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.