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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.

The set of all rotations that can be implemented physically  in three dimensions form a group. These most important and frequently used rotations are the rotations which can be implemented physically. These rotations, do not change the handedness of the coordinate axes and are called proper rotations. The improper  rotations take left handed systems to right handed systems, or vice  versa. The statement of  Euler's theorem about rotations is given.
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Rotations about a fixed axis form a one parameter subgroup of  rotations.

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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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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 \(SO(3)\).

TO BE FINALIZED

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.

In the phase space formulation, the constant of motion \(G\) given by Noether's theorem, expressed in terms of coordinates and momenta generates the infinitesimal symmetry transformation.