Editing Jacobi field (section)

==Solving the Jacobi equation==

Let <math>e_1(0)=\dot\gamma(0)/|\dot\gamma(0)|</math> and complete this to get an [[orthonormal]] basis <math>\big\{e_i(0)\big\}</math> at <math>T_{\gamma(0)}M</math>. [[Parallel transport]] it to get a basis <math>\{e_i(t)\}</math> all along <math>\gamma</math>. 
This gives an orthonormal basis with <math>e_1(t)=\dot\gamma(t)/|\dot\gamma(t)|</math>. The Jacobi field can be written in co-ordinates in terms of this basis as <math>J(t)=y^k(t)e_k(t)</math> and thus
:<math>\frac{D}{dt}J=\sum_k\frac{dy^k}{dt}e_k(t),\quad\frac{D^2}{dt^2}J=\sum_k\frac{d^2y^k}{dt^2}e_k(t),</math> 
and the Jacobi equation can be rewritten as a system
:<math>\frac{d^2y^k}{dt^2}+|\dot\gamma|^2\sum_j y^j(t)\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\rangle=0</math> 
for each <math>k</math>. This way we get a linear ordinary differential equation (ODE). 
Since this ODE has [[smooth function|smooth]] [[coefficient]]s we have that solutions exist for all <math>t</math> and are unique, given <math>y^k(0)</math> and <math>{y^k}'(0)</math>, for all <math>k</math>.