||Message]Content page
| Title | Name |
Date  |
|---|---|---|
[NOTES/CM-10010] Group Structure of Canonical TransformationsThe set of all canonical transformations given by a generator of one of the four types form a group. |
|
24-06-15 05:06:04 |
[NOTES/CM-10004] Infinitesimal Canonical Transformations --- ExamplesThe transformations
form important canonical transformations.
|
|
24-06-15 04:06:15 |
[NOTES/CM-10001] Canonical Transformations Defined$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}} \newcommand{\EqRef}[1]{Eq.\eqref{#1}} \newcommand{\eqRef}[1]{\eqref{#1}}$ Canonical transformation is defined in three different,equivalent ways. |
|
24-06-15 04:06:17 |
[NOTES/CM-05009] Keplar Orbit ParametersThe 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. |
|
24-06-14 18:06:12 |
[NOTES/CM-05007] Differential Equation of Orbit$\newcommand{\Label}[1]{\label{#1}}$ \(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\) 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. |
|
24-06-14 18:06:38 |
[NOTES/CM-05008] Keplar Problem --- Solving Differential EquationDifferential 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.$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$ |
|
24-06-14 12:06:57 |
[NOTES/CM-05001] Cyclic coordinates and constants of motionCyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion. |
|
24-06-13 19:06:54 |
[NOTES/CM-06001] Scattering Theory --- Basic DefinitionsWe define solid angle, flux and the scattering angle and flux and cross section are defined. |
|
24-06-12 05:06:58 |
[NOTES/CM-06009] Cross Section in Terms of ProbabilityThe 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. |
|
24-06-12 05:06:16 |
[NOTES/CM-02006] Generalized Force and Lagrange Equations -- An Example$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} 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. |
|
24-06-11 22:06:18 |