Can I substitute the new coordinates in the old hamiltonian?

In summary, when transforming from a set of ##q_i## and ##p_i## to ##Q_i## and ##P_i##, the new Hamiltonian ##K(Q_i, P_i)## will obey the same equations if the transformations are canonical. However, finding the new Hamiltonian is not as simple as just substituting the transformations into the old coordinates. This is because the Hamiltonian is a function on phase space and can be expressed in any coordinates. Therefore, one cannot simply substitute the old coordinates into the old Hamiltonian to get the new one.
  • #1
Coffee_
259
2
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?
 
Physics news on Phys.org
  • #2
The Hamiltonian is a function on phase space. It does not matter what coordinates you use to express it.
 
  • #3
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?
 

Related to Can I substitute the new coordinates in the old hamiltonian?

1. Can I substitute the new coordinates in the old Hamiltonian without any modifications?

No, the new coordinates will most likely result in a different form of the Hamiltonian, and modifications will be necessary.

2. How do I know which modifications to make to the Hamiltonian?

This depends on the specific situation and the type of coordinates being substituted. It is important to consult relevant literature and consult with experts in the field.

3. Are there any cases where the new coordinates can be directly substituted without modifications?

In some special cases, such as when the new coordinates are canonical, direct substitution may be possible. However, this is not always the case and it is important to carefully consider the specific situation.

4. What are the consequences of using the wrong Hamiltonian with substituted coordinates?

Using the wrong Hamiltonian can result in incorrect calculations and predictions, leading to inaccurate conclusions and potentially hindering scientific progress. It is important to carefully consider and verify the Hamiltonian being used.

5. Can I substitute the new coordinates in the Hamiltonian at any point during the calculation?

No, the substitution should be done at the beginning of the calculation, before any other steps are taken. This ensures that the resulting calculations are accurate and reliable.

Similar threads

I Proof that canonical transformation implies symplectic condition
    • Mechanics
Replies
7
Views
1K
Canonical transformations, poisson brackets
    • Classical Physics
Replies
1
Views
954
Are Canonical Transformations the Key to Understanding Hamiltonian Dynamics?
    • Classical Physics
Replies
4
Views
1K
Given a set of equations, show if it is a Hamiltonian system
    • Advanced Physics Homework Help
Replies
9
Views
1K
When can I substitute transformations into the Hamiltonian?
    • Introductory Physics Homework Help
Replies
2
Views
723
A Recover Hamilton equation from 2-form defined on phase space
    • Mechanics
Replies
3
Views
342
Question about canonical transformation
    • Classical Physics
Replies
1
Views
1K
I How can we transform a Lagrangian to obtain a new set of equations of motion?
    • Classical Physics
Replies
2
Views
2K
The restricted canonical transformation group
    • Advanced Physics Homework Help
Replies
1
Views
2K
Proving Hamiltonian Invariance with Goldstein Problems
    • Advanced Physics Homework Help
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top