- #1

cwasdqwe

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

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

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.

(See attempted 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

[itex]\left\{Q,P\right\}=1[/itex]

[itex]\left\{ P,Q \right\}=-1[/itex]

[itex]\left\{ Q,Q \right\}=0[/itex]

[itex]\left\{ P,P \right\}=0[/itex]

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

[itex]\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}[/itex]

[itex]\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}[/itex]

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

[itex]-\frac{\partial H}{\partial Q}= \dot{P}[/itex]

[itex]-\frac{\partial H}{\partial P}= \dot{Q}[/itex]

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 [itex]\left\{ H,f \right\}=0[/itex], i.e., if commutes with the Hamiltonian, but yet I've not it clear.

Thank you very much!

## Homework Statement

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

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

[itex]\left\{Q,P\right\}=1[/itex]

[itex]\left\{ P,Q \right\}=-1[/itex]

[itex]\left\{ Q,Q \right\}=0[/itex]

[itex]\left\{ P,P \right\}=0[/itex]

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

[itex]\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}[/itex]

[itex]\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}[/itex]

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

[itex]-\frac{\partial H}{\partial Q}= \dot{P}[/itex]

[itex]-\frac{\partial H}{\partial P}= \dot{Q}[/itex]

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 [itex]\left\{ H,f \right\}=0[/itex], i.e., if commutes with the Hamiltonian, but yet I've not it clear.

Thank you very much!

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