Teschl ODE

Problem 6.5 (One-parameter Lie groups). Suppose d(t, x) is differen tiable and satisfies (6.11). Then the family Di(x) is known as a local one- parameter Lie group of transformations (the term local is omitted if W =R XM). Show that & is the flow of the vector field f(x) = 0(0,x). The differential operator X = f(x) . grad = >f,(x) 3 (6.13) is known as the infinitesimal generator of d,(x). Suppose f(x) is analytic in x and recall from Theorem 4.1 that d(t, I) is analytic in t. Show that & can be recovered from X via its Lie series p(t, x) = exp(tX)x = > -xx. (6.14) 1=0 iminary version made available with permission of the publisher, the American Mathematical S 192 6. Dynamical systems Here the right-hand side is to be understood as the definition of exp(tX):. (Hint: The Taylor coefficients are the derivatives which can be obtained by differentiating the differential equation.)