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[DOC/LOG-2024-03] LOG, HISTORY and INFORMATIONNode id: 6086pageThe workflow, time allotment to different activities of Proofs program are divided into several compartments.
This is a log file |
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24-05-12 17:05:38 |
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[NOTES/CM-08001] The Group of Special Orthogonal Matrices in \(N\) DimensionsNode id: 6232page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\U}[1]{\underline{\sf #1}}$
The definition of special orthogonal group in \(N\) dimensions is given. For \(N=3\) this is just the group of all proper rotations. |
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24-05-10 05:05:05 |
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[DOC/Latex/Macros] Freq-Used-MacrosNode id: 6231page |
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24-05-10 05:05:03 |
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[NOTES/CM-08013] Rotation of a Vector about Coordinate AxesNode id: 6230pageWe derive the transformation rules for a rotation about \(X_3\)- axis. The concept of active and passive rotations is briefly explained.$\newcommand{\Label}[1]{\label{#1}}, \newcommand{\Prime}{^\prime}\newcommand{\eqRef}[1]{\eqref{#1}}$ |
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24-05-09 22:05:49 |
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[NOTES/CM-08014] Active and Passive RotationsNode id: 6229pageThe active and passive view of rotations are defined. |
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24-05-09 12:05:46 |
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[NOTES/CM-08010] Motion in Frames with Linear AccelerationNode 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. |
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24-05-09 12:05:47 |
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[NOTES/CM-08003] Euler AnglesNode 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. |
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24-05-09 12:05:39 |
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[NOTES/CM-08006] Axis Angle Parametrization of Rotation Matrix Node id: 6218page$\newcommand{\Prime}{{^\prime}}\newcomman{\U}[1]{\underline{\sf #1}}$
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 (\mu \theta) (\hat{n}\cdot \vec{I})+(1-\cos \mu \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} |
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24-05-09 11:05:36 |
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[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. |
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24-05-09 10:05:41 |
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[NOTES/CM-08004] Motion in Non Inertial FramesNode id: 6220page
$\newcommand{\U}[1]{\underline{\sf #1}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
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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24-05-09 06:05:58 |
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[NOTES/CM-08009] Effect of Earth's RotationNode id: 6222pageThe 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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24-05-09 04:05:54 |
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[WHER/CM-07001] Where does Harmonic Oscillator Appear In Engineering ?Node id: 6228collection |
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24-05-06 04:05:08 |
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[WHER/GT-08001] Where do Rotations Appear in Physics?Node id: 6226page |
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24-05-05 08:05:05 |
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[WHER/GT-ALL] Group Theory >> Where does a Topic Appear in Physics and ...?Node id: 6227collection |
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24-05-05 07:05:55 |
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[DOC/RESOURCES-ALL] Classification of ResourcesNode id: 6099page |
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24-05-05 07:05:16 |
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[NOTES/CM-08012] Matrices for Rotations about Coordinate AxesNode id: 6225pageThe rotation matrices for rotations about the three axes are listed. |
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24-05-03 08:05:13 |
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[NOTES/CM-08011] Rotations about a fixed axisNode id: 6224pageRotations about a fixed axis form a one parameter subgroup of rotations. |
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24-05-03 05:05:24 |
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[NOTES/CM-08007] A Note on Fundamental InteractionsNode id: 6221pageA short discussion of pseudo forces and fundamental interactions is given. |
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24-05-03 04:05:03 |
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[NOTES/CM-08002] An explicit form of rotation matrixNode id: 6214pageFor 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. |
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24-05-03 04:05:37 |
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[NOTES/CM-08005] Finite Rotations of Vectors about an Arbitrary AxisNode 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 \(\hat n\). |
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24-05-02 20:05:20 |
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