1. In two-dimensional space, define rotating coordinate axes by

   

   ( and might describe a set of axes painted on a flywheel which is rotating with respect to an inertial observer.)

   

   NOTE: For the questions below, work in basis, where the position is given by

   

   1. Find the velocity in terms of the , axes.

      

   2. Find the acceleration in terms of the , axes.

      

   3. What force is required to keep both u and v constant?

      

   4. If and describe axes on a flywheel rotating with frequency , why does your answer to part (c) make sense?

      

2. For the potential
   1. Sketch the equipotential curves in the xy-plane.

      

   2. Sketch the gradient field in your map above.

      

   3. At the point (x,y) = (3,3),
      1. What is the force ?

         

      2. Show that the force component in the direction vanishes, where

         

         soln

         

      3. What does this tell you physically about the direction of , in terms of your map in parts (a) and (b)?

         

3. 1. For the following forces, state whether the force is conservative or non-conservative. If the force is conservative, display the potential. If not, prove it's not, by calculating either or a relevant work integral .

      
      
       [0.25in]
      Constant force 		   
      
      YES 		 V = -ax-by-cz
       [0.5in]
      The force 		    (for some function G)
      
      YES 		 V = -G(r)
       [0.5in]
      The force 		    (r,    are polar coordinates in the xy-plane)
      
      NO 		 work integral around closed circular loop
      
      		  
       [0.5in]
      The force 		   
      
      NO 		    
       [0.2in]
      
      

   2. 

      

   -1.2in

4. 1. Let variables r and describe the motion of on the plane, and z the motion of . Write the Lagrangian for this system, in terms of the variables r and .

      

   2. What is the Euler-Lagrange equation for ? What does it mean, physically?

      and appear in the Lagrangian only in the kinetic energy term for , which depends only on . Thus E-L gives

      

      the angular momentum l of is conserved.

   3. Find an Euler-Lagrange equation for r, which depends only on constants and the single function r(t).

      

   4. At what radius does undergo a circular orbit?

      when , thus

      

   5. What is the frequency of small radial oscillations about this circular orbit?

      

      The frequency of radial oscillations is . The denominator is because that is the effective mass for radial oscillations; see the left hand side of the radial equation of motion (c). Thus

      

   6. Show that the near-circular orbits are closed, when .

      

      so

      

      For , so that the periods in r and are equal; the orbit closes on itself and repeats when .

5. A particle in the plane experiences a central force, with potential .
   1. What is the effective potential for the radial coordinate?

      

   2. Which term dominates this effective potential,
      1. as ?

         

      2. as ?

         

   3. Sketch qualitatively, labeling (but not calculating) any important energies and radii.

      

   4. 1. For what initial energies and radii is the orbit bound?

         for (this includes all physical r when E;SPMlt;0)

      2. For what initial energies and radii is the orbit unbound?

         for

         for for all r

   5. Write down the Lagrangian for this planar particle, in potential .

      

   6. Legendre transform to obtain the Hamiltonian for this particle.

      

   7. Derive Hamilton's equations for this particle.

      

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