My professor in class went a little over chaos theory, and basically said that Newtonian determinism no longer applies, since as time goes to infinity, no matter how close together two initial points are, the distance between them will increase greatly. But why isn't this merely a matter of the imprecision of our measuring instruments? If we can somehow know our initial conditions exactly, wouldn't we still be able to calculate what the system will be like at some time t in the future?
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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 always be some uncertainty in the initial conditions, and it makes sense to characterize the behavior of a system in terms of its response to this uncertainty. Basically, a chaotic system is one in which any uncertainty in the state at time $t=0$ leads to exponentially larger uncertainties in the state as time goes on, and a non-chaotic system is one in which any initial uncertainty in the state decays away or at least stays steady with time. In the former (chaotic) case, given that we can't know the initial conditions to infinite precision, there will always be some time after which predictions of the behavior of the system become essentially meaningless - the uncertainty becomes so large that it fills up most of the state space. This is effectively similar to the behavior of a truly non-deterministic (e.g. quantum) system, in that our ability to make predictions about it is limited, so some people call chaotic systems non-deterministic. |
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I think there is some debate about this, at least according to Steven Strogatz. I paraphrase a part of his lecture that I'm watching: Even systems with perfectly known, perfectly deterministic laws, where all the positions of all the particles, and all the forces, are known, can still be unpredictable. It is such systems that are called chaotic. This is his view of it. I wish I could provide a link to the video, but it's on my hard drive. I think this is an important debate to have, if it is still up in the air, because it has a lot of implications for the notion of free will. It's a philosophically deep question. From my point of view, which is that of a lay person, Strogatz's view seems a bit hard to imagine, but then again, he's the expert. |
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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 prize for a solution to the stability of the solar system by demonstrating no such solution existed. This is what opened the door for chaos theory, where Poincare developed methods of separatrices and perturbations on them. For general systems it turns out that Newtonian mechanics is not integrable, Classical mechanics for systems with three or more bodies can\rq t be solved in closed form. For any $N$ body problem there are 3 equations for the center of mass, $3$ for the momentum, $3$ for the angular momentum and one for the energy. These are $10$ constraints on the problem. An N-body problem has $6N$ degrees of freedom. For $N~=~2$ this means the solution is given by a first integral with degree $2.$ For a three body problem this first integral has degree $8$. This runs into the problem that Galois illustrated which is that any root system with degree $5$ or greater generally have no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for $8$ solutions there is some eight order polynomial $p_8(x)~=~\prod_{n=1}^8(x~-~\lambda_n)$, with $8$ distinct roots$\lambda_n$ that are constant along the $8$ solutions. Since $p_8(x)~=~p_5(x)p_3(x)$, a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding its roots, or a set of solutions that are algebraic. This means that any system of degree higher than four are not in general algebraic. At the root of the $N$-body problem Galois theory tells us there is no algebraic solution for $N~\ge~3$. A problem in classical mechanics is the vanishing denominator problem for three bodies. This has a Hamiltonian $H( J,~\theta)~=~H_0(J,~\theta)~+~\epsilon H_1(J,~\theta)$ for $J~=~(J_1,~J_2)$ and $\theta~=~(\theta_1,~\theta_2)$. Here a generating function written according to the variable $J^\prime$ The number of resonance conditions that exist on the real line are dense. Within any $\epsilon$ neighborhood there will exist a countably infinite number of possible resonance conditions that correspond to rational numbers. As the orbit of a planet drifts it will pass through these resonance conditions and be chaotically perturbed. It is to be expected that for simple rational numbers, such as $1/12$ for Earth and Jupiter, rather than $1003/12000$ strong resonances occur. For more complex rational numbers it might be expected that the instability will be weaker. In other words if the ratio of frequencies are \lq\lq sufficiently irrational\rq\rq$~$so that $$ \Big|{{\omega_1}\over{\omega_2}}~-~{m\over s}\Big|~>~{{k(\epsilon)}\over{s^{2.5}}},~\lim_{\epsilon~\rightarrow~0}k(\epsilon)~\rightarrow~0 $$ the orbit is more stable. So an orbit that is removed from a “strong resonance”” condition near a simple rational number will be more stable than an orbit that is near an orbit with a simple rational ratio of frequencies. This is the basis for Greenberg’s Hamiltonian approach to chaos theory. This is called deterministic for the differential equations are time reverse invariant so the motion of a particle is absolutely determined. However. if you have a slight variation in the initial conditions of that particle is may in general end up arbitrarily far away from its starting point. The tiny variation $\delta z~=~(\delta q,~\delta p)$ becomes amplified by an exponential map $\delta z~\rightarrow~exp(\lambda t)\delta z$, for $\lambda$ the Lyapunov exponent. Any error in the specification of the initial conditions of a body results in the amplification of this error. From an algorithmic perspective a truncation results in numerical overflow errors which grow. So the dynamics of a particle can’t be integrated by computer to arbitrary accuracy into the future, even though nature actually does determine its dynamics. W. Zurek took this a bit further and considered how quantum fluctuations, where $\delta z$ is set by the Heisenberg uncertainty principle. |
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I think the following from the wikipedia entry clears up well the terminology:
Bold mine. Note that chaos results in completely deterministic systems when small errors in initial conditions yield highly divergent solutions. It is the "exactly" in your question that is unattainable, that will be a tiny bit off in chaotic situations,(highly nonlinear response to input parameters) even in computer solutions, because one cannot be more accurate than the computer bits. Note also that this does not mean that there are no mathematical methods to study the behavior of such systems. There are, and can be predictive in bulk. I would give as an example the study of Tsonis et al who have studied climate with a neural net chaotic model using as inputs bulk behavior of atmospheric and ocean currents. |
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