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49

On predicting planetary orbits A number of studies have shown that the inner solar system is chaotic, with a Lyapunov time scale of about 5 million years. This 5 million year time scale means that while one can somewhat reasonably create a planetary ephemeris (a time-based catalog of where the planets were / will be) that spans from 10 million years into ...


20

An orbit is stable because of conservation of angular momentum. Suppose we start with an object in an exactly circular orbit and slow it down slightly. That means it is moving at less than orbital velocity so it starts to fall inwards. However as its distance to the Sun decreases the tangential component of its velocity has to increase to conserve angular ...


20

Chaotic is not the same as random. A chaotic system is entirely deterministic, while a random system is entirely non-deterministic. Chaotic means that infinitesimally close initial conditions lead to arbitrarily large divergences as the system evolves. But it's impossible, practically speaking, to reproduce the same initial conditions twice. Given ...


16

It has been shown by Eichhorn, Linz and Hänggi in 2000 that the numerical values of Lyapunov exponents are invariant under any invertible variable transform. This is just a reformulation of the fact that they are metric invariant, because the authors presume the norm $|\cdot|$ to be an arbitrary norm in the given coordinates - just it's basic properties such ...


15

This question was studied fairly recently by a team at Edinburgh University. Their paper is available here, though I'm not sure if you can get it without having to hand over some cash. The bottom line is that in principle the trajectory of a die can be calculated, but it is a chaotic system and that means tiny inaccuracies in the measured initial conditions ...


13

Well, yes. In a purely mathematical world where you can specify initial conditions exactly, chaotic systems are fully deterministic. It's not like a quantum system with wavefunction collapse, whose evolution can never be specified exactly by the initial conditions. But in practice, we can never specify (or know) the initial conditions exactly. So there will ...


12

The answer is sort of yes and no. YES: If you have two perfectly identical panes of glass and two perfectly identical projectiles, and you throw the two projectiles in a perfectly identical way, then the two panes will shatter in a perfectly similar fashion. This is really just by construction, you did the same thing twice. NO: Shattering glass involves ...


11

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


10

If we accept that the system (the Earth's atmosphere in this case) is chaotic and adopt the usual definitions of a chaotic system, e.g. one by Edward Lorenz Chaos: When the present determines the future, but the approximate present does not approximately determine the future. we immediately see the answer to your question. The small (impossible to ...


10

I think the following from the wikipedia entry clears up well the terminology: Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is ...


10

Classical mechanics is perfectly integrable for two bodies as a closed or isolated system. However, early on it was found that problems existed, where Newton found he could not find a solution for the motion of the planets in a complete form. He made his famous statement that God had to readjust the solar system now and them. Poincare solved the Sweden ...


8

The gravitational potential is what is known as a central force, which means that "how strong" the potential is only depends on how far away you are, and not on what angle you are relative to it. Having said that, gravitational systems are often treated in terms of an effective potential (full explanation provided on the Wikipedia page) which look like this ...


7

If you take a well-behaved physical system and perturb it a little bit, then you expect the total behavior or your system to be changed only a little bit. You can quantify this by saying that if your initial perturbation is $\delta$, then the final perturbation can never exceed $\gamma \delta$ for some constant $\gamma$. In many cases, such perturbations ...


7

Numerical simulations are not always meaningful, as chaos theory belongs to the large subject of dynamical systems theory. Although the definitions differ, chaos generally occurs in three contexts: Sensitive dependence on initial conditions (SDIC). The set is topologically transitive. Periodic points are dense in the set. Think of two particles having ...


7

Chaotic trajectories are perfectly deterministic, it's just that they demonstrate an extreme sensitivity to initial conditions. This is to say that if you start with the exact, precisely same initial conditions, you will get the exact, precisely same trajectories. But if you are even a tiny bit off from the initial conditions, the resulting trajectories will ...


6

The basic idea is that statistical properties of complex physical systems fall into a small number of universal classes. A very known example of this phenomenon is the universal law implied by the central limit theorem where the sum of a large number of random variables belonging to a large class of distrubutions converges to the normal distribution. Please ...


6

David Bar Moshe's answer is fine, but I wanted to go into more detail. The main reason that random matrices show up in dynamical systems is because they describe the level statistics of classically chaotic motions. In classically integrable systems, there is a semiclassical formula for the level-spacing, determined by the Bohr-Sommerfeld rule. If you know ...


6

Good question. You are correct in that without any restoring force, an object balanced precariously in an equilibrium position will be unstable. In physics, we use the scalar quantity of "potential" to find the equilibrium positions. These will be the maxima and minima in the potential field. The negative gradient of the potential gives the force. You've ...


6

Qualitative discussion (almost math free) The real key to understand orbits is the conservation of angular momentum. A two body orbit is nice this way insofar as it is a planar system and we get an easy expression for the angular momentum (we'll assume a satellite much, much less massive than the primary and not bother with the canonical transformation ...


5

I think that this Wikipedia article pretty much summarizes all of this.


5

I'm following Steven H. Strogatz's explanation in his book Nonlinear Dynamics and Chaos throughout this answer. Great book, worth a read. See also the detailed explanation on the wolfram website. Let's use the logistic map $x_{n+1} = rx_n(1-x_n)$ as an example. Its bifurcation diagram looks like this (courtesy of wikipedia): The easiest window to ...


5

Strictly speaking, there is no quantum chaos. Time evolution is unitary, which implies that small changes in state are not magnified in size. Thus the sensitive dependence on initial conditions, the prerequisite for chaos in classical mechanics, is absent in quantum mechanics. Even stronger, in discrete quantum systems with a finite-dimensional Hilbert ...


5

Jack Wisdom at MIT has extensively studied the question of the stability of the solar system. He has a list of papers with links to freely-readable PDF files on his website: http://groups.csail.mit.edu/mac/users/wisdom/ A good starting point might be "Is the Solar System Stable? and Can We Use Chaos to Make Measurements?" (PDF) (in Chaos, proceedings ...


5

It is true that the double pendulum exhibits integrable behavior, when the initial angles are very small, however, in general, it is very difficult to characterize the chaotic behavior of the double pendulum in terms of the initial angles. There are other representations which provide a clearer picture of its chaotic behavior. The introductory section of ...


5

How far ahead can we predict solar and lunar eclipses? NASA has uncertainty calculations that show how certain we are about when eclipses happen. From a back of the envelope, the eclipses will likely vary by a full day 35 thousand years from now. That said, we have eclipse seasons, so we know eclipses will continue to happen, and at roughly which time of ...


4

To start things off I'd say that noting the $L_z$ component is conserved seems to mean pretty much nothing, since you're considering the motion as restricted to the $\mathcal{X}\mathcal{Y}$ plane. If you had assumed the motion along the $\mathcal{Z}$ axis to be possible, then we'd be talking about the spherical double pendulum instead of the planar one ...


4

Three different points of views on essentially the same thing: Chaotic systems are not only sensitive to numerical errors, but also to any other small perturbations, such as dynamical noise, which may simulate real conditions. Though tiny perturbations affect the detailled, microscopic future of a system, its qualitative dynamics is unaffected. And the ...


4

In general, no. It is possible to recognise a (system of) differential equation(s) as being nonlinear purely by inspection, but there are plenty of non-chaotic nonlinear systems. Chaos is a stronger (and, unfortunately, not well-defined) condition. Also, many (most?) systems, including the famed Lorenz attractor, only exhibit chaos under certain conditions. ...


4

The best reference for this is Feigenbaum's original article, reprinted in "Universality in Chaos" by Cvitanovic. The point is that when you iterate a map, every time you period double, you fold up the function one more time. The behavior is dominated by the solution to the following equation: $$ \alpha g(g(x/\alpha)) = g(x)$$ Which says that g iterated ...


4

I'm guessing that when you talk about randomness you're thinking about the collapse of the wavefunction and that the the result of the collapse is apparently random. If so, most us currently believe that the randomness is only apparent and is the result of decoherence. Decoherence describes the interaction of a quantum system with the environment around it. ...



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