Can I substitute the new coordinates in the old hamiltonian?

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SUMMARY

The discussion centers on the transformation of Hamiltonians in classical mechanics, specifically the transition from old coordinates (q_i, p_i) to new coordinates (Q_i, P_i). It is established that if the transformations are canonical, the new Hamiltonian K(Q_i, P_i) will satisfy the same equations as the original Hamiltonian. However, the process of deriving K(Q_i, P_i) is not simply a matter of substituting the new coordinates into the old Hamiltonian; it requires a deeper understanding of the phase space and canonical transformations.

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  • Understanding of canonical transformations in Hamiltonian mechanics
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  • Knowledge of Hamiltonian functions and their properties
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This discussion is beneficial for physics students, researchers in classical mechanics, and anyone interested in advanced topics related to Hamiltonian dynamics and canonical transformations.

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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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