Chris Bishop's book "Pattern Recognition and Machine Learning" has some stuff on applying variational methods to machine learning. You might have a look at that and follow his references.

Also, to me, "A Primer on the Calculus of Variations and Optimal Control Theory" by Mike Mesterton-Gibbons looks nice. You can get a sample of it (table of contents, the first few pages) at http://www.ams.org/bookstore-getitem/item=STML-50