On averaging, reduction, and symmetry in hamiltonian systems

Churchill, R. C.; Kummer, Martin; Rod, David L.

doi:10.1016/0022-0396(83)90003-7

Cited by 109 publications

(93 citation statements)

References 18 publications

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“…This is one of the simplest nontrivial examples where reduction can be explicitly carried out with applications to such problems as the H6non-Heiles Hamiltonian (see [5], [13,14], and [21]). A special case of such a system is the twodimensional isotropic harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

“…In the case that the periodic orbits originate as the energy increases from an equilibrium point of the Hamiltonian at energy zero, this can often be done by some perturbation scheme that then gives their existence in the original Hamiltonian flow at low positive energies (see for example [19]). Such a program is carried out in detail in [5], [13,14], and [21].…”

Section: Introductionmentioning

confidence: 99%

“…500-502] and the references therein). This latter method was applied to the H6non-Heiles Hamiltonian in [5].…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

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Reduction of the semisimple 1:1 resonance

Cushman

Rod

1982

Physica D: Nonlinear Phenomena

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“…We recall the following result, a complete proof of which can be found for instance in [6], [7] and [12]. It proceeds by constructing a sequence of symplectic "normal form transformations".…”

Section: The Birkhoff Normal Formmentioning

confidence: 99%

“…For Hamiltonian systems with symmetry, the following elegant and well-known result is often useful, see [6] and [12]: Theorem 4.2 Let H = H 2 + H 3 + . .…”

Section: The Birkhoff Normal Formmentioning

confidence: 99%

An Integrable Approximation for the Fermi–Pasta–Ulam Lattice

Rink

The Fermi-Pasta-Ulam Problem

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This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular this proves Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal Birkhoff normal form computations of Nishida, the KAM theorem and discrete symmetry considerations.

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Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

Ortega

1996

Journal of Differential Equations

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On averaging, reduction, and symmetry in hamiltonian systems

Cited by 109 publications

References 18 publications

Reduction of the semisimple 1:1 resonance

Reduction of the semisimple 1:1 resonance

An Integrable Approximation for the Fermi–Pasta–Ulam Lattice

Periodic Solutions of a Newtonian Equation: Stability by the Third Approximation

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