HW#1  The first four homework problems may be found in my Lambert oscillator notes.

HW#2  [1]  Find an integral invariant under the rms-arithmetic mean iteration:  a_n , b_n → a_n+1 = (a_n²/2+b_n²/2)^1/2 , b_n+1 = a_n/2+b_n/2.
             [2]  Critique the latest "physicist's proof" of the Riemann hypothesis, as given here.

HW#3  [1]  Compute Lyapunov exponents for the logistic map in the special case, x_n+1 = 4x_n(1-x_n).
             [2]  Find limit cycles and bifurcations, numerically, for the damped, driven Lambert oscillator.
             [3]  Show that upon averaging phase space volume shrinks, <dV/dt> < 0,  for any driven oscillator with damping.
             [4]  Show the Schwarzian derivative of F is negative for various maps on the unit interval:  F = ρx(1-x); F = 1-μ|x|^r; F = a sin(πx).  All parameters here are positive, with r > 1.  Also recall the Schwarzian derivative is defined as D[F] = F'''/F' - 3 (F''/F')²/2.

HW#4  [1]  For a charged particle in a magnetic dipole field, consider trajectories which are in the equatorial plane containing the dipole, with the dipole normal to the plane.  Numerically compute and plot examples of "trapped" and "tightly wound" trajectories, as sketched in lecture.
             [2]  Again for equatorial plane trajectories of a charged particle in a magnetic dipole field, express the azimuthal angle φ(x) in terms of elliptic integrals, where ρ_circle ρ ≡ x, and ρ_circle is the radius of the stable circular orbit.  Find at least one particular case where the elliptic integrals reduce to elementary functions (logarithms).
             [3]  For any central force problem, with potential V(r), show that H = p²/2m + V(r), L_z, and L²  are in involution (i.e. for any pair of them, {A,B} = 0).  Determine whether these are functionally independent for the Coulomb potential, V(r) = - α/r, for constant α.
             [4]  Read M A Almeida et al 1992 J. Phys. A: Math. Gen. 25 L227-L230, "On the nonintegrability of the Stormer problem."

HW#5  [1]  As a trivial exercise in Floquet theory, determine the standard basis functions and the matrix R (as defined in your class notes) for the simple harmonic oscillator with time-independent frequency ω. 
             [2]  Read Frenkel and Portugal, J. Phys. A:  Math. Gen. 34 (2001) 3541-3551, "Algebraic methods to compute Mathieu functions".
             [3]  Find a Lagrangian to describe a particle moving on the xy-plane in a time-dependent but spatial-independent magnetic field perpendicular to the plane, i.e. B(t) = B(t) e_z .  Obtain the equations of motion from this Lagrangian.
             [4]  Solve numerically for some solutions of the driven pendulum.  Take the pivot to move with a single frequency, y(t) = A sin(νt) .  Determine some numerical values of A and ν so that stable "inverted" solutions exist, and plot one of these inverted solutions as a function of time.
             [5]  Read Blackburn, Smith, and Jensen, Am. J. Phys. 60 (1992) 903, "Stability and Hopf Bifurcations in the Inverted Pendulum".

HW#6  [1]  Determine a classical generating function F[φ,θ] that leaves invariant the "hyperbolic" Mathieu Hamiltonian H.   Show that H[φ] and H[θ] have the same effect when acting on exp(iF).
             [2]  Use the integral equation satisfied by the 2π periodic Mathieu functions to compute the first few terms in their Fourier series representations.  Also determine the integral equation eigenvalues to the same order.
             [3]  Find an on-line version of Theory and Application of Mathieu Functions by N. W. McLachlan.
             [4]  Show the consistency of the canonical transformation generated by
                        F[φ,θ] = ∫dx κ₁ cos(φ(x,t)-θ(x,t)) - κ₂ cos(φ(x,t)+θ(x,t)) + φ(x,t)∂_xθ(x,t)
                   requires that both φ and θ obey the same sine-Gordon equation, the latter equation involving only the product κ₁κ₂.