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Title+Summary	Name	Date
[NOTES/CM-10004] Infinitesimal Canonical Transformations --- Examples Node id: 6259page The transformations Space translations Rotations Time evolution form important canonical transformations.	kapoor	24-06-15 04:06:15	n
[NOTES/CM-10001] Canonical Transformations Defined Node 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.	kapoor	24-06-15 04:06:17	n
[NOTES/CM-05009] Keplar Orbit Parameters Node id: 6298page The 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.	kapoor	24-06-14 18:06:12	n
[NOTES/CM-05007] Differential Equation of Orbit Node 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.	kapoor	24-06-14 18:06:38	n
[NOTES/CM-05008] Keplar Problem --- Solving Differential Equation Node id: 6191page Differential 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.$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$	kapoor	24-06-14 12:06:57	n
[NOTES/CM-05001] Cyclic coordinates and constants of motion Node id: 6176page Cyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion.	kapoor	24-06-13 19:06:54	n
[NOTES/CM-05002] Reduction of Two Body Problem Node id: 6177page It 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.	kapoor	24-06-13 18:06:33	n
[LECS/CM-06] Scattering Node id: 6293page	kapoor	24-06-12 11:06:51	n
[NOTES/CM-06001] Scattering Theory --- Basic Definitions Node id: 6192page We define solid angle, flux and the scattering angle and flux and cross section are defined.	kapoor	24-06-12 05:06:58	n
[NOTES/CM-06009] Cross Section in Terms of Probability Node id: 6205page The 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.	kapoor	24-06-12 05:06:16	n
[NOTES/CM-06010] Two Particle Scattering Node id: 6294page An 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.	kapoor	24-06-12 04:06:59	n
[LECS/CM-02] From Newtonian Mechanics to Euler Lagrange Form of Dynamics Node id: 6287page $\newcommand{\pp}[2][ ]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][ ]{\frac{d #1}{d #2}}$	kapoor	24-06-11 22:06:28	n
[NOTES/CM-02009] Cyclic Coordinates Node 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.	kapoor	24-06-11 22:06:56	n
[NOTES/CM-02006] Generalized Force and Lagrange Equations -- An Example Node 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.	kapoor	24-06-11 22:06:18	n
[LECS/CM-04] Hamiltonian Form of Dynamics Node id: 6282page	kapoor	24-06-11 20:06:27	n
[XMP/CM-04004] Short Examples --- Computing Hamiltonian and Lagrangian Node id: 6292page $\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$	kapoor	24-06-11 20:06:08	n
[LECS/CM-11] Hamilton Jacobi Theory Node id: 6291page [toc:0] $\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$	kapoor	24-06-11 16:06:32	n
[LECS/CM-03] Action Principles Node id: 6275page	kapoor	24-06-11 14:06:35	n
[LECS/CM-04003] Node id: 6285page	kapoor	24-06-10 04:06:56	n
[LECS/CM-04002] Node id: 6284page	kapoor	24-06-10 04:06:48	n

The transformations

Space translations
Rotations
Time evolution

form important canonical transformations.

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}} \newcommand{\EqRef}[1]{Eq.\eqref{#1}} \newcommand{\eqRef}[1]{\eqref{#1}}$

Canonical transformation is defined in three different,equivalent ways.

The 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.

$\newcommand{\Label}[1]{\label{#1}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
$\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.

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

Cyclic coordinates are defined; canonical momentum conjugate to a cyclic coordinate is shown to be a constant of motion.

It 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.

We define solid angle, flux and the scattering angle and flux and cross section are defined.

The 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.

An 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.

$\newcommand{\pp}[2][ ]{\frac{\partial #1}{\partial #2}}\newcommand{\dd}[2][ ]{\frac{d #1}{d #2}}$

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.

$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}
\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\mHighLight}[1]{\mbox{#1}}$

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.

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$

[toc:0]

$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$

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Browse & Filter

[NOTES/CM-10004] Infinitesimal Canonical Transformations --- Examples

[NOTES/CM-10001] Canonical Transformations Defined

[NOTES/CM-05009] Keplar Orbit Parameters

[NOTES/CM-05007] Differential Equation of Orbit

[NOTES/CM-05008] Keplar Problem --- Solving Differential Equation

[NOTES/CM-05001] Cyclic coordinates and constants of motion

[NOTES/CM-05002] Reduction of Two Body Problem

[LECS/CM-06] Scattering

[NOTES/CM-06001] Scattering Theory --- Basic Definitions

[NOTES/CM-06009] Cross Section in Terms of Probability

[NOTES/CM-06010] Two Particle Scattering

[LECS/CM-02] From Newtonian Mechanics to Euler Lagrange Form of Dynamics

[NOTES/CM-02009] Cyclic Coordinates

[NOTES/CM-02006] Generalized Force and Lagrange Equations -- An Example

[LECS/CM-04] Hamiltonian Form of Dynamics

[XMP/CM-04004] Short Examples --- Computing Hamiltonian and Lagrangian

[LECS/CM-11] Hamilton Jacobi Theory

[LECS/CM-03] Action Principles

[LECS/CM-04003]

[LECS/CM-04002]

Pages