Helliwell & Sahakian Modern Classical Mechanics at least seems to do a much better job of explaining the Legendre transform than Goldstein, but it still never mentions the convexity requirement on f.
I feel like understanding the general convex conjugate and then seeing the Legendre transform as a special case is almost more intuitive.
I feel like understanding the general convex conjugate and then seeing the Legendre transform as a special case is almost more intuitive.