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Let me show you part of a book "Mechanics From Newton’s Laws to Deterministic Chaos" by Florian Scheck.

I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals?

It seems this point is crucial for me to get into generating functions (and then to Hamilton-Jacobi functions) but I won't proceed untill I fully understand derviations. So can anyone explain mathematical story of the derviation?

I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals?

It seems this point is crucial for me to get into generating functions (and then to Hamilton-Jacobi functions) but I won't proceed untill I fully understand derviations. So can anyone explain mathematical story of the derviation?