Lecture 2: Realizations of the reduced phase space of a hamiltonian system with symmetry

Kummer, Martin

doi:10.1007/bfb0018326

Cited by 40 publications

(71 citation statements)

References 8 publications

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“…** Research supported in part by NSERC Grant ~A8507. [7] found that for the harmonic oscillator the space of all orbits of a given energy (which is topologically a 3-sphere in R 4) could be mapped by the Hopf mapping to the standard 2-sphere in R 3 (see also [9, p. 169] and [13][14][15]). The Hopf mapping (given in (1.5) below) maps distinct orbits of the oscillator (which are great circles on the 3-sphere) to distinct points of the 2-sphere which is then the reduced space (orbit space) for this oscillator.…”

Section: Introductionmentioning

confidence: 99%

Reduction of the semisimple 1:1 resonance

Cushman

Rod

1982

Physica D: Nonlinear Phenomena

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The method of "averaging" is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated "reduced" Hamiltonian on a "reduced space". This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1 : 1 resonance case. The reduced space will be a 2-sphere in R 3, and the reduced differential equations will be Euler's equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on "degeneracies" due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.

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Section: Introductionmentioning

confidence: 99%

Reduction of the semisimple 1:1 resonance

Cushman

Rod

1982

Physica D: Nonlinear Phenomena

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“…We begin by reviewing reduction theory for Hamiltonian systems with symmetry on principle fiber bundles. As the material is partially available in [4,16], we shall provide only a sketch here using notation that is to be employed in the sequel. Let G denote a Lie group with the unity element e ∈ G and G ≃ T e (G) be its Lie algebra.…”

Section: 2mentioning

confidence: 99%

“…Other aspects of dynamical systems related to properties of reduced symplectic structures were studied in [16,17,18] where, in particular, the reduced symplectic structure was completely described within the framework of the classical Dirac scheme, and several applications to nonlinear (including celestial) dynamics were given.…”

Section: Introductionmentioning

confidence: 99%

The electromagnetic Lorentz condition problem and symplectic properties of Maxwell- and Yang–Mills-type dynamical systems

Bogolubov¹,

Prykarpatsky²,

Taneri³

et al. 2009

J. Phys. A: Math. Theor.

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Abstract. Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups.This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac -Fock -Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered.

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“…This has the structure of a Lie-Poisson bracket (see Marsden and Weinstein [6] or Marsden et al [7] for background and references on Lie-Poisson structures) plus a canonical bracket, although the variables used in these two terms are not independent. The second representation, given in section 4, gives the bracket as a special case of the Poisson bracket on the reduction of the cotangent bundle of a principal bundle by its group due to Montgomery, Marsden and Ratiu [8] (see also Kummer [9]). These brackets have the following general structure (sche- In other examples, the curvature term can represent Coriolis or magnetic forces (see for example, Kummer [9] and Krishnaprasad and Marsden [10]).…”

Section: Introductionmentioning

confidence: 99%

“…The second representation, given in section 4, gives the bracket as a special case of the Poisson bracket on the reduction of the cotangent bundle of a principal bundle by its group due to Montgomery, Marsden and Ratiu [8] (see also Kummer [9]). These brackets have the following general structure (sche- In other examples, the curvature term can represent Coriolis or magnetic forces (see for example, Kummer [9] and Krishnaprasad and Marsden [10]). In our example the Lie-Poisson bracket represents the internal fluid contribution decoupled from the canonical bracket for the boundary motion.…”

Section: Introductionmentioning

confidence: 99%

The Hamiltonian structure for dynamic free boundary problems

Lewis¹,

Marsden²,

Montgomery³

et al. 1986

Physica D: Nonlinear Phenomena

131

120

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Hamiltonian structures for 2-or 3-dimensional incompressible flows with a free boundary are determined which generalize a previous structure of Zakharov for irrotational flow. Our Poisson bracket is determined using the method of Arnold, namely reduction from canonical variables in the Lagrangian (material) description. Using this bracket, the Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated. The structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained. In the case of two-dimensional flows, the vorticity bracket is determined and the generalized enstrophy is shown to be a Casimir function, This investigation also clears up some confusion in the literature concerning the vorticity bracket, even for fixed boundary flows.

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Lecture 2: Realizations of the reduced phase space of a hamiltonian system with symmetry

Cited by 40 publications

References 8 publications

Reduction of the semisimple 1:1 resonance

Reduction of the semisimple 1:1 resonance

The electromagnetic Lorentz condition problem and symplectic properties of Maxwell- and Yang–Mills-type dynamical systems

The Hamiltonian structure for dynamic free boundary problems

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