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The Lagrange rigid body motion

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The Lagrange rigid body motion ANNALES DE L’INSTITUT FOURIER TUDOR RATIU P. VAN MOERBEKE The Lagrange rigid body motion Annales de l’institut Fourier, tome 32, no 1 (1982), p. 211-234. <http://www.numdam.org/item?id=AIF_1982__32_1_211_0> © Annales de l’institut Fourier, 1982, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 32, 1 (1982), 211-234 THE LAGRANGE RIGID BODY MOTION by T. RATIU and P. van MOERBEKE In this paper, we discuss the three-dimensional rigid body motion about a fixed point under the influence of gravity; the main emphasis will be put on its symplectic structure, its constants of the motion and the origin of these from a group theoretical point of view. It can be expressed as a Hamiltonian vector field on the 6-dimensional space SO (3) x so (3). The in variance of the problem under rotation about the direction of gravity and about the axis of symmetry, leads to conservation of angular momentum with regard to the gravity axis. This invariant together with a trivial extra-invariant leads to a reduc- tion of this problem to a smooth manifold symplectically diffeomor- phic to a four-dimensional submanifold of so (3 ) x so (3). The Hamiltonian vector field in this reduced manifold leads precisely to the customary Euler-Poisson equations M = M x f t + r x x and r = r x Sl, where M, i2, F and \ denote respectively the angular momentum, angular velocity, the coordinates of the unit vector in the direction of gravity and the coordinates of the center of

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