So I was reading two books: “A Primer on the Calculus of Variations and Optimal Control” by Mike Mesterton-Gibbons, and “The Calculus of Variations” by Bruce van Brunt. When it came down to free endpoints, Gibbons only did the case when the independent and dependent variable are related aka transversality (Lecture 11), where as Brunt did both unrelated and related (Chapter 7.2-7.3), albeit in a “hand-wavy” way. Brunt used big-O notations throughout, as well as some variables like $X_k$ and $Y_k$ which somehow made no sense to me as they came out of the blue. Gibbons, on the other hand, used notation that is much easier to interpret.
Is there at least a book that derives the equations for optimizing a functional with free unrelated endpoints using a method that’s closer to Gibbons’? If so, can one please point out the chapters and sections if not showing how it’s derived here?
Update 1: From my impression, many calculus of variations references discuss the free related endpoints while they seem to avoid discussing the free unrelated endpoints.