Reading:

Read quantitatively B & O Sections 3.7 and 5.1; KB 361 lecture notes 31 -- 33.
Read qualitatively B & O Sections 5.2 -- 5.3; KB 361 lecture note summaries 34 -- 36.

Problems for Review:

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1. A spherical pendulum consists of a bob of mass m and a weightless rod of length l. One end of the rod is fixed at the origin, while the other is attached to the bob, so that the bob pivots in any direction about the origin, yet always remains at fixed radius l. The bob thus moves on a spherical surface of radius l, subject to gravity.
   1. Write the Lagrangian for the spherical pendulum in spherical coordinates, eliminating any constraints.
   2. Legendre transform to obtain the Hamiltonian for the spherical pendulum, in terms of .
   3. Show that is conserved.
   4. Combine the Hamiltonian term depending on with the ordinary potential energy term to define an effective potential energy . Sketch for varying values of . What does this tell you about the allowed motions of constant ? Sketch these allowed motions for varying values of .

2. Consider a particle of mass m constrained to move on the surface of a cylinder defined by . The particle is attracted toward the origin by a radial force .
   1. Write the Lagrangian in cylindrical coordinates , eliminating any constraints. Find the Euler-Lagrange equations for and z.
   2. Legendre transform to obtain the Hamiltonian, in terms of .
   3. Find Hamilton's equations and show that they reproduce the Euler-Lagrange equations in (a).
   4. Sketch the allowed motions of the particle. Under what conditions are these motions periodic?

3. (essentially B & O 5-3) A particle is attracted toward a force center at the origin via the central force .
   1. Find the potential V(r) corresponding to this central force.
   2. Find and sketch the effective potential . For what values of the energy are there bound orbits? scattering orbits?
   3. At what radius is there a circular orbit? What is its period?
   4. Find the frequency of small radial oscillations about the circular orbit. Relate it to the angular frequency . What does this tell you about whether near-circular orbits close?
4. (essentially B & O 5-2) Find the condition for stable circular orbits for a potential energy of the form

   

   where is nonzero and is positive. Under what condition on do perturbed near-circular orbits close? Sketch the orbits for two values of meeting your criterion.

Problems to Solve:

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1. B & O 3-14.
2. In a child's toy, a bead, subject to gravity, moves along a spiraling wire of shape , in cylindrical coordinates with k constant.
   1. Write the Lagrangian in terms of z and only, eliminating constraints. Derive the Euler-Lagrange equation for z.
   2. Legendre transform your Lagrangian of (a) to obtain the Hamiltonian in terms of .
   3. Find Hamilton's equations and show that they reproduce the Euler-Lagrange equation in (a).
   4. Comment on the sensibility of the equation of motion obtained in (a) and (c), focusing on how varies with changing spiral slope .

3. A particle is attracted toward the origin via the radial potential V= cr.
   1. Find and sketch the effective potential .
   2. At what radius is there a circular orbit? What is its period?
   3. Find the frequency of small radial oscillations about the circular orbit. Relate it to the angular frequency . What does this tell you about whether near-circular orbits close?
4. A particle is attracted toward the origin via the radial potential .
   1. Find and sketch the effective potential .
   2. Show that the only allowed circular orbit has the period expected from simple harmonic motion, .
   3. Consider a small radial perturbation of the circular orbit. Show that the motions remains periodic, with the same harmonic oscillator period T derived above. Sketch the motion.

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