Solving Hamilton Equations with Limited Knowledge

[psycho_dad]

Homework Statement

[PLAIN]http://img411.imageshack.us/img411/4412/sssa.jpg

Homework Equations

The Attempt at a Solution

Actually I have very basic knowledge of university physics and math, so the only things I've done are calculating Hamilton equations (I hope correctly)

q'=p2+p1
p'=-(q2+q1)

and f3 as I guess it was poisson brackets

f3=4q2p2-4q1p1

Now I don't have enough theoretical mechanics knowledge to move on, any help would be very welcome. Thanks!

 

[vela]

You didn't get the equations right. Hamilton's equations are

$$\dot{p_i} &=& -\frac{\partial H}{\partial q_i}$$

$$\dot{q_i} &=& \frac{\partial H}{\partial p_i}$$

So you should get four equations in total.

 

[psycho_dad]

Ok, so I get four equations (thank you for pointing it out):
q'1=p2
q'2=p1
p'1=-q2
p'2=-q1

Now, should I integrate them, and if yes, what should I do next to get solution?

 

[vela]

The problem now is that you have differential equations involving two functions, q1 and p2 for example. You want to combine equations so that you get a differential equation that involves only one function, which you can then solve.

 

[psycho_dad]

I'm not sure if I understand you correctly, but ok, this is what I get, but still can't see the whole idea

p1=-p''1
p2=-p''2
q1=-q''1
q2=-q''2

 

[vela]

Those are differential equations you can solve. You actually only need two of them. If you find q1, for instance, you can use one of the original equations, p2=q1', to solve for p2.

 

[psycho_dad]

I can't :D Well I think q1 and q2 should be trigonometric functions (cosines) while p1 and p2 (-sines), but I thought that q and p are some kind coordinate and moment functions

 

[vela]

You should review how to solve basic differential equations. Mechanics is already difficult enough to learn on its own, but not having a good grounding in mathematics makes it even more so.

 

[psycho_dad]

Yeah, I know, I actually more needed than wanted to solve this problem. I eventually managed to integrate equations (yeah, it turned out to be easy task after one glance into math book), but the easier way to show that functions were of the same system, was just to calculate poisson brackets of hamilton function and all other functions
{H;f1}={H,f2}={H,f3}=0

Anyway, thanks for help :)

 

[vela]

Ah, of course. It's been so long since I've taken classical mechanics I had forgotten all about that.