Units problem with my Hamilton's equations

[fluidistic]

Homework Statement

Let the Hamiltonian with canonical variables be H(q,p)=\frac{\alpha ^3 e^{2\alpha q }}{p^3} where alpha is a constant.
1)Given the generating function F(q,Q)=\frac{e^{2\alpha q }}{Q}, find the expression of the new coordinates in function of the old ones: Q(q,p) and P(q,p).
2)Find the expression of K(Q,P) and the corresponding Hamiltonian equations.
3)With the initial conditions Q(t=0)=Q_0 and P(t=0)=P_0, solve these equations for times t<P_0^2/2.
4)Find q(t) and p(t) for the initial conditions q(0)=0 and p(0)=1.

Homework Equations

Lots of.

The Attempt at a Solution

1)I found out Q=\frac{\alpha e^{\alpha q}}{p} and P=\frac{p^2}{\alpha ^2 e^{\alpha q}}.
2)The Hamiltonian in function of the new variables gave me K=\frac{Q}{P}. This simple expression makes me feel I didn't make any mistake yet.
3)Hamilton equations gave me \dot P=-\frac{1}{P} and \dot Q = -\frac{Q}{P^2}.
Solving the first equation gave me \frac{P^2}{2}=-t+\frac{P_0^2}{2}. But... I am adding a time with a linear momentum squared ( kg times m /s )^2. How can this be right? Even in the problem statement, they write "t<P_0^2/2", does this even make sense?
By the way I do not know how to answer to question 3. Can someone help me?

 

[Liquidxlax]

http://en.wikipedia.org/wiki/Generating_function_(physics)

you're are using the rules of F₁ right?

it should fall out pretty easy from that

 

[fluidistic]

http://en.wikipedia.org/wiki/Generating_function_(physics)

you're are using the rules of F₁ right?

it should fall out pretty easy from that

Yes I do, I actually solved part 1 and 2 (stuck on part 3).
Just a question... P and Q can in theory have any units? Because the problem statement compares time unit vs P^2 units. In other words, can P have units of \sqrt s?

 

[Liquidxlax]

Yes I do, I actually solved part 1 and 2 (stuck on part 3).
Just a question... P and Q can in theory have any units? Because the problem statement compares time unit vs P^2 units. In other words, can P have units of \sqrt s?

I can't say for sure as I'm currently learning this as well, but i think you're right so long as the transformation is canonical

MJM^T=J the units are negligible

I just had my midterm and 2 of the canonical transforms didn't have proper units yet i did get the questions right

 

[fluidistic]

Ah ok thanks a lot.
Assuming what I found is right then for part 3) I find P(t)=\sqrt {-2t+P_0^2} and Q(t)=e^{\frac{\ln (2t+P_0^2 )-\ln (P_0^2)}{2}+\ln Q_0}.
For 4), I assume they meant q(t) and p(t) as written and not Q(t) and P(t) that I just found. Otherwise the condition Q_0=0 would be a real problem.

 

[Liquidxlax]

i figured since i can't explain it i'd actually do the problem and i did not get the same P as you

i got P = (p²e^-2αq)/4α²

P = -dF/dQ = e^2αq/Q²

maybe that is why you're not getting your desired units?

i'm not going to finish the problem because I've suffered enough with my homework and midterms lol

 

[fluidistic]

My bad I made a typo when writing F here. It should be F(q,Q)=\frac{e^{\alpha q}}{Q}. I do not see any other typo for now... sorry about that.
P.S.:No problem if you don't solve the problem. :) But now I'm convinced P and for that matter, p can have almost any possible units. Not necessarily the ones of linear momentum or so, as I previously thought.

 

[ygolo]

These are generalized coordinates, so p and P don't necessarily have to be linear momentum.

 

[fluidistic]

These are generalized coordinates, so p and P don't necessarily have to be linear momentum.

I see, thank you. Their name/letter mislead me.