The Journal of Geometric Mechanics (JGM)

Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion

Pages: 223 - 241, Volume 2, Issue 3, September 2010      doi:10.3934/jgm.2010.2.223

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Sebastián Ferrer - Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo, Spain (email)
Martin Lara - Real Observatorio de la Armada, ES-11 110 San Fernando, Spain (email)

Abstract: The Hamilton-Jacobi equation in the sense of Poincaré, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction and perturbation theory. We illustrate our approach dealing with attitude and orbital dynamics. Based on the use of Andoyer and Whittaker symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation of the free rigid body motion and Kepler problem, respectively, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in attitude and orbital dynamics. In addition, new canonical transformations are demonstrated.

Keywords:  Hamiltonian mechanics, Hamilton-Jacobi equation, Poincaré regularization, Andoyer variables, Whittaker variables, symplectic transformations, rotational dynamics, orbital dynamics.
Mathematics Subject Classification:  Primary: 34C20, 37N05, 47A55; Secondary: 70F15, 70E20, 70H20.

Received: September 2009;      Revised: August 2010;      Available Online: November 2010.