- #1
Emanuel84
- 14
- 0
Hi, I tried to solve this problem, but I was unsuccessful
Here is the problem:
Given the transformation:
[itex] \left \{ \begin{array}{l} Q = p^\gamma \cos(\beta q) \\ P = p^\alpha \sin(\beta q) \end{array} \right. [/itex]
a) Determine the values of the constants [itex] \alpha [/itex], [itex] \beta [/itex] and [itex] \gamma [/itex] for which this transformation is canonical.
b) In correspondence of these values, find a generating function of the transformation.
How can I solve this problem? Firstly, I used the Poisson bracket condition for canonicity:
[itex] [Q,P]_{q,p} = \frac{\partial Q}{\partial q}\frac{\partial P}{\partial p}-\frac{\partial Q}{\partial p}\frac{\partial P}{\partial q} [/itex].
Afterwards I supposed:
[itex] pdq-PdQ [/itex]
to be an exact differential.
Still, I didn't manage to find [itex] \alpha [/itex], [itex] \beta [/itex] and [itex] \gamma [/itex], as if I missed a condition...
Can you help me, please?
Here is the problem:
Given the transformation:
[itex] \left \{ \begin{array}{l} Q = p^\gamma \cos(\beta q) \\ P = p^\alpha \sin(\beta q) \end{array} \right. [/itex]
a) Determine the values of the constants [itex] \alpha [/itex], [itex] \beta [/itex] and [itex] \gamma [/itex] for which this transformation is canonical.
b) In correspondence of these values, find a generating function of the transformation.
How can I solve this problem? Firstly, I used the Poisson bracket condition for canonicity:
[itex] [Q,P]_{q,p} = \frac{\partial Q}{\partial q}\frac{\partial P}{\partial p}-\frac{\partial Q}{\partial p}\frac{\partial P}{\partial q} [/itex].
Afterwards I supposed:
[itex] pdq-PdQ [/itex]
to be an exact differential.
Still, I didn't manage to find [itex] \alpha [/itex], [itex] \beta [/itex] and [itex] \gamma [/itex], as if I missed a condition...
Can you help me, please?