Canonical transformations, generating function

In summary, the conversation discusses finding canonical transformations using the given generating function and verifying that the transformations of generalized coordinates are canonical. It also raises the question of the meaning of a canonical transformation originated by a specific generating function. The notation used and the specific steps to be taken are unclear and require further clarification.
  • #1
fluidistic
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Homework Statement


Given the generating function [itex]F=\sum _i f_i (q_j,t)P_i[/itex],
1)Find the corresponding canonical transformations.
2)Show that the transformations of generalized coordinates are canonical transformations.
3)What meaning does the canonical transformation originated by the generating function [itex]\Phi (q,P)=\alpha qP[/itex] has?

Homework Equations


[itex]p_i=\frac{\partial F }{\partial q_i}[/itex], [itex]P_i=-\frac{\partial F }{\partial Q_i}[/itex], [itex]H'=H+\frac{\partial F }{\partial t}[/itex].

The Attempt at a Solution


I don't know how to start. The notation confuses me, particularly the j. Should the sum be a sum over i and j?

Edit:1)[itex]Q_i=\frac{\partial F }{\partial P_i}=f_i(q_j,t)[/itex]. A canonical transformation is such that [itex]\dot Q_i=\frac{\partial H'}{\partial P_i}[/itex] and [itex]\dot P_i =-\frac{\partial H'}{\partial Q_i}[/itex].
Therefore I guess I must verify that [itex]\frac{\partial ^2 F}{\partial t \partial P_i}=\dot Q_i[/itex] and that... oh well I'm totally confused.
 
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  • #2
2)This is not really clear to me. I mean, I know that canonical transformations preserve the form of the Hamiltonian equations of motion, but how does this relate to the generating function?3)I'm guessing this means that the momentum and position are related by an exponential function, but I'm not sure.
 

1. What are canonical transformations?

Canonical transformations are mathematical transformations that preserve the fundamental equations of motion in Hamiltonian mechanics. They allow us to transform from one set of canonical variables to another without changing the dynamics of the system.

2. What is a generating function in canonical transformations?

A generating function is a mathematical function that helps us perform canonical transformations. It is a function of the old and new canonical variables and has the property that its partial derivatives with respect to the old variables are equal to the new canonical variables and vice versa.

3. How do you determine the type of a canonical transformation?

The type of a canonical transformation is determined by the type of generating function used. If the generating function depends only on the old canonical variables, it is a Type I transformation. If it depends only on the new canonical variables, it is a Type II transformation. And if it depends on both the old and new variables, it is a Type III transformation.

4. What is the physical significance of canonical transformations?

Canonical transformations have a physical significance in that they allow us to simplify the equations of motion for a system. They can help us eliminate unwanted variables or simplify the Hamiltonian, making it easier to solve for the dynamics of the system.

5. Are there any limitations to using canonical transformations?

While canonical transformations are powerful tools in Hamiltonian mechanics, they do have limitations. They can only be used for systems with a Hamiltonian that is a function of the canonical variables, and they cannot change the form of the Hamiltonian. Additionally, they may not always lead to simpler equations of motion and may introduce new variables that are difficult to interpret physically.

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