Ivancevic_Applied-Diff-Geom

254 Applied Differential Geometry: A Modern Introductioncorresponding local one–parameter group exp(εv). The nth prolongationof v, denoted pr (n) v, will be a vector–field on the n−jet space M (n) , andis defined to be the infinitesimal generator of the corresponding prolongedon–parameter group pr (n) [exp(εv)]. In other words,pr (n) v| (x,u (n) ) =d dε∣ pr (n) [exp(εv)](x, u (n) ) (3.69)ε=0for any (x, u (n) ) ∈ M (n) .For a vector–field v on M, given byv = ξ i (x, u) ∂∂x i + φα (x, u) ∂∂u α ,the nth prolongation pr (n) v is given by [Olver (1986)](i = 1, ..., p, α = 1, ..., q),pr (n) v = ξ i (x, u) ∂∂x i + φα J (x, u (n) ) ∂∂u α ,Jwith φ α 0 = φ α , and J a multiindex defined above.For example, in the case of SO(2) group, the corresponding infinitesimalgenerator iswithv = −u ∂∂x + x ∂∂u ,exp(εv)(x, u) = (x cos ε − u sin ε, x sin ε + u cos ε) ,being the rotation through angle ε. The first prolongation takes the form(pr (1) [exp(εv)](x, u, u x ) = x cos ε − u sin ε, x sin ε + u cos ε, sin ε + u )x cos ε.cos ε − u x sin εAccording to (3.69), the first prolongation of v is get by differentiating theseexpressions with respect to ε and setting ε = 0, which givespr (1) v = −u ∂∂x + x ∂∂u + (1 + u2 x) ∂∂u x.3.9.2.5 General Prolongation FormulaLetv = ξ i (x, u) ∂∂x i + φα (x, u) ∂ , (i = 1, ..., p, α = 1, ..., q), (3.70)∂uα