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[NOTES/CM-08004] Equation of Motion in Non Inertial Frames

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 We derive an expression for Lagrangian for motion of a charged particle in a rotating frame, It is shown that the equation of motion can be written as

\begin{eqnarray} m\ddot{\vec{x}}=\vec{F_{e}}-2m\vec{\omega}\times{\dot{\vec{x}}}-m\vec{\omega} \times(\vec{\omega}\times\vec{x}) \end{eqnarray} where \(\vec {F}_e\) is the external force. As seen from the rotating frame, the particle moves as if it is under additional forces

• $-2m\vec{\omega}\times{\dot{\vec{x}}}$ is called Coriolis force
• $-m\vec{\omega}\times(\vec{\omega}\times\vec{x})$ is known as centrifugal force 

[NOTES/CM-08001] The Group of Special Orthogonal Matrices in \(N\) Dimensions

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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 \(SO(3)\).

[NOTES/CM-09011] General Displacement of a Rigid Body

TO BE FINALIZED

[NOTES/CM-09012] Why Two Sets, Body and Space Sets, of Axes?

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.