Math 135 Spring 2007 :: Extra question
This question is a serious level of difficulty above that of the homework. Solving it may positively
affect your grade, but is completely optional.1 Only attempt it if you are interested, if you have
time, and if it will not negatively affect your success in any of your other activities. If you find a
solution, please present it to me during my office hours.
A Hamiltonian flow is a system of ODEs that can be written in the form
ẋ(t) = ∇p H(x(t), p(t)),
x(0) = x0 ,
ṗ(t) = −∇x H(x(t), p(t)),
p(0) = p0 ,
for some function H(x, p) called the Hamiltonian. Both x and p are vectors with n components,
denoted xi and pi respectively.
Assume that
v
uX
n
u n X
pi Aij (x)pj ,
H(x, p) = t
i=1 j=1
where the Aij (x) are smooth functions of x. [The Hamiltonian system would then govern the
dynamics of light rays undergoing refraction in an anisotropic physical medium, for instance. In
this case x is position, and p is called slowness.]
Show that, for all times t ≥ 0, for all 1 ≤ i ≤ n,
n
X
∂xj (t)
j=1
∂p0i
pj (t) = 0,
(1)
where p0i are the components of p0 . In other words, show that ∇p0 x(t) and p(t) are perpendicular—
their dot product is zero. [You should think of the solution as a vector field x(t; x0 , p0 ), function
of time and the initial conditions. The notation ∇p0 x(t) means the gradient of each component of
the solution x(t) with respect to p0 . In equation (1) we represent it as a matrix with column index
j corresponding to the components of x(t), and row index i corresponding to the components of
P
∂x (t)
p0 . Then the expression nj=1 ∂pj0i pj (t) is a matrix-vector product.]
1
Not solving the problem will not prevent you from getting A+. The grade of whoever does not solve the problem
will not be lowered in case some people do solve the problem. Etc.