docu5_lie-groups.nb

Exponential map

The exponential map in the context of Lie groups is exp:G→G, or equivalently exp:g→G. We continue with discussion of the Lie group R.

$G = Setup[{(x_1 y_1 + x_2 y_3)/(1 + x_3 y_2), (x_2 + x_1 y_2)/(1 + x_3 y_2), (x_3 y_1 + y_3)/(1 + x_3 y_2)}, {1, 0, 0}] ;$

The exponential of a tangent vector {,,}∈G is determined by the integral curve of the left-invariant vector field induced by v through e∈G. Recall the general form of a left-invariant vector field on G.

${v_3 x_2 + x_1 (v_1 - v_2 x_3), v_2 (x_1 - x_2 x_3), v_3 + x_3 (v_1 - v_2 x_3)}$

Even for "simple" instances for v∈G, Mathematica has trouble finding exact solutions. We try the three basis vectors ,, of g.

$( {{{{x_1[] →^, x_2[] →0, x_3[] →0}}}, {{}}, {{{x_1[] →1, x_2[] →0, x_3[] →}}}} )$

For we yield the (unique) solution exp t ={1,t,0}, since

$E0[ΧIntegral[ΤL[G, {0, 1, 0}, 1]], Χ0[0, G[0]]]/.Ft[sol, ∂_sol, sol/.→0]$

${x_1[] →1, x_2[] →, x_3[] →0}$

Created by Mathematica (September 30, 2006)