There are a number of sources.  If you are comfortable using differential forms, then one source that you may find useful is 

L. Hsu, *Calculus of variations via the Griffiths formalism*, J. Differential Geom. **36**, 551–589 (1992).

He explains how to do this calculation very explicitly there.

Another source is 

R. Montgomery, *A Tour of Subriemannian Geometries, Their Geodesics and Applications*,  AMS Mathematical Surveys and Monographs, Volume 91. 2002.  MathSciNet review: MR1867362

You can get a pdf download from the American Mathematical Societ web page https://www.ams.org/books/surv/091/