Course : PH31006/PH41008: Mathematical Methods II : 2011–2012(II) Assignment 1 1. Consider the motion of a particle of mass m moving under the influence of the following potentials:
1 (i) V (x) = 2 kx2
;
(ii) V (x) = αx.
(a) Reduce each of the second order equations arising out of Newton’s second law (m¨ = x − ∂V ) to a pair of first order equations (a dynamical system). ∂x (b) Solve the second order equations assuming x(t = 0) = x0 and p(t = 0) = p0 . (c) Draw the fixed energy curves in xp space (phase space). 2. Consider the simple traffic flow model discussed in the class. Given the choice of
↧