Sophus Lie, the mathematician

Example.Consider the differential equationThe equation can be writtendy y + x(x 2 + y 2 )dx -x y(X 2 + y 2 )xdzThe left hand side equals tan a where a is the angle at (x, y) betweenthe integral curve through (x,y) and the radius from (0,0) to (x,y).The right hand side depends only on the distance from the origin sothe angle a is constant on the circle with center (0,0) passing through(x, y). Consequently, the rotation groupfor whichOt : (x. y) -- (x cos t - y sin t, x sin t + y cos t)T = -y&/&x + xz/dyas an inte­leaves the equation stable. The theorem gives (x 2 + y 2 ) - 1grating factor to the original equation.Another example is the equationdu(d2v)2which occurs as an example for different purposes in a paper [17] byHilbert. Cartan in [6. 7] showed that this equation has the exceptionalgroup G2 as a stability group (transformations in the independent variablex and dependent variables u and v allowed). See Anderson, Kamranond Olver [1] for a modern treatment.Generalizations of the theorem above to systems ([21]) led Lie todevelop a theory for r-parameter local groups of transformations in R':T(t): i = fi(Xl .. ,n; tl,... tr) (t) = (tl t,). (5)Here it is assumed that the fi are analytic functions, depending essentiallyon all the ti and thatT(o) = I, T(t)o T(s) = T(U), (6)IZ