Question

Show that angular momentum is conserved for a force law F = kr and find the allowed orbits.

Answer 1

The angular momentum is defined as
  
  And so, the time rate change of the angular momentum is
   

  
   
  
  
where, we have used the fact that for any vector
   
And for the given force field
  
And so,
  
  
  
And so, the angular momentum is conserved.

And from the equation of the motion of a particle in a orbit under a force field , is given by
  
where,
  
And so, for the force field
  
the orbital equation is
  

  

And this is difficult to solve. However, in terms of the total energy is
  

And for the given force , the potential energy is
  
So, this is the simple harmonic oscillator potential. So, the energy expression is given by
  
  
where, the effective potential energy is
  
The plot of this potential is given by
   
The lowest of this effective potential is
  
  
   

And so, the minimum value of the effective potential is
  
  
  
And so, for the total energy greater than this minimum value of the effective potential energy, i.e., for
   
there are two turning points as seen in the plot. So, the orbits are bounded orbits. And more careful analysis shows that the orbits are elliptical in nature.