Grald Teschl ODE Problem 6.5 (One-parameter Lie groups). Suppose (t, x) is differentiable and satisfies (6.11). Then the family t(x) is known as a local one parameter Lie group of transformations (the term local is omitted if W = R M).

Show that is the flow of the vector field f(x) = (0 , x). The differential operator X = f(x) grad = Xn j=1 fj (x) xj (6.13) is known as the infinitesimal generator of t(x). Suppose f(x) is analytic in x and recall from Theorem 4.1 that (t, x) is analytic in t. Show that can be recovered from X via its Lie series (t, x) = exp(tX)x = X j=0 t j j! Xjx