Reduction of the semisimple 1:1 resonance

Cushman, Richard; Rod, David L.

doi:10.1016/0167-2789(82)90008-2

Cited by 64 publications

(32 citation statements)

References 16 publications

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“…The reduced phase space of the two-mode system (K = 2) with resonance 1:1 is a 2-sphere S 2 which is isomorphic to CP 1 [9]. This basic case has, of course, been studied in great detail, notably in application to the Hénon-Heiles system [10] and its molecular analogues [11], and 1:1 resonant vibrational subsystems of polyatomic molecules [12].…”

Section: Reduction Of Dynamical Symmetrymentioning

confidence: 99%

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Assigning vibrational polyads using relative equilibria: application to ozone

Kozin

Sadovskií

Zhilinskii

2005

Spectrochimica Acta Part A: Molecular and Biomolecular Spectros

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We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy E(n) of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or "hyperpolyads" should be distinguished from the wellknown polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

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Section: Reduction Of Dynamical Symmetrymentioning

confidence: 99%

“…In this case, the polyad space is the space CP 1 which is diffeomorphic to a 2-sphere S 2 [9]. This space is often called the polyad phase sphere [12].…”

Section: A4 Reminder On the Two Mode Casementioning

confidence: 99%

Assigning vibrational polyads using relative equilibria: application to ozone

Kozin

Sadovskií

Zhilinskii

2005

Spectrochimica Acta Part A: Molecular and Biomolecular Spectros

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“…Exploiting the well-known equivalence of the two-dimensional 1:1 harmonic oscillator and an angular momentum system, we introduce vibrational angular momenta v 1 , v 2 , v 3 [69,20,21]. The internal structure of vibrational polyads formed by the doubly degenerate vibrational mode E can be described in terms of these dynamical variables.…”

Section: 42mentioning

confidence: 99%

Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

Efstathiou¹,

Sadovskií²,

Zhilinskii³

2004

SIAM J. Appl. Dyn. Syst.

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Abstract. We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. 1. Introduction. This paper unifies modern methods of classical theory of symmetric Hamiltonian dynamical systems and quantum theory of molecules (and other isolated finiteparticle systems). Considerable progress was achieved in both directions in the last decades and deep relations between these seemingly distant theories became evident. Significant effort by mathematicians and molecular physicists to converge the two fields resulted in the qualitative theory of highly excited quantum molecular systems based on recent mathematical developments. We join the two approaches and demonstrate what kind of concrete results can be immediately obtained in molecular systems [1,2,3,4] by applying powerful methods of symmetric Hamiltonian systems [5,6,7,8,9,10,11]. We choose a concrete problem of rotation-vibration of a four-atomic molecule with tetrahedral equilibrium configuration [12,13] in order to explain the details of our approach.

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“…Restricting the reduction mapping to the three spheres given by the energy surfaces, the orbit mapping becomes a reduction of the energy surface S 3 to the reduced phase space S 2 , which mapping is exactly the Hopf fibration S 3 → S 2 . See [3]. In section 2. we will make this precise starting from the classical description of the Hopf fibration in complex coordinates.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore {G i , G j } = ε ijk G k , thus these functions form a basis for the Lie algebra so(3) and G = SO (3).…”

mentioning

confidence: 99%