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The definition of infnitesimal transformations is given.

The transformations

1. Space translations
2. Rotations
3. Time evolution

form important canonical transformations.
 

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.

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

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.

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.

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.