I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations (which are second order).


I know how to derive the Euler Poincare equations (which are first order) for the geodesics. However, I can't find anywhere or work out how to convert the boundary conditions into boundary conditions for the corresponding EP equation. I am expecting to obtain one condition on the derivative of the generator at $t=0$ to the geodesic. Any advice or references would be great!



Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton
There is no easy way to do this, since you have to solve an ODE between the E-L equations and the E-P equations. For numerical solutions you can minimize the functional
$$
F(X,g\_0,g\_1) = \int\_0^1 \|X(t)\| dt + \epsilon d(g(1).g\_0 - g\_1)
$$
with respect to $X:[0,1]\to \mathfrak g$, where 
$$
\partial\_t g(t) = X(t).g(t),\quad g(0)=1
$$
and where $d$ is an arbitrary (easy) metric on $G$ which measures how well you hit when you shoot with right invariant geodesics from $g\_0$ to $g\_1$. 
This method is used for image registration (for the diffeomorphism group instead of $G$) and is called LDDMM (see references given in 8.1 of [here](http://www.mat.univie.ac.at/~michor/shape-overview.pdf)).


