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[NOTES/CM-10007] Generator of a Canonical TransformationNode id: 6248pageThe definition finite and infinitesimal canonical transformation are given. Using the action principle we define the generator of a canonical transformation.
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[NOTES/CM-10006] Two Simple Examples of Canonical TransformationsNode id: 6247page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$ |
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24-05-18 20:05:43 |
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[NOTES/CM-10005] What is a Canonical Transformation?Node id: 6245page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref[#1]}$
A canonical transformation is a change of variable in phase space such that the equations of motion in the new variables are of the Hamiltonian form. |
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24-05-17 16:05:18 |
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[NOTES/CM-09014] Angular Momentum of a Rigid BodyNode id: 6244pageThe angular momentum of a rigid body is given by where \(\mathbf I\) is moment of inertia tensor and \(\vec \omega \) is the angular velocity.\begin{eqnarray} \vec{L}&=&\int dv \rho(\vec{X})\vec{X}\times(\vec{\omega}\times\vec{X})\\ &=&\int dV \rho(\vec{X})\Big[(\vec{X}\cdot\vec{X})\vec{\omega}-(\vec{X}\cdot\vec{\omega} )\vec{X}\Big] \end{eqnarray} or \(\vec L=\mathbf I\, \vec \omega\). |
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24-05-15 19:05:17 |
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[NOTES/CM-09008] Specifying Orientation Using Body AxesNode id: 6238pageA possible way of specifying the orientation of a rigid body is to give orientation of body fixed axes w.r.t. a space fixed axes. Euler angles are a useful set generalized coordinates to specify orientation of the body axes relative to a space fixed axis. |
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[NOTES/CM-09001] Degrees of Freedom of a Rigid BodyNode id: 6234pageBy considering possible motions of a rigid body with one, two or three points fixed, we show that a rigid body has six degrees of freedom. |
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24-05-15 07:05:56 |
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[NOTES/CM-09005] Heavy Symmetrical Top with One Point FixedNode id: 6236page$\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref{#1}}\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}}$
We set up Lagrangian for a heavy symmetrical top and show that the solution can be reduced to quadratures. |
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24-05-14 17:05:13 |
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[NOTES/CM-09004] Kinetic Energy of Rigid BodyNode id: 6235pageAn expression for the kinetic energy in terms of the moment of inertia tensor and the angular velocity w.r.t the body frame of reference is obtained. It is shown that \begin{equation}
\text{KE}=\sum_{ij} \omega_{bi} I_{ij}^{(b)} \omega_{ij}
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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-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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[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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[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-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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[NOTES/CM/08001] The Group of Orthogonal Matrices in Three DimensionsNode id: 6215pageThe groups of all orthogonal matrices is defined It has a subgroup of matrices with determinant +1, This subgroup is called specail orthogonal group.
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[NOTES/CM/The Group of Special orthogonal Matrices Three Dimensions]Node id: 6212page |
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[LSN/EM-ALL] Electromagnetic Theory ---Stockpile of Lessons Node id: 5906collectionThis page is under construction
Last Updated May 8, 2023
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