1. Plane Pendulum
   1. Show that the (nonlinear) equation of motion for a pendulum displaced from equilibrium by an angle is:

      

      Here x marks the position of the center of mass, I is the moment of inertia, and the second form for is appropriate for a ``simple'' pendulum with ..

   2. Solve for the angular displacement and the angular velocity as functions of time for arbitrary (input) initial values of and . In particular, solve for energies less than and greater than 2mgx.
      (It is suggested that you use the order Runge-Kutta integration algorithm or Maple.)
   3. Test the stability of your solution by checking for long times how the kinetic-, potential-, and total energies of the system change (this may be hard to do with Maple).
2. 1. Use the energy integral for a plane pendulum to determine the velocity as a function of position,

      

   2. Convert this last relation into an integral expression for the period as a function of .
3. Phase Space Portraits Draw the phase space portraits for the planar pendulum. This should look like Fig.1.10 on page 34 of Scheck. Use either the direct solution of the differential equation or the energy integral.
   1. Obtain orbits for small energies as well as for energies large enough for the pendulum to go over the top.
   2. Indicate the hyperbolic points where trajectories flow in and out.
   3. Sketch the potential as a function of .
4. (Scheck 1.11) Consider the 1-D harmonic oscillator with (viscous) friction:

   

   1. Solve this differential equation by making the substitution

      

      and determining the constants. N. B., are the initial position and momentum only for the frictionless case.

   2. For and , plot .

5. (like Scheck 1.15) Kepler perturbation Consider the motion of a particle with angular momentum l in an inverse-square force field subjected to an inverse-cube perturbation:

   

   1. Derive the general orbit equation

      

   2.   Describe the motion for , , and . (Hint: Use the orbit equation and change variables to u=1/r. You should get a spiral and ellipses of sorts.)
   3. Which of the orbits of () are stable? (By ``stable'' we mean orbits in which the particle remains localized, that is, in which it does not fly away or fall into the center.)
   4. Show that the equation of the orbit can be put into the form:

      

   5. Show that for this is the equation for an ellipse, but that for the ellipse precesses.
   6. Show that for close to 1 (i.e. for orbits which are nearly elliptical) the rate of precession of the perihelion is

      

   7. Optional: The perihelion of Mercury is observed to precess at the rate of 40'' of arc per century. Show that this precession can be accounted for (nonrelativistically) if the value of n were as small as . (The eccentricity of Mercury's orbit is 0.206 and its period is 0.24 years).
6. Perturbed circular orbit (Like Scheck 1.21) Consider a circular orbit about some attractive potential U(r).
   1. Find the frequency of oscillations for small radial perturbations about the circular orbit.
   2. An almost-circular orbit can be considered as a perturbed circle. For , with n an integer, show that the angle through which the orbit shifts as r varies from maximum to minimum is . (This means that only if n=2 will the orbit return back to itself, i.e. not precess.)
   3. Plot the phase space portraits corresponding to these oscillations about a circular orbit.
7. Square well orbits (Related to Scheck 2.18) Solve for the orbits of a particle of mass m in a a square well potential of depth W and radius R. Present your results as a polar plot. [One approach evaluates (1.68), another is geometric and can be done with a ruler!]
8. Non-Rutherford scattering (Scheck 1.23 with hint) Let a two-body system be subject to the potential

   

   with r the inter-particle distance and positive.

   1. Calculate the scattering orbits .
   2. For fixed angular momentum, for what values of does the particle make one (two) revolutions about the center of force? (Hint: The particle will make n rotations about the force center when , where is the radial distance to the perihelion.)
   3. Follow and discuss an orbit that collapses to r=0. Hint: Show that for the orbit has the solution

      

9. Action-Angle (Scheck 2.1 + hint; same as 2.27) The energy E(p,q) is an integral of finite 1D, periodic motion. The area of the periodic orbit is

   

   1. Why is the portrait symmetric with respect to the q-axis?
   2. Show that the period of the motion must be the derivative of F wrt E

      

      (Hint: Deduce an integral expression for dF(E)/dE and show that it is proportional to the integral expression in the text for the period T.

   3. Calculate F and T for the harmonic oscillator

      

      and draw the analogy to the quantum mechanical result .

10. Masses And String Two masses and are connected by a string passing through a hole in a smooth table so that rests on the table and hangs suspended.
    1. What are the conditions for this system to execute circular motion?
    2. How do the conditions for circular motion differ from those which lead to the lower mass just pulling the upper one down?
    3. If the system is executing circular motion, what happens as ?
    4. Assuming that moves only in a vertical line, indicate a set of generalized coordinates which completely specify the motion.
    5. What are Lagrange's equations for this system and their physical significance.
    6. Use these equations to immediately obtain the first integrals related to energy and angular momentum conservation.
    7. What is the frequency for small oscillations about a circular orbit?
11. Masses And Spring (Scheck 2.5 mod) Two equal masses m move without friction along a rail while connected by a massless spring of spring constant k and nonstretched length l.
    1. Deduce the generalized coordinates for this system which immediately lead to a first integral related to total momentum.
    2. Express the Lagrangian in terms of these generalized coordinates and deduce the frequency of oscillation .
    3. Calculate the deviations and from equilibrium positions for the following initial conditions:

       

12. Use Hamilton's Principle to prove that the shortest distance between two points in a 3-D space is a straight line. ( Hint The Euler Lagrange equations.)
13. Hamilton's Principle, 2 (similar to Scheck 2.8) Given the experimental fact that a particle falls a known distance in a known time , guess that the functional dependence of distance on time is

    

    Show that for reasonable conditions, the integral

    

    is an extremum only when a=0 and b=g/2.

14. Lorentz Force You are given the experimental fact that a particle within an electromagnetic field experiences the Lorentz force:

    

    Prove that this same force arises from a Lagrangian including the (scalar) potential

    

    where is the Coulomb (4th component of 4-vector) potential, and is the ``vector'' potential.

15. Hamilton's Principle (Scheck 2.16) A particle of mass m is described by the Lagrangian

    

    Here is the z-component of angular momentum and is not necessarily a constant, yet is a constant, yet is not necessarily .

    1. Deduce the equations of motion.
    2. Express the equations in terms of the complex variable x+iy and of z, and solve them.
    3. Construct the Hamiltonian function and find the kinematic and canonical momenta.
    4. Show that the particle has only kinetic energy and that the latter is conserved.
16. L and H The Lagrangian for a particle of mass m projected vertically upwards from the surface of the earth is

    

    where M is the mass of the earth, R is the radius of the earth, and z is the vertical coordinate of the particle.

    1. Deduce the equations of motion.
    2. Find the corresponding Hamiltonian and find the kinematic and canonical momenta..
    3. Sketch the phase space plane identifying bounded and unbounded regions.
    4. Find the equation of the separatrix.
17. L and H Consider a plane pendulum whose length l decreases uniformly with time:

    

    1. Determine the Lagrangian and Hamiltonian for this system.
    2. How many degrees of freedom are there?
    3. Show that and that E is not a constant of motion and that H is not a constant of the motion. Explain.
18. Midterm Problem A bead of mass m moves without friction on a circular hoop of wire of radius R. The hoop's axis is vertical in a uniform gravitational field and the hoop is rotating at a constant angular velocity .
    1. List the coordinates needed to specify the location of the bead and the k constraints.
    2. Determine, but do not solve, the Lagrange equations of motion for this system.
    3.   Determine the equilibrium position of the rotating bead (i.e., an angle which does not change with time).
    4. For small perturbations about this equilibrium configuration, what is the frequency of oscillations?

    5. What is the condition, in terms of and , for perturbed circular orbits to close?

    6. Determine if the Hamiltonian equals the total energy.
    7. What are the conditions on for the equilibrium angle of part to be real?
19. Liouville's Theorem Work out for yourself Figures 2.13-2.15 of Schnick (Liouville's theorem for the plane pendulum).
    1. Verify that different points on the starting circle take different times to orbit through phase space.
    2. Examine the orbits for at least 5 times: 0, where is the period for small oscillations (but not for your problem).
    3. Include a frictional torque and note its effect on the orbits.
20. Logistics Map Study the logistics map

    

    (If you have already studied the logistics map, use the ecology map .)

    1. Plot up population versus generation number for
    2. Explain the meaning of and identify: transients, steady states, extinction, attractors, period doubling, seed independence, and chaos.
    3. Predict the values of for the first 2 cycle and verify your predictions.
    4. Make a bifurcation diagram, a plot of versus for , and identify the period doublings.
21. Chaotic Pendulum Consider a pendulum moving through a viscous medium while a sinusoidal external torque is applied to its bob:

     

    Here is the natural frequency of the system arising from the gravitational restoring force, the term arises from friction, and the f term measures the strength of the driving torque. The difficulty with the computer study of this system is that with four parameters the parameter space is immense.

    1. Determine analytically the value of f for which the average energy dissipated by friction during one cycle is balanced by the energy put in by the driving force during that cycle. This is a stable configuration.
    2. Show by computation that when this condition is met, there arises a limit cycle near the origin in phase space.
22. Computer Study of Chaotic Pendulum In this problem you will reproduce the phase-space diagrams given below.
    1. Set and and plot .
    2. Indicate which part of the orbits are transients.
    3. Correlate phase-space structures with the behavior of by also plotting versus t (preferably next to versus ).
    4. Gain some physical intuition about the flow in phase space by watching how it builds up with time.

23. For the second part of the study, use the same parameters as in first part, but now sweep through a range of values.
    1. Use initial conditions: , and .
    2. For , you should find a period-three limit cycle where the pendulum jumps between three orbits in phase space. (More precisely, there are three dominant Fourier components.)
    3. For , you should find running solutions where the pendulum keeps going over the top. Try to determine how many rotations are made before the pendulum settles down.
    4. For , you should find chaotic motion in which the paths in phase space become bands of motion and the Fourier spectrum becomes broad (if you let the solution run long enough). Try to determine just how small a difference in values separates the regular and the chaotic behaviors.
    5. Decrease your time step and try to determine how the bands get filled. Try to distinguish short term and long term behaviors in phase space.

24. Fourier Analysis of Chaotic Pendulum Determine the Fourier spectrum of the chaotic pendulum for motion which is 1 cycle, 3 cycle, 5 cycle and chaotic. You may use a canned analysis program such as that in xmgr.
25. Poisson bracket (Scheck 2.21) Consider a single particle.
    1. Evaluate the Poisson brackets:
    2. If the Hamiltonian function in its natural form H=T+U is invariant under rotations, what quantities can U depend on?
26. linear stability analysis (Scheck 6.1) Suppose we have a 2-D system of equations, , obtained by linearizing a dynamical system in the neighborhood of an equilibrium position. Consider the three cases in which has one of the Jordan normal forms:

    

    Case (i) corresponds to the situations shown in Figs. 6.2a-c.

    1. For these three cases determine the characteristic exponents and the flow (6.13) with s=0.
    2. Sketch figures similar to Fig.6.2 for case (ii) with (a=0, b>0), and case (iii) with .
27. Non-linear and non-Hamiltonian (Scheck 6.4) Consider the system

    

    where is a damping term and with is a detuning parameter.

    1. Show that if the system is Hamiltonian and find the Hamiltonian.
    2. Draw the projections of the phase portraits for onto the -plane and determine the position and nature of the critical points.
    3. Study the change in the critical points for and show that the picture obtained above is structurally unstable for positive .
28. Liapunov function (Scheck 6.6) Consider the flow of the equations of motion

    

    Determine the Liapunov coefficients and from these the position and nature of the critical points.

29. Liapunov coefficients Examine the behavior of the logistics map for .
    1. Plot up the population as a function generation number.
    2. Run again with where is close to machine precision ( for single precision, for double precision), and place the results on the same plot as before.
    3. Note how many generations it takes before the populations begin to differ from those of the previous part, and when the results appear completely unrelated.
    4. Determine the Liapunov coefficients for values of in this range.
30. Fractal dimension Determine the approximate fractal dimension of the bifurcation diagram for several values of .
31. Mapping
    1. If you have not yet done it, study the logistics map instead of the one in this problem. (The Computational Physics notes are a superb reference for this.)
    2. (Scheck 6.12) Consider the mapping

       

       Show that there are no stable fixed points.

    3. Calculate numerically 50,000 iterations of this mapping for various initial values . (Scheck suggests [but I do not see his point too well]:) plot the histogram of the points that land in one of the intervals

       

       It may be better to just plot versus to obtain a visualization. See too, Lichenberg and Lieberman.

    4. Follow the development of two close initial values and verify that they diverge in the course of the iteration. (Ref: Collet, Eckmann (1980).)
32. Roessler's model (Scheck 6.13) (Ref: Bergè, Pomeau, Vidal(1984).) Study by numerical integration:

    

    1. Make plots of (x,y,z) as functions of time as well as projections of the motion onto the (x,y) and planes.
    2. Make a Poincaré mapping of the transverse section .
    3. When , x has an extremum on the section. Plot the value of the extremum as a function of the previous extremum .
33. Pendulum with vibrating pivot (parametric pendulum) This is similar to the case we studied, only now the driving term depends on x

    

    An analytic study is discussed by Landau and Lifschitz (Secs.25-30), while results of numerical work are found in McLaughlin, Percival and Richards, and Gould and Tobochnik.

34. Duffing Oscillator Here we have a nonlinear oscillator which is damped and driven

    

    This is similar to the pendulum system we studied, but has some advantage in the ease with which the multiple attractor sets can be found. It has been studied by Moon and Li (1985).

35. Lorentz Attractor In 1962 Lorenz deduced three equations as a simple model for weather prediction

    

    The solution of these nonlinear equations does not always give simple results, in fact, they have helped encourage the study of chaotic systems. See if you too can create a Lorentz attractor of the type given in the figures. Check that the chaotic solutions are very sensitive to the initial conditions. A order Runge Kutta method should be good.

36. A Computer Fly

    

    The parameter e controls the degree of randomness.

37. Hénon-Heiles potential (HJ) Examine the classical mechanical phase space for the Hénon-Heiles potential:

    

    This is similar to the anharmonic oscillator we have already studied in that it binds particles near the origin, but frees them if they move far out. If we look at x=0 we have

    

    Accordingly, bound orbits occur for 0 < E < 1/6. The available part of phase space is then

    

    The equations of motion for this problem follow from the Hamiltonian equations

    

    1. Numerically solve for the position (x(t),y(t)) for a particle in the Hénon-Heiles potential.
    2. Plot (x(t),y(t)) for a number of initial conditions. Check that the initial condition E< 1/6 leads to a bounded orbit.
    3. Calculate the energy H as a function of time and check that it stays constant (else you have numerical problems).
    4. Produce a Poincare section in the plane. Each time you find x = 0 (at least to some precision), plot . For a fixed energy, try this plot for various initial conditions.
    5. Try to isolate a smaller region of phase space and examine the structure of the orbits there.
38. (F & W 7.5) String waves A string of uniform mass density and length l hangs under its own weight in the earth's gravitational field, Consider small transverse displacements u(x,t) in a plane.
    1. Compute the equilibrium tension in the string , where x is the distance from the point of suspension.
    2. Show that the normal modes satisfy Bessel's equation. Hint: Make the substitution .
    3. What are the boundary conditions?
    4. What are the normal mode frequencies? (Find the first three from tables or computer.)
    5. What are the normal modes? (Sketch the first three.)
    6. Construct the general solution to the initial-value problem.
39. Damped Waves Consider the function

    

    as a possible solution to the 1-D wave equation. The argument is the phase of the wave.

    1. Determine the phase velocity V for this wave, that is, the velocity with which we must move along the x-axis to keep the phase stationary ( )
    2. Explain the physical significance of a complex frequency and a complex velocity V.
    3. Determine the relation between a complex and V, and give an explicit expression for V.
40. (Fetter & Walecka 8.8) Membrane Waves Consider the wave equation for a membrane in the shape of a sector of a circle with opening angle and radius a, subject to the condition u=0 on the boundary.
    1. Show that the problem is separable in plane polar coordinates and that the eigenfunctions are

       

       with eigenvalues determined by

       

    2. Sketch the normal modes and nodal likes for a few typical low-lying n=modes.
    3. Use tables or computer to find the three lowest eigenfrequencies for wach of the special cases .
    4. Construct the general solution to the initial-value problem

       

41. KdeV Equation Consider the KdeV equation

      

    1. Look for steady-state solutions which have the form of a traveling wave:

        

       The traveling wave form () means that if we move with a constant speed c, the wave we are keeping up with will have a constant magnitude for all time. Show that this leads to the equation

       

    2. Verify that this has the inverse solution

        

       where d and e are integration constants.

    3. Verify that if we demand as boundary conditions that f, , and as , we can invert () to obtain the solution

        

       We see in () an amplitude which is proportional to the wave speed c, and a function which gives a lump-like wave. This is a characteristic mathematical form for a soliton.

    4. Sketch or plot the form of this function.

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