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The active and passive view of rotations are defined and relationship between them is described.

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Rotations about a fixed axis form a one parameter subgroup of  rotations.

The effect of Coriolis force on the force acting on a body on the earth is computed at the poles, equator and at a general point. he banking of railway tracks and of roads is briefly discussed.

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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 

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A closed form expression for rotation matrix is derived for rotations about an axis by a specified angle \(\theta\).

\begin{equation}
R_{\hat{n}}(\theta)=\widehat {Id}-\sin (\theta) (\hat{n}\cdot \vec{I})+(1-\cos \theta)(\hat{n}\cdot\vec{I})^{2}
\end{equation}Here  \(\widehat{Id} \) is the identity matrix. and \(\vec{I}=(I_1,I_2,I_3)\) is given by 
\begin{equation} I_1=\left[\begin{array}{clc} 0 &0 &0\\ 0 &0 &-1\\ 0 &1 &0 \end{array}\right],I_2=\left[\begin{array}{clc} 0 &0 &1\\ 0 &0 &0\\ -1 &0 &0 \end{array}\right],I_3=\left[\begin{array}{clc} 0 &-1 &0\\ 1 &0 &-1\\ 0 &1 &0 \end{array}\right] \end{equation}. 

Also the components of the position vector a point transform a

\begin{equation} {\vec{x}}\Prime=(\hat{n}\cdot{\vec{x}})\hat{n}+\cos\theta\big(\vec{x}-(\vec{x}\cdot\vec{n})\hat{n}\big)-\sin\theta (\hat{n}\times \vec{x})\end{equation}