Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 255be a vector–field defined on an open subset M ⊂ X × U. The nth prolongationof v is the vector–field [Olver (1986)]pr (n) v = v + φ α J (x, u (n) ) ∂∂u α , (3.71)Jdefined on the corresponding jet space M (n) ⊂ X × U (n) . The coefficientfunctions φ α J are given by the following formula:φ α (J = D J φ α − ξ i u α )i + ξ i u α J,i , (3.72)where u α i = ∂u α /∂x i , and u α J,i = ∂uα J /∂xi . D J is the total derivative withrespect to the multiindex J, i.e.,D J = D j1 D j2 ...D jk ,while the total derivative with respect to the ordinary index, D i , is definedas follows. Let P (x, u (n) ) be a smooth function of x, u and derivatives of uup to order n, defined on an open subset M (n) ⊂ X ×U (n) . the total derivativeof P with respect to x i is the unique smooth function D i P (x, u (n) )defined on M (n+1) and depending on derivatives of u up to order n + 1,with the recursive property that if u = f(x) is any smooth function thenD i P (x, pr (n+1) f(x)) = ∂ x i{P (x, pr (n) f(x))}.For example, in the case of SO(2) group, with the infinitesimal generatorv = −u ∂∂x + x ∂∂u ,the first prolongation is (as calculated above)whereAlso,pr (1) v = −u ∂∂x + x ∂∂u + ∂φx ,∂u xφ x = D x (φ − ξu x ) + ξu xx = 1 + u 2 x.φ xx = D x φ x − u xx D x ξ = 3u x u xx ,thus the infinitesimal generator of the second prolongation pr (2) SO(2) actingon X × U (2) ispr (2) v = −u ∂∂x + x ∂∂u + (1 + u2 x) ∂∂+ 3u x u xx .∂u x ∂u xx