Geometrical properties of the Euler-Poisson equations of a rigid body about a fixed point

Mathematics

Scientific paper

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Body Kinematics, Euler Equations Of Motion, Fixed Points (Mathematics), Poisson Equation, Rigid Structures, Classical Mechanics, Geometry, Gyroscopes, Structural Weight, Transformations (Mathematics)

Scientific paper

The geometrical properties of the Euler-Poisson equations of the motion of a heavy rigid body about a fixed point are studied. Two cases are studied. In the first one the rigid body has a gyrostatic moment, constant with respect to itself. It is shown, from the reduction of the Euler-Poisson equations, that there exists a presymplectic structure for the initial Euler equations, this fact being due to the existence of an invariant volume. Secondly the rigid body is considered without its gyrostatic moment. A new property is revealed which is the existence of an infinitesimal transformation, due to the homogeneity of the equations. This study is illustrated by the integrable case of the symmetrical top.

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