A heavy top is a rigid body moving under influence of gravity with one of its points fixed. A brief description of four interesting cases of a heavy top is given.
TO BE FINALIZED
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We discuss some general questions about, choice of frames of reference, generalized coordinates and constants of motion.
The Newton's laws hold in an inertial frame. However the equations of motion involve the moment of inertia tensor which in turn depends on the orientation of the body and varies with time. This make it solution to the motion of a rigid body problem impossible. This difficulty is absent in the body fixed axes, the moment of inertia tensor depends only on the the geometry of the problem. So whether we use space axes ,or the body axes, depends on the problem to be solved, we use axes which makes the solution of the problem simpler.
The angular momentum of a rigid body is given by where \(\mathbf I\) is moment of inertia tensor and \(\vec \omega \) is the angular velocity.\begin{eqnarray} \vec{L}&=&\int dv \rho(\vec{X})\vec{X}\times(\vec{\omega}\times\vec{X})\\ &=&\int dV \rho(\vec{X})\Big[(\vec{X}\cdot\vec{X})\vec{\omega}-(\vec{X}\cdot\vec{\omega} )\vec{X}\Big] \end{eqnarray} or \(\vec L=\mathbf I\, \vec \omega\).
A possible way of specifying the orientation of a rigid body is to give orientation of body fixed axes w.r.t. a space fixed axes. Euler angles are a useful set generalized coordinates to specify orientation of the body axes relative to a space fixed axis.
By considering possible motions of a rigid body with one, two or three points fixed, we show that a rigid body has six degrees of freedom.
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We set up Lagrangian for a heavy symmetrical top and show that the solution can be reduced to quadratures.
An expression for the kinetic energy in terms of the moment of inertia tensor and the angular velocity w.r.t the body frame of reference is obtained. It is shown that \begin{equation}\text{KE}=\sum_{ij} \omega_{bi} I_{ij}^{(b)} \omega_{ij}\end{equation}
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