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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.

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.

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.

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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All  orthogonal  all \(N\times N\) orthogonal matrices  form a group called \(O(N)\). The set of all orthogonal  matrices with unit determinant  form a subgroup \(SO(N)\) . The group of all proper rotations coincides with \(SO(3)\).