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[NOTES/CM-10004] Infinitesimal Canonical Transformations --- ExamplesNode id: 6259pageThe transformations
- Space translations
- Rotations
- Time evolution
form important canonical transformations.
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24-06-15 04:06:15 |
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[NOTES/CM-10001] Canonical Transformations DefinedNode id: 6246page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}} \newcommand{\EqRef}[1]{Eq.\eqref{#1}} \newcommand{\eqRef}[1]{\eqref{#1}}$
Canonical transformation is defined in three different,equivalent ways. |
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24-06-15 04:06:17 |
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[NOTES/CM-05009] Keplar Orbit ParametersNode id: 6298pageThe equation for the orbit involves two constants of integration. We determine these constants and obtain an expression for the eccentricity in terms of energy angular momentum etc.. Conditions on energy for different types of possible orbits , elliptic, parabolic and hyperbolic, are written down. |
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24-06-14 18:06:12 |
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[NOTES/CM-05007] Differential Equation of OrbitNode id: 6299page$\newcommand{\Label}[1]{\label{#1}}$ \(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\) \(\newcommand{\DD}[2][]{\frac{d^2 #1}{d #2^2}}\)\(\newcommand{\dd}[2][]{\frac{d #1}{d #2}}\)
Two methods of obtaining the differential equation of the orbit, in a sphericall symmetric potential, are given using the Euler Lagrange equations and conservation law. |
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24-06-14 18:06:38 |
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[NOTES/CM-05008] Keplar Problem --- Solving Differential EquationNode id: 6191pageDifferential equation for orbits is solved. The orbits are shown to be conic sections. Kepler's three laws are proved. Some properties of hyperbolic orbits are derived.$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$ |
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24-06-14 12:06:57 |
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[NOTES/CM-05001] Cyclic coordinates and constants of motionNode id: 6176pageCyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion. |
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24-06-13 19:06:54 |
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[NOTES/CM-05002] Reduction of Two Body ProblemNode id: 6177pageIt is proved that the two body problem with central potential \(V(|\vec{r}_1-\vec{r}_2|)\) can be reduced to one body problem. In the reduced problem the body has reduced mass and moves in spherically symmetric potential \(V(r)\). In this case the center of mass moves like a free particle. |
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24-06-13 18:06:33 |
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[LECS/CM-06] Scattering Node id: 6293page |
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24-06-12 11:06:51 |
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[NOTES/CM-06001] Scattering Theory --- Basic DefinitionsNode id: 6192pageWe define solid angle, flux and the scattering angle and flux and cross section are defined. |
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24-06-12 05:06:58 |
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[NOTES/CM-06009] Cross Section in Terms of ProbabilityNode id: 6205pageThe definition of cross section is formulated in probabilistic terms. This interpretation turns out to be useful for interpretation of the cross section as an area, and also for quantum mechanical problems. |
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24-06-12 05:06:16 |
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[NOTES/CM-06010] Two Particle ScatteringNode id: 6294pageAn important class of two particle scattering is when the forces are central, {\it i.e.} the potential depends on the separation \(\vec r_1-\vec r_2\) between the particles: \[ V(\vec r_1, \vec r_2)= V(\vec r_1-\vec r_2).\] The two particle elastic scattering problem becomes equivalent to scattering of a particle, of reduced mass, from a potential \(V(\vec r)\), with \(\vec r= \vec r_1-\vec r_2,\) in the centre of mass frame. |
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24-06-12 04:06:59 |
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[LECS/CM-02] From Newtonian Mechanics to Euler Lagrange Form of DynamicsNode id: 6287page$\newcommand{\pp}[2][ ]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][ ]{\frac{d #1}{d #2}}$ |
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24-06-11 22:06:28 |
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[NOTES/CM-02009] Cyclic CoordinatesNode id: 6046page A generalized coordinate,\(q_k\), is called cyclic if the Lagrangian is independent of the coordinate \(q_k\). It is shown that the corresponding canonical momentum is a constant of motion. A simple example of cyclic coordinate is given.
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24-06-11 22:06:56 |
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[NOTES/CM-02006] Generalized Force and Lagrange Equations -- An ExampleNode id: 6049page$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\mHighLight}[1]{\mbox{#1}}$
Whenever the generalized force \(Q_k\) \begin{equation}\label{AB01} Q_k=\sum_\alpha F^\text{(e)}_\alpha \pp[\vec{x}_\alpha]{q_k}. \end{equation} can be written in the form \begin{equation}\label{AB02} Q_k = \dd{t}\Big(\pp[U]{\dot{q}_k}\Big)-\pp[U]{q_k} . \end{equation} the equations of motion take the Euler Lagrange form with \(L=T-U\). It is demonstrated that the Lorentz force,the force on a charged particle in e.m. fields, has the form as in \eqref{AB02}. An expression for the generalized potential \(U\) is derived. |
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24-06-11 22:06:18 |
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[LECS/CM-04] Hamiltonian Form of DynamicsNode id: 6282page |
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24-06-11 20:06:27 |
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[XMP/CM-04004] Short Examples --- Computing Hamiltonian and LagrangianNode id: 6292page$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$ |
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24-06-11 20:06:08 |
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[LECS/CM-11] Hamilton Jacobi TheoryNode id: 6291page[toc:0]
$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$ |
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24-06-11 16:06:32 |
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[LECS/CM-03] Action PrinciplesNode id: 6275page |
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24-06-11 14:06:35 |
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[LECS/CM-04003] Node id: 6285page |
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24-06-10 04:06:56 |
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[LECS/CM-04002] Node id: 6284page |
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24-06-10 04:06:48 |
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