Can I substitute the new coordinates in the old hamiltonian?

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Transforming from a set of coordinates (q_i, p_i) to (Q_i, P_i) in Hamiltonian mechanics maintains the canonical structure, ensuring the new Hamiltonian K(Q_i, P_i) follows the same equations of motion. The process of finding K involves substituting the old coordinates expressed in terms of the new coordinates into the original Hamiltonian. While this substitution is straightforward, it may not always yield a simple form for K, depending on the complexity of the transformation. The key point is that the Hamiltonian remains a function on phase space, regardless of the coordinate system used. Thus, the transformation does not change the underlying physics, but care must be taken in the substitution process.
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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?
 
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The Hamiltonian is a function on phase space. It does not matter what coordinates you use to express it.
 
Orodruin said:
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?
 
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