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Title+Summary	Name	Date
[NOTES/CM-08009] Centrifugal force --- Effect of Earth's Rotation Node id: 6222page 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.	kapoor	24-06-19 16:06:55	n
[NOTES/CM-08015] Rotations in Three Dimensions Node id: 6301page 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. $\newcommand{\Prime}{{^\prime}}$	kapoor	24-06-18 05:06:55	n
[NOTES/CM-08013] Rotation of a Vector about \(X_3\)- Axis Node id: 6230page We derive the transformation rules for a rotation about \(X_3\)- axis. The concept of active and passive rotations is briefly explained.$\newcommand{\Label}[1]{\label{#1}}, \newcommand{\Prime}{^\prime}\newcommand{\eqRef}[1]{\eqref{#1}}$	kapoor	24-06-17 22:06:43	n
[NOTES/CM-08011] Rotations about a fixed axis Node id: 6224page Rotations about a fixed axis form a one parameter subgroup of rotations.	kapoor	24-06-17 22:06:47	n
[NOTES/CM-08006] Axis Angle Parametrization of Rotation Matrix Node id: 6218page $\newcommand{\Prime}{{^\prime}}\newcommand{\U}[1]{\underline{\sf #1}}$ 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}	kapoor	24-06-17 22:06:34	n
[NOTES/CM-08004] Equation of Motion in Non Inertial Frames Node id: 6220page $\newcommand{\U}[1]{\underline{\sf #1}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\Label}[1]{\label{#1}}$ 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	kapoor	24-06-17 20:06:56	n
[NOTES/CM-08014] Active and Passive Rotations Node id: 6229page The active and passive view of rotations are defined and relationship between them is described. $\newcommand{\Label}[1]{\label{#1}}$	kapoor	24-06-17 19:06:18	n
[NOTES/CM-08001] The Group of Special Orthogonal Matrices in \(N\) Dimensions Node id: 6232page $\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\U}[1]{\underline{\sf #1}}$ 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 \(SO(3)\).	kapoor	24-06-16 15:06:35	n
[NOTES/CM-08010] Motion in Frames with Linear Acceleration Node id: 6223page $\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$ The equations of motion in a linearly accelerated are are derived and an expression for pseudo force is obtained.	kapoor	24-05-09 12:05:47	n
[NOTES/CM-08003] Euler Angles Node id: 6216page $\newcommand{\Prime}{^{\prime}}$ Euler angles are an important ways of parametrization of rotations. The definition of Euler angles and and expression of the rotation matrix in terms of the Euler angles are given.	kapoor	24-05-09 12:05:39	n
[NOTES/CM-08008] Proper Rotations and $SO(3)$ Node id: 6219page $\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$ The definition and properties of proper rotations are presented.	kapoor	24-05-09 10:05:41	n
[NOTES/CM-08012] Matrices for Rotations about Coordinate Axes Node id: 6225page The rotation matrices for rotations about the three axes are listed.	kapoor	24-05-03 08:05:13	n
[NOTES/CM-08002] An explicit form of rotation matrix Node id: 6214page For two sets of coordinate axe \(K\) and \(K^\prime\) having common origin, an explicit form of the rotation matrix connecting them is obtained in terms of direction cosines.	kapoor	24-05-03 04:05:37	n
[NOTES/CM-08005] Finite Rotations of Vectors about an Arbitrary Axis Node id: 6217page $\newcommand{\Prime}{{^\prime}}$ Using geometrical arguments, we will derive the result \begin{equation}\vec{A}^\prime = \vec{A} - (\hat{n}\times\vec{A})\, \sin\alpha + \hat{n}\times (\hat{n}\times\vec{A})\, (1-\cos\alpha ) \end{equation}between components of vectors related by a rotation by and angle \(\theta\) about an axis \(\hat n\).	kapoor	24-05-02 20:05:20	n

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.
$\newcommand{\Prime}{{^\prime}}$

We derive the transformation rules for a rotation about $X_3$- axis. The concept of active and passive rotations is briefly explained.$\newcommand{\Label}[1]{\label{#1}}, \newcommand{\Prime}{^\prime}\newcommand{\eqRef}[1]{\eqref{#1}}$

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

$\newcommand{\Prime}{{^\prime}}\newcommand{\U}[1]{\underline{\sf #1}}$

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}

$\newcommand{\U}[1]{\underline{\sf #1}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\Label}[1]{\label{#1}}$

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

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

$\newcommand{\Label}[1]{\label{#1}}$

$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\U}[1]{\underline{\sf #1}}$

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 $SO(3)$.

$\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$

The equations of motion in a linearly accelerated are are derived and an expression for pseudo force is obtained.

$\newcommand{\Prime}{^{\prime}}$

Euler angles are an important ways of parametrization of rotations. The definition of Euler angles and and expression of the rotation matrix in terms of the Euler angles are given.

$\newcommand{\Prime}{{^\prime}}\newcommand{\Label}[1]{\label{#1}}$

The definition and properties of proper rotations are presented.

The rotation matrices for rotations about the three axes are listed.

For two sets of coordinate axe $K$ and $K^\prime$ having common origin, an explicit form of the rotation matrix connecting them is obtained in terms of direction cosines.

$\newcommand{\Prime}{{^\prime}}$

Using geometrical arguments, we will derive the result \begin{equation}\vec{A}^\prime = \vec{A} - (\hat{n}\times\vec{A})\, \sin\alpha + \hat{n}\times (\hat{n}\times\vec{A})\, (1-\cos\alpha ) \end{equation}between components of vectors related by a rotation by and angle $\theta$ about an axis $\hat n$.

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[NOTES/CM-08009] Centrifugal force --- Effect of Earth's Rotation

[NOTES/CM-08015] Rotations in Three Dimensions

[NOTES/CM-08013] Rotation of a Vector about \(X_3\)- Axis

[NOTES/CM-08011] Rotations about a fixed axis

[NOTES/CM-08006] Axis Angle Parametrization of Rotation Matrix

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

[NOTES/CM-08014] Active and Passive Rotations

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

[NOTES/CM-08010] Motion in Frames with Linear Acceleration

[NOTES/CM-08003] Euler Angles

[NOTES/CM-08008] Proper Rotations and $SO(3)$

[NOTES/CM-08012] Matrices for Rotations about Coordinate Axes

[NOTES/CM-08002] An explicit form of rotation matrix

[NOTES/CM-08005] Finite Rotations of Vectors about an Arbitrary Axis