Why does chaos preclude exact solutions?

Question

It is sometimes said that the n-body problem (using the initial positions and velocities of n point masses to calculate their future paths) has no general closed-form solution because the system is chaotic: A small change in the initial conditions produces large changes in the subsequent motion. Why should that fact preclude an exact solution?

Answer 1

You are right about this wording being sloppy. A chaotic system is a deterministic system whose solutions at late times are exponentially sensitive to early times. Nothing about this definition precludes an exact solution. See https://www.lpthe.jussieu.fr/~viallet/solvable_chaos_pla.pdf for examples of chaotic systems that do have an exact solution.

Answer 2

I think Connor is correct. For what it's worth, there are three different words people are using here. "Analytical" means it can be represented by a convergent power series. "Closed-form" (usually) means it is analytical AND the expression contains no infinite series, products, etc. They are both "exact."

Karl Sundman actually demonstrated that, given non-zero angular momentum, all initial conditions of the three-body problem have an analytical (so, exact) solution. It is in the form of an infinite (Puiseux) series, so it is not closed-form.

The reason people don't talk about it often is that it's not that useful. For one, it is incredibly slowly converging. For a modern introduction, see here.

Answer 3

I think there might be two issues being conflated.

In a strictly mathematical sense, a system could be chaotic and yet have exactly calculable future states. This would mean that for two initial states that differ only very slightly their future states eventually are wildly different, but for each of them we can say exactly what the future states will be.

But when we're trying to do predictive physics rather than pure maths, a system being chaotic means that we cannot predict its future states even if the mathematical model we're using has exact solutions. This is because to make predictions about the evolution of a physical system we need to provide inputs to the mathematical model describing the initial state. Those inputs will be based on measurements of the real world (i.e. the masses and positions of the planets in a solar system), and those measurements are never perfect.

If the measurements are not perfect, then the initial state we provide to the model will never perfectly correspond to actual reality; there will be small differences from the true state. And if the model is chaotic those small differences matter; no matter how much we improve our measurement accuracy an arbitrarily small difference between the state that we measure and the true state will eventually lead to very different predictions of the future. It doesn't matter if we can exactly calculate the future of every initial state if we cannot tell which one of those very different exact predictions is going to be close to what happens in reality!

Non-chaotic systems can be more easily used to make predictions in physics because when our measurements provide a description of the initial state that is slightly off from true reality, the prediction will only be slightly off from the true future.

So chaos doesn't necessarily preclude exact solutions in the mathematics, it precludes using any solutions to make arbitrarily long-term predictions.

Answer 4

As the previous answers explained, chaos does not preclude the existence of simple analytic solutions. A simple counter is the logistic map. Say you want to solve the dynamic system: $$ x_{n+1} = 4x_n(1-x_n) $$ It ticks all the usual boxes of a chaotic system: sensitivity to initial conditions (i.e. a positive Lyapunov exponent which turns out to be $\ln2$), dense periodic orbits etc. At the same time, you can easily solve the problem by recognising a trigonometric identity: $$ x_n = \sin^2\theta_n \\ x_n = \sin^2\left(2^n\arcsin(\sqrt{x_0})\right) $$

Hope this helps.

Answer 5

It all depends on your definition of "solvable" and "chaos". In classical mechanics (and I believe that is the case your are interested in), the statement is true, if you are talking about Liouville integrability (as definition of solvable): see Chaos and integrability in classical mechanics