanical system under study has a Lie group as configuration space, and that its dynamics is described by a Lagrangian invariant under the lift to the tangent bundle of the action of this group on itself by translations either on the right or on the left.…”

Section: Introductionmentioning

confidence: 99%

On Henri Poincaré's Note "Sur une forme nouvelle des équations de la Mécanique"

Marle¹

2013

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The Euler-Poincaré equations and double bracket dissipation

Cited by 179 publications

References 64 publications

Hamiltonian Reduction by Stages

Hamiltonian Reduction by Stages

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

On Henri Poincaré's Note "Sur une forme nouvelle des équations de la Mécanique"

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