MATH 557 Homework Set #3
Fall 2015
12. The equation for the motion θ(t) of a simple damped pendulum is
θ00 + kθ0 + q sin θ = 0
for some constants k ≥ 0, q > 0. Rewrite this ODE as a first order system and verify the
necessary continuity and Lipschitz conditions needed to guarantee local existence and
uniqueness for the corresponding initial value problem.
13. A system of 2n ordinary differential equations is said to be a Hamiltonian system
if it is possible to label the unknown functions as (q, p) = (q1 , . . . qn , p1 . . . pn ) so that
there exists a function H = H(t, q, p) (the Hamiltonian function) for which the system
may be written
dqi
∂H dpi
∂H
=
=−
i = 1, . . . n
dt
∂pi dt
∂qi
A scalar equation of order 2n is Hamiltonian if one can find an equivalent Hamiltonian
system.
a) Show that the pendulum equation in the previous problem is Hamiltonian if k = 0,
2
with a time independent Hamiltonian function. (Hint: Let q = θ, p = θ0 , H = p2 −cos q).
b) Show that any second order equation of the form x00 + f (x) = 0 is Hamiltonian.
c) If H = H(q, p) only (i.e. it is independent of time), show that H(q(t), p(t)) is
constant in time if (q(t), p(t)) is a solution.
14. Read the definitions of weakly Lipschitz and locally Lipschitz functions on page 63
of the text.
a) Show that if f is Lipschitz then it is weakly Lipschitz.
b) Show that if f is weakly Lipschitz then it is locally Lipschitz.
c) Show by examples that the converse to both of these statements is false.
15. Let f = f (x) be positive and locally Lipschitz on R. Show that the solution of the
scalar problem
x0 = f (x)
x(t0 ) = x0
R ∞ ds
exists for all t ≥ t0 if and only if x0 f (s) = ∞.