A general problem in (q,p) -> (Q,P) for Hamiltonian

[cwasdqwe]

Hello everyone, this is my first thread. Hope to be helpful here, as well as to find some help! :D

Homework Statement

Given a Hamiltonian $H(q,p)$(known) and given a transformation of coordinates $(q,p)\rightarrow (Q,P)$:

a) Show that it is a canonic transformation
b) Solve the equations of motion
c) Is this a symmetry of H? Find the constants of motion.

Homework Equations

(See attempted solution)

The Attempt at a Solution

Just need to know if I'm wrong in some of these, and in that case being sure I'm doing right

a) By using the Poisson brackets, they must verify all of these

$\left\{Q,P\right\}=1$
$\left\{ P,Q \right\}=-1$
$\left\{ Q,Q \right\}=0$
$\left\{ P,P \right\}=0$


b) I tried by using the chain rule in the Hamiltonian

$\frac{\partial H}{\partial p}=\frac{\partial H}{\partial P}\frac{\partial P}{\partial p} + \frac{\partial H}{\partial Q}\frac{\partial Q}{\partial p}$

$\frac{\partial H}{\partial q}=\frac{\partial H}{\partial P}\frac{\partial P}{\partial q} + \frac{\partial H}{\partial Q}\frac{\partial Q}{\partial q}$

then I have a system of two equations and the two $\frac{\partial H}{\partial Q}$ and $\frac{\partial H}{\partial P}$, which will give me the equation of motion by using Hamilton's:

$-\frac{\partial H}{\partial Q}= \dot{P}$
$-\frac{\partial H}{\partial P}= \dot{Q}$


c) This point I haven't got clear. I've tried to relate it with 1, knowing that a quantity f is a constant of motion if the Poisson Bracket $\left\{ H,f \right\}=0$, i.e., if commutes with the Hamiltonian, but yet I've not it clear.

Thank you very much!

 

[mark726]

Thank you for your post and welcome to the forum! It's great to have another scientist join our community. Let's take a look at your questions and see if we can help.

a) Your approach to showing that the transformation is canonical is correct. The Poisson brackets must satisfy the four properties you listed in order for the transformation to be canonical.

b) Your approach to solving the equations of motion using the chain rule is also correct. However, you should also consider the inverse transformation (Q,P)\rightarrow (q,p) to get the complete set of equations of motion.

c) Your understanding of a constant of motion is correct. In this case, the constants of motion will be functions of the coordinates and momenta in the new coordinates (Q,P). To find them, you can use the fact that the Poisson bracket of the Hamiltonian with the constant of motion must be zero.

I hope this helps! Let us know if you have any further questions or if you need clarification on anything. Good luck with your work!