1. Using polar coordinates in the plane, infinitesimal distances are

   

   For the central force potential V(r), we thus have

   

   The conjugate momenta are

   

   Thus

   

   is the conserved energy.

   Hamilton's equations give

   

   The equations on the right just reproduce the relations between and , and . Using them to eliminate momenta and substitute into the Hamilton's equations on the left gives

   

   reproducing the Euler-Lagrange equations in Equation 3.40.

2. 1. In cylindrical coordinates, infinitesimal distances are

      

      Here is constant and so we have

      

      For the gravitational potential mgz, we thus have

      

      This Lagrangian has partial derivatives

      

      giving Euler-Lagrange equation

      

      or

      

   2. The conjugate momentum is

      

      so

      

      is the conserved energy.

   3. Hamilton's equations give

      

      The equation on the right just reproduces the relations between and . Eliminating in Hamilton's equation on the left gives

      

      reproducing the Euler-Lagrange equation in part (a).

   4. 
      

      

      Only the component of gravity parallel to the wire gives an unopposed force accelerating the bead; that component is . Then only the vertical component of the unopposed force accelerates the z-component; this is . Thus wise application of Newton II leads to

      

      in agreement with the equation of motion from parts (a) and (c).

3. 1. 

      which has asymptotes decreasing from infinity for small r, and cr diverging to infinity as . In between the decreasing behavior at small r and the increasing behavior at large r must be a turnover point where has a minimum. The potential thus looks as sketched above, with bound orbits for all allowed energies.

   2. The circular orbit occurs at the minimum of the effective potential:

      

      giving a minimum at

      

      It has period

      

   3. 

      which is positive, indicating a stable orbit. Thus

      

      Since these frequencies are not rational multiples, the near-circular orbits never close (become periodic).

4. 1. 

      which has asymptotes decreasing from infinity for small r, and diverging to infinity as . In between the decreasing behavior at small r and the increasing behavior at large r must be a turnover point where has a minimum. The potential thus looks as sketched above, with bound orbits for all allowed energies.

   2. The circular orbit occurs at the minimum of the effective potential:

      

      giving a minimum at

      

      It has period

      

   3. 

      Evaluating this at gives

      

      so that

      

   

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