I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. Additionally, the distribution is defined by right invariant vector fields.


Due to the abundance of right invariance, I am hoping for an equation which holds only in $\mathfrak{su}(n)$, for the generator to such a geodesic. I have attempted to use Lagrange multipliers to achieve what I want, but I am confused about how exactly to do this at the algebra level.



equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
OK. While I could write an exposition of the now-standard method of setting up and solving these sorts of right-invariant constrained variational problems on Lie groups, I think that it will be better for you to see a full exposition. 


I recommend the book *Exterior Differential Systems and the Calculus of Variations*, by P. A. Griffiths, as a good place to read about these kinds of problems. Griffiths works out several examples of exactly the kind of thing you are looking for in that book, and, if you read that, you should have no trouble setting up your problem along these lines and writing down the appropriate Euler-Lagrange equations and extracting the Euler-Poincaré equations from them, which are exactly the equations on the dual of the Lie algebra that describe the evolution on the Lie algebra part. 


