Recent content by Random137

From the poincare points, it doesn't look it will form a completely closed curve, is it safe to assume it will form a completely closed curve as time tends to infinity?

Can you explain what do you mean by tori in the poincare plots?

Does that mean the system is chaotic? Would you say in the case where epsilon tending to zero would be easier/feasible to do?

What I wanted to answer is, for a given set of initial conditions (x(0), y(0), p_x(0), p_y(0)) can I roughly estimate the values of <p_x^2> and <p_y^2> without running the numerical integrations. Actually, I know the value of <p_x^2> + <p_y^2> (= constant * energy) for a given set of initial...

Well, I still don't really know how to find the invariant. I still don't know how to tackle the questions that I want to answer. Do you have any recommendations?

Looking at the 500 trajectories, I calculated <p_x^2> and <p_y^2> for each of the trajectories by evolving the trajectories over a long period of time. Then I arrange the trajectories according to their <p_x^2> value (from smallest to largest). There seems to be a pattern, similar values of...

Doing the same thing with a smaller epsilon, it seems like the behaviour is quite different? How can I extract information from these Poincare plots?

The following contains 500 different trajectories with initial condition satisfying: x(0) = 0, y(0) = 1 with p_x^2(0) + p_y^2(0) = 1

The reason I use Mathematica because it has built in function to solve differential equations numerically, so it doesn't require too much coding and understanding numerical algorithms. What I am really interested is the case when epsilon = 0.25 (ultimate goal). The limit when epsilon = 1 is...

I am a grad student but my knowledge in this area is a bit weak, I guess what I really need is something that bridge the gap between undergrad and grad level. Can you explain a bit more about "if a trajectory is really on a torus or merely squeezed in between two tori"? Roughly how many...

I tried to implement the velocity-verlet algorithm by myself and I had the same problem with energy conservation when epsilon is small. Would Runge-Kutta fix such problem? Do you happen to use Mathematica?

My understanding in this area of Hamiltonian dynamics (ergodicity, invariant torus, symplectic form etc) is lacking. I tried to read about this topic but most things I found are written in some rigorous mathematical language involving things like manifolds which I am not really familiar with. Do...

Currently, I am using something called Symplectic Partitioned Runge Kutta method in Mathematica (which I don't really understand how it works), it allows me to set invariant quantities and it will try and keep the invariant quantities almost constant throughout the integration over many time steps.

I created the same plots for epsilon = 0.02, I have some numerical issues trying to reduce epsilon even more as it gets more difficult keeping the energy "constant" (within some reasonable amount of error). It does seem to get simpler as you expected!

Let me elaborate on this: According to Bertrand's theorem, there are only 2 central potentials - k / r (Kepler) and k r^2 (harmonic) with all bound orbits which are also closed orbits. Therefore, one can assume/prove <p_x^2> = <p_y^2> for other central potentials. According to Virial Theorem...