[LSN/ME-12001] General Features of Two Body Problem

\(\S\) Recall and Understanding

1. It is known that the orbits for a spherically symmetric potentials lie in a plane. This statement is a consequence of _________________
2. It is known that the areal velocity for motion in a spherically symmetric potentials is constant. This statement is a consequence of _________________ 
3. What is Kepler's second law about areal velocity? Does it hold for elliptic orbits only?
4. What is Kepler's second law about areal velocity? Does it hold for motion in orbits of every spherically symmetric potential, or only for \(-k/r\) potential?
5. State Kepler's law about planetary motion. Will the law be valid for bounded orbits in any spherically symmetric potential other than \(-k/r\)?
6. Formulate a condition for circular orbits in a spherically symmetric potential in terms of \(r, \dot{r}, \theta, \dot{\theta}\).
7. Write the equation of orbit for motion of a body in \(-k/r, k>0\) potential. Explain each symbol appearing in the equation.
8. What is effective potential for the radial motion in a spherically symmetric potential?
9. Give a sufficient condition so that a potential problem in three dimension can be reduced to motion of a particle in one dimension.
10. Write a condition on the potential between two bodies so that so that the solution can be reduced to motion of one body in an effective potential in one dimension.
11. Which of the three Kepler's laws, are true for any type, bounded and unbounded, orbits in (-k/r) potential?
12. The motion of all planets in sun's gravitational potential is bounded. Give an example of real (physics example) of unbounded motion in \(-k/r\) potential.
13. Is is necessary the energy should be negative that for a bounded motion in any spherically potential.
14. Is energy being negative is a sufficient condition that for a bounded motion in any spherically potential.
15. Is the condition that energy be negative an unnecessary and insufficient requirement so that motion in a spherically symmetric potential be bounded?

• In absence of external forces, the center of mass moves like a free particle. The total linear momentum and the angular momentum about the centre of mass are constants of motion.
• When the all external forces do not add to zero, the motion of the centre of mass is governed the sum of all external forces acting on the two bodies.
• In case the interaction potential is spherically symmetric, the equation of motion has interpretation of a single particle of reduced mass \(\mu\) moving in a potential \(V\text{eff}(r)\).