The Journal of Geometric Mechanics (JGM)

Andoyer's variables and phases in the free rigid body

Pages: 25 - 37, Volume 6, Issue 1, March 2014      doi:10.3934/jgm.2014.6.25

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Sebastián Ferrer - Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain (email)
Francisco J. Molero - Departamento de Matemática Aplicada, Universidad de Murcia, Murcia, 30071 Espinardo, Spain (email)

Abstract: Using Andoyer's variables we present a new proof of Montgomery's formula by measuring $\Delta\mu$ when $\nu$ has made a rotation. Our treatment is built on the equations of the differential system of the free rigid solid, together with the explicit expression of the spherical area defined by the intersection of the surfaces given by the energy and momentum integrals. We also consider the phase $\Delta\nu$ of the moving frame when $\mu$ has made a rotation around the angular momentum vector, and we give the formula for its computation.

Keywords:  Dynamical systems, integrable Hamiltonian systems, free rigid body, Montgomery's phase, Andoyer's variables.
Mathematics Subject Classification:  Primary: 70E15.

Received: May 2013;      Revised: November 2013;      Available Online: April 2014.