Asymmetric Heavy Top

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A top is a rigid body moving with one fixed point. It is classified as symmetric when the moments of inertia with respect to the and axes are equal, which produces periodic motion. In the case of an asymmetric top, the motion becomes chaotic. This Demonstration plots the solutions of the six first-order nonlinear differential equations describing the dynamics of an asymmetric top.

Contributed by: Enrique Zeleny (April 2013)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The equations for the velocity vector (a vertical unit vector in a laboratory frame relative to the positions of the axes of the body frame) are given by

,

,

,

and the Euler equations for the changes of angular momentum produced by the gravitational torque are

,

,

,

where is the angular velocity vector, the mass of the body, the acceleration of gravity, (a vector) represents the position of the center of gravity, and , , are the principal moments of inertia.

Reference

[1] T. Té́l and M. Gruiz, Chaotic Dynamics, An Introduction Based on Classical Mechanics, New York: Cambridge University Press, 2006.



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