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!