Proving a Hamiltonian satisfies Hamilton's equations

[CMJ96]

Homework Statement

Homework Equations

Given in the above picture

The Attempt at a Solution

I have tried to rearrange the relationship between P and R to gain an expression for R, in terms of P. I subbed that into the expression for E and attempted to differentiate. I ended up with this expression.

I can't see how this is going to lead to the relationship for V.. any help would be appreciated here because I'm not sure where I am going with this question

 

[stevendaryl]

Homework Statement

View attachment 195547

Homework Equations

Given in the above picture

The Attempt at a Solution

I have tried to rearrange the relationship between P and R to gain an expression for R, in terms of P. I subbed that into the expression for E and attempted to differentiate. I ended up with this expression.
View attachment 195548
I can't see how this is going to lead to the relationship for V.. any help would be appreciated here because I'm not sure where I am going with this question

I think the expression for $E$ is wrong. I get the right answer for $V$ if

$E = \frac{\rho \kappa^2 R}{2} (ln(\frac{8R}{a_0}) - \frac{3}{2})$

The E given doesn't work out, unit-wise. (It has the same units as momentum, while it should have the units of momentum times velocity)

 

[CMJ96]

I think the expression for $E$ is wrong. I get the right answer for $V$ if

$E = \frac{\rho \kappa^2 R}{2} (ln(\frac{8R}{a_0}) - \frac{3}{2})$

The E given doesn't work out, unit-wise. (It has the same units as momentum, while it should have the units of momentum times velocity)

Oh dear... this is really concerning because this is a problem that my lecturer gave my class from a textbook that he edited.

 

[George Jones]

How are $dE/dP$ and $dE/dR$ related?

I don't know about the typo, as I haven't done the calculation.

 

[CMJ96]

How are $dE/dP$ and $dE/dR$ related?

I don't know about the typo, as I haven't done the calculation.

Is it a chain rule? dE/dR * dR/dP=dE/dP?

 

[George Jones]

Yes.

How is dR/dP related to dP/dR?

Then, put everything together.

 

[CMJ96]

Okay so I have attempted to apply the chain rule, I have the following equations, it seems close but not quite there, am I along the right lines?

 

[George Jones]

What is

$$\frac{d}{dx} \ln \left( ax \right)?$$

 

[stevendaryl]

Okay so I have attempted to apply the chain rule, I have the following equations, it seems close but not quite there, am I along the right lines?
View attachment 195655

You're making a little mathematical mistake:

$\frac{d}{dR} ln(AR) = 1/R$, not $\frac{A}{R}$

 

[CMJ96]

Ahhh yes!, so instead of -3/2 + a_0/8 it would be -3/2 + 1, hence the -1/2. Thank you, this is very helpful!

 

[Farang]

Hiiiiiii