dmf is kind of df/dp
and d\phai /dt is kind of dp/dt
so you're right.
I just need help understanding one line of the proof in this pdf. It's top of page 5, how does the chain rule lead us to the second line from the bottom (the line where the t=0 moves to the end) of the proof?
Why is the line not df/dp dp/dt? I thought that's what the chain rule was...