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.
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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.
Using angular momentum conservation it is shown that orbits for a spherically symmetric potential lie in a plane; This makes it possible to work in plane polar coordinates. The equation for radial motion becomes similar to that in one dimension with potential replaced by an effective potential. An expression for the effective potential is obtained.
It is proved that the Runge-Lenz vector \begin{equation}\vec{N} = \vec{v} \times \vec{L} - \frac{k\vec{r}}{r}\end{equation} is a constant of motion for the Coulomb potential \(-k/r\).
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