Generalized Hopf fibration and geometric SO(3) reduction of the 4DOF harmonic oscillator

J.C. van der Meer; F. Crespo; S. Ferrer

doi:10.1016/S0034-4877(16)30021-0

Generalized Hopf fibration and geometric SO(3) reduction of the 4DOF harmonic oscillator

J.C. van der Meer, F. Crespo, S. Ferrer

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Abstract

It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.

Original language	English
Pages (from-to)	239-249
Number of pages	11
Journal	Reports on Mathematical Physics
Volume	77
Issue number	2
DOIs	https://doi.org/10.1016/S0034-4877(16)30021-0
Publication status	Published - 2016

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title = "Generalized Hopf fibration and geometric SO(3) reduction of the 4DOF harmonic oscillator",

abstract = "It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.",

author = "{van der Meer}, J.C. and F. Crespo and S. Ferrer",

year = "2016",

doi = "10.1016/S0034-4877(16)30021-0",

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AU - van der Meer, J.C.

AU - Crespo, F.

AU - Ferrer, S.

PY - 2016

Y1 - 2016

N2 - It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.

AB - It is shown that the generalized Hopf map ℍ×ℍℍ×ℍ → ℍ×ℝ×ℝℍ×ℝ×ℝquaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7S7 → S4S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibration and associated twistor fibration play a role in the geometry of the Kepler problem and the rigid body problem.

U2 - 10.1016/S0034-4877(16)30021-0

DO - 10.1016/S0034-4877(16)30021-0

M3 - Article

SN - 0034-4877

VL - 77

SP - 239

EP - 249

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

IS - 2

ER -

Generalized Hopf fibration and geometric SO(3) reduction of the 4DOF harmonic oscillator

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