Unperturbed chandler motion and perturbation theory of the rotation motion of deformable celestial bodies

Barkin, Yu. V.

doi:10.1080/10556799808232092

Cited by 8 publications

(6 citation statements)

References 5 publications

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“…In that case, even the standard successive approximations method is suitable for finding the direct integration of the equations of motion by simple quadratures (see, for instance, [9] and references therein). In the triaxial case, however, because of the elliptic integrals and functions on which the solution of the unperturbed problem rests, expanding the disturbing function as a trigonometric series may require a nontrivial preprocessing in which elliptic integrals and functions are conveniently expressed in terms of Jacobi theta functions [4,10]. Notwithstanding, some exceptions are found where the solution of the perturbed problem can be computed directly in elliptic functions, at least up to the first order, without need of resorting to series expansions [11,12,7].…”

Section: Introductionmentioning

confidence: 99%

Short-axis-mode rotation of a free rigid body by perturbation series

Lara¹

2014

Celest Mech Dyn Astr

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A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton-Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita's corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter.

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Section: Introductionmentioning

confidence: 99%

Short-axis-mode rotation of a free rigid body by perturbation series

Lara¹

2014

Celest Mech Dyn Astr

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“…Also they are used in rotational dynamics as (1) the periodic solutions of a Kovalevskaya top (El-Sabaa 1992), (2) the integrable case of a rotational motion of a gyrostat with and without a rotor (Cavas and Vigueras 1994;Elipe and Lanchares 2008), (3) the descriptions of torque-free rotation of a triaxial rigid body (Barkin 1999;Fukushima 2008a), (4) the construction of symplectic integrator for rotation (Breiter and Buciora 2000;Fukushima 2009a), and (5) the application of the implicit midpoint integrator for satellite attitude dynamics (Hellstrom and Mikkola 2009).…”

Section: Jacobian Elliptic Functionsmentioning

confidence: 99%

Fast computation of complete elliptic integrals and Jacobian elliptic functions

Fukushima

2009

Celest Mech Dyn Astr

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As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K (m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9-19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K (1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K (m) and E(m) by means of Jacobi's nome, q, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K (m)/4, with the help of the new method to compute K (m). The new procedure is 25-70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K (m) is not taken into account.

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“…the general solution to the considered problem has the form (9) Quantities , and in formula (9) are the initial values determined through Andoyer variables at t = 0 by expressions (7). The phases w 1 and w 2 and frequencies n 1 and n 2 correspond to the Chandler motion of the poles and to daily rotation of the deformable Earth, respectively.…”

Section: Study Of Chandler Motion Of the Polesmentioning

confidence: 99%

“…High-precision data of experimental observations and measurements of the trajectory of motion of the Earth's poles testify that very complex dynamical processes take place in the Earth-Moon-Sun system [1][2][3][4][5][6][7][8][9]. The creation of an adequate mathematical model which would allow one to describe satisfactorily the actual trajectories of the rotational axis (instantaneous positions of the angular velocity vector) in some convenient frame of reference associated with the Earth is an interesting problem of theoretical and celestial mechanics.…”

Section: Introductionmentioning

confidence: 99%

A Celestial Mechanics Model of Oscillations of the Poles of a Deformable Earth

Акуленко

Kumakshev

Markov³

2005

Cosmic Res

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The problem of Chandler motion of the Earth's poles is studied in the context of the model of a viscoelastic body. The Earth-Moon system is considered as a binary planet rotating around their barycenter. Numerical values of the period and amplitudes of oscillations of the poles are obtained by estimating the elastic deformation of the Earth and the variation of its inertia tensor, and they agree well with observational data. An evolution model of the Earth-Moon-Sun system is constructed by taking into account tidal forces of dissipation character. By means of the method of averaging, the qualitative properties of motion on asymptotically large intervals of time (comparable and essentially longer than the period of precession of the Earth's axis) are established and commented on.

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Unperturbed chandler motion and perturbation theory of the rotation motion of deformable celestial bodies

Cited by 8 publications

References 5 publications

Short-axis-mode rotation of a free rigid body by perturbation series

Short-axis-mode rotation of a free rigid body by perturbation series

Fast computation of complete elliptic integrals and Jacobian elliptic functions

A Celestial Mechanics Model of Oscillations of the Poles of a Deformable Earth

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