Can I substitute the new coordinates in the old hamiltonian?

[Coffee_]

We went over this concept quite fast in class and there is one thing that confused me:

When transforming from a set of $q_i$ and $p_i$to $Q_i$ and $P_i$, if one checks that the transormations are canonical the new Hamiltonian $K(Q_i, P_i)$ obeys exactly the same equations.This has been proven.

Question: How does one in general find this new Hamiltonian $K(Q_i, P_i)$? I have gotten the impression that it's not as easy as just substituting the transformations into the old coordinates, or is it?

 

[Orodruin]

The Hamiltonian is a function on phase space. It does not matter what coordinates you use to express it.

 

[Coffee_]

The Hamiltonian is a function on phase space. It does not matter what coordinates you use to express it.

This means that I can just substitute my old coordinates in function of the new coordinates into my old hamiltonian to get the new hamiltonian?