Strategies for solving Hamilton's equations

[jjustinn]

When solving virtually any non-trivial system via Hamilton's equations, it seems that I'm ending up with equations of the form

$∂H/∂p = dx/dt = p/m$
$∂H/∂x = -dp/dt = -F(x)$
$p(t) = ∫(dp/dt)dt$
$d^2x/dt^2 = F(x)$
E.g., everything is given as a function of x, but we need to solve for t. I can change the variable of integration to x and get e.g. p(x), but that doesn't really help (especially since you can often directly solve for p(x)). One particular problem that's been irking me is where F(x) is either a delta function or a step function.

So...are there any tricks for solving Hamilton's equations (or Euler/Lagrange) for systems that are NOT like the $d^2x/dt^2 = -Kx$ that invariably show up in textbooks?

 

[dipole]

Well, there's really only a very small number of systems that have exact analytical solutions. Most systems have to be integrated numerically. If you post a specific example someone might be able to help you solve it, but I don't think there's really much general advice one can give.

 

[jjustinn]

Well, there's really only a very small number of systems that have exact analytical solutions. Most systems have to be integrated numerically. If you post a specific example someone might be able to help you solve it, but I don't think there's really much general advice one can give.

Actually, the system that brought me here is one where the equations of motion are well known: elastic collisions in one dimension -- either between two particles or between a particle and a potential barrier (possibly infinite).

I'm afraid to give too specific / complete a derivation, as it will probably get deleted as "homework-like", but one problem I've been wrestling with has $H = p^2/2m + Kθ(x)$, where θ(x) is a step function, and K → ∞...so in the above example, F(x) would be -Kδ(x), and the equations of motion would be the particle moving from -∞ toward zero at constant momentum $p_0$, then at x = 0, it immediately changes to $-p_0$, and moves back towards -∞. So, $dp/dt = -2p(x)δ(x)$, and $x(t) = -|p_0t/m|$. But, for the reasons stated above, I can't get from H to the (known) solution...and I'm having similar problems with "simpler" problems (e.g. a ramp instead of a delta function). If you're really interested, there's another thread on that particular problem here: https://www.physicsforums.com/showthread.php?p=4568733#post4568733...but I think the root of the problem there is I suck at integrals -- so if I get this figured out, that should follow.

Thanks,
Justin