Transforming Lagrangian without changing the equations of motion.

[alemsalem]

Main Question or Discussion Point

I know that it works with adding a total time derivative and multiplying the Lagrangian by a constant.
are these the only things that can be done to a Lagrangian such that the equations of motion have the same solutions q(t).

Thanks!!

 

[Dickfore]

There is a wider class of so called canonical transformations.

 

[alemsalem]

There is a wider class of so called canonical transformations.

These are the ones i'm having a problem with. the modified Hamilton's principle gives this definition for a canonical transformation:
pq - H = PQ - K + (total time derivative).. and that's because you can add a total time derivative inside the integral for the action.
I also want to know if there is more transformations on the Lagrangian without a change of variables.

Thanks :)