# Units problem with my Hamilton's equations

by fluidistic
Tags: equations, hamilton, units
 PF Gold P: 3,225 1. The problem statement, all variables and given/known data 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 $t  P: 322 http://en.wikipedia.org/wiki/Generat...tion_(physics) you're are using the rules of F1 right? it should fall out pretty easy from that PF Gold P: 3,225  Quote by Liquidxlax http://en.wikipedia.org/wiki/Generat...tion_(physics) you're are using the rules of F1 right? it should fall out pretty easy from that Yes I do, I actually solved part 1 and 2 (stuck on part 3). Just a question... P and Q can in theory have any units? Because the problem statement compares time unit vs [itex]P^2$ units. In other words, can P have units of $\sqrt s$?

P: 322
Units problem with my Hamilton's equations

 Quote by fluidistic 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

MJMT=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
 PF Gold P: 3,225 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.
 P: 322 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 = e2αq/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
 PF Gold P: 3,225 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.
 P: 30 These are generalized coordinates, so p and P don't necessarily have to be linear momentum.
PF Gold
P: 3,225
 Quote by ygolo 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.

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