Generating function for canonical transformation

In summary, the conversation discusses finding the generating function for a given transformation with two independent variables. The attempt at a solution involves expressing the transformation in terms of the generating function and using equations for the third kind of generating function. A possible solution for the generating function is given at the end.
  • #1
Loxias
29
0

Homework Statement


Given the transformation

[tex] Q = p+iaq, P = \frac{p-iaq}{2ia} [/tex]


Homework Equations


find the generating function


The Attempt at a Solution



As far as I know, one needs to find two independent variables and try to solve. I couldn't find such to variables.

I've tried expressing it in terms of F(Q,P), and F(q,p) but always had one more term in the equation that prevented me from getting to [tex] H(q,p) = -H(Q,P) + \frac{\partial F}{\partial t} [/tex]

I'm pretty clueless as to what is needed here. Can someone help me get started?

Thanks.
 
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  • #2
Ok, this is what I did :

[tex] Q = 2ia(P + 2q) , P = \frac{p-iaq}{2ia} [/tex]
which means that Q and p are independent coordintes, which means the generating function will be of the third kind, [tex] F_3(Q,p) [/tex].

for the third kind,
[tex] q = -\frac{\partial F_3}{\partial p} = \frac{p}{ia}-2P [/tex]
[tex] P = -\frac{\partial F_3}{\partial Q} = \frac{Q}{2ia}-2q [/tex]

from the first equation we get
[tex] F_3 = 2pP - \frac{p^2}{2ia} + F(Q) [/tex]
and from the second
[tex] F_3 = 2Qq - \frac{Q^2}{4ia} + F(p) [/tex]

summing both I get
[tex] F_3 = 2Qq + 2pP - \frac{1}{2ia} (p^2 + \frac{Q^2}{2}) [/tex]

Does this seem right??
 
Last edited:

Related to Generating function for canonical transformation

What is a generating function for a canonical transformation?

A generating function for a canonical transformation is a mathematical function that allows us to map a set of canonical variables to a new set of canonical variables. It is used to simplify the process of performing canonical transformations in classical mechanics.

How is a generating function related to Hamiltonian mechanics?

A generating function is closely related to Hamiltonian mechanics as it is used to generate new Hamiltonian equations of motion from a given set of canonical variables. It allows us to transform between different sets of canonical variables while preserving the Hamiltonian structure of the system.

What are the properties of a generating function for a canonical transformation?

The two main properties of a generating function for a canonical transformation are that it is a function of the old and new canonical variables and that its partial derivatives with respect to these variables are first-order homogeneous functions. These properties ensure that the transformation is canonical and preserves the Hamiltonian structure of the system.

How do we determine the appropriate generating function for a given canonical transformation?

The appropriate generating function for a given canonical transformation can be determined by using a set of equations known as the canonical transformation equations. These equations relate the old and new canonical variables and their respective momenta, and the generating function can be derived from these equations.

Can a generating function be used for non-canonical transformations?

No, a generating function is specific to canonical transformations and cannot be used for non-canonical transformations. For non-canonical transformations, different mathematical methods, such as the Lagrangian formalism, must be used.

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