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.

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    $\begingroup$ What do you mean by 'the geodesics [that] are parallel to a linear affine distribution on $\mathrm{SU}(n)$'? Do you simply mean the geodesics of a (right-invariant) sub-Finsler structure on $\mathrm{SU}(n)$ or do you really mean the geodesics of a (right-invariant) Finsler structure that happen to have their velocity vectors lie in a given (right-invariant) $k$-plane field on $\mathrm{SU}(n)$? Your question literally asks for the latter curves (of which, there might not be any), but I suspect that you actually want the former (of which there are bound to be plenty). $\endgroup$ Aug 24 '14 at 17:59
  • $\begingroup$ I meant the former thing. Would they still called be called sub finsler geodesics in the case of an affine linear distribution rather than just a linear distribution? I wasn't sure. $\endgroup$
    – Benjamin
    Aug 24 '14 at 21:58
  • $\begingroup$ So, by 'parallel to an affine linear' distribution, do you mean that you require that the tangent vectors actually lie in a right-invariant subset of the tangent bundle that, at the identity, is an affine subspace of the tangent space at the identity? I'm sorry to be so picky about the language, but I want to be sure that I understand what you mean by affine linear distribution, which, if I am right, is what most people call an affine distribution. $\endgroup$ Aug 25 '14 at 0:48
  • $\begingroup$ Yes, I am hoping for what you said. I found it really hard to achieve clarity with this question! $\endgroup$
    – Benjamin
    Aug 25 '14 at 16:23

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.

  • $\begingroup$ Would it be ok to ask for another reference as I can't get hold of that book without paying a billion pounds more than I have as it's not in my university library. $\endgroup$
    – Benjamin
    Aug 26 '14 at 0:32
  • $\begingroup$ Actualy, I got it after all. Thanks, it had the answer I needed. $\endgroup$
    – Benjamin
    Aug 26 '14 at 8:24
  • $\begingroup$ I've read through the relevant parts of the link you sent. However, I must be confused by something. I seem to be deriving that the equations I need are the standard EP equations for the constrained geodesics simultaneously with the equations imposing the constraints (i.e. that the curve is parallel). These equations seem to be over determined in the case of $SU(2)$ and a two dimensional affine distribution which is right invariant. Should I expect this or have I made an error? I thought that controlability of this system is guaranteed by the Frobinious theorem/Hormander condition. $\endgroup$
    – Benjamin
    Sep 1 '14 at 21:45

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