# Does the $3$-body problem imply that determinism is false?

Classical mechanics is famous for its supposed ability to theoretically calculate the state of a system at any given time provided all the necessary initial conditions. I believe this is the definition of determinism.

But a simple system if 3 particles interacting gravitationally fails to admit such a description. Some people use the arguments that one can still determine the state up to a certain time for any given system, however this is only an approximate prediction leaving even in this limited time window some uncertainty about the system.

EDIT :

This is purely a methematical physics question. Can you formally prove classical mechanics is deterministic from classical mechanics. It's not a question about the universe but about the theory..

• See the Picard–Lindelöf theorem. Nov 13, 2020 at 21:44
• No closed form solution $\ne$ non-deterministic Nov 14, 2020 at 5:26
• Several unfriendly comments removed. Please remember to be kind to each other, everyone. Nov 15, 2020 at 12:57

But a simple system if 3 particles interacting gravitationally fails to admit such a description.

No it doesn't. Given some set of initial conditions, the gravitational three-body problem can be solved for any time you like, to whatever degree of precision you like. It's true that the result can't be expressed in a neatly-packaged "closed form," but that's more a statement about how we write formulas down than it is a statement about the behavior of the system.

More interesting is the fact that the gravitational three-body problem is generically chaotic, meaning (among other things) that nearby initial conditions diverge from each other exponentially. From a practical point of view, this limits how far in the future we can reliably predict the motion of an observed three-body system. Because our measurements have finite resolution, we can't precisely pin down the true initial state of the system, and two initial conditions which are both compatible with our measurements will eventually evolve to completely different final states. Even if exact measurement of initial conditions were possible, our calculations necessarily have finite precision, which would give us the same problem.

However, this is not the same as non-deterministic evolution. Given exact initial conditions, the evolution of a chaotic system is completely deterministic - it's just that if the model is based on real-world observations, those predictions quickly become relatively meaningless because of uncertainty in the initial conditions and finite numerical precision.

• Nov 14, 2020 at 14:00

Classical mechanics is deterministic for any system. Given a perfectly known set of initial/boundary conditions, you can calculate the state of the system at any time.

However, this says nothing about the sensitivity to those initial conditions. For a chaotic system, choose a slightly different set of initial conditions (as close to the initial ones as you wish) and after some time the solutions will be significantly different. This doesn't make it less deterministic, it just forbids you to approximate the initial conditions. Or to approximate the calculation itself (when solving numerically).

So, classical mechanics is deterministic, in a desperately useless kind of way (for many systems).

• Comments are not for extended discussion; this conversation has been moved to chat. Nov 15, 2020 at 11:03

Most models of physical systems based on classical mechanics you can find are deterministic, in the sense the equations have locally only one possible solution.

There may still be special cases of particular initial state, where many different evolutions are consistent with the model. If the system ever reaches that state, the model ceases to be deterministic.

Three body system is usually considered deterministic, because for usual initial conditions (finite masses, coordinates, velocities, forces) the equations have locally only one possible solution. It is possible that in course of evolution of such system, particles get arbitrarily close to each other (this can't happen for 2 body system, but it can for three (and more) body system), which may destroy the one solution property.

There is much easier example of non-determinism in classical mechanics than special states of many body problem. It is called Norton's dome.

https://en.wikipedia.org/wiki/Norton%27s_dome

I don't see the question in the OP as related to chaotic behavior and nonlinear systems. I think it express the idea that "deterministic" is related to the existence of an analytic solution to the equations describing the evolution of the system. This idea comes probably from the introductory physics courses where we make our example systems as simple and ideal as possible so that most of the time the solution is simple enough. The reality is that the existence of an analytic solution is an exception while the rule is that most of the time we only have numerical methods to solve the equations. But this is not something bad and does not spoil the deterministic character of classical physics. Actualy if you look at the evolution of a mechanical system in time by numerical methods is more obvius that the state of the system in the n+1 iteration depends on the n iteration. What is more detrministic than this? An eye opener may be Feymans's lectures series. In the very begining he shows how to find the elliptical trajectory of a planet gravitationally interacting with the Sun just by applying Newton's laws step by step, using small time intervals. No integrals, no differential equations. Numerical solution are the core of physics, analitic solutions are some rare events. For Shrodinger equation, the only atomic system with analytic solution is the hydrogen atom. But the energy levels can be calculated for all the atoms. So again, the analytic solution is an exception but usually the only one discussed in introductory QM courses. If you ever played with an astronomy software, you could see that the positions of the planets in the solar system can be detrmined for thousands of years in the past and future with great accuracy. And it's not just three body. They can include comets and asteroids and so on. So the three body problem is no problem from the point of view of predictive power of classical mechanics.

Whether the universe as defined by modern physics is deterministic or not is an open question, but its substantially deterministic.

The universe would be deterministic if there was an injective function between the state at one time and the state infinitesimally later than that. If that function existed, we could simply apply it repeatedly to determine the state of the system in the future.

For the vast majority of situations (and I mean VAST), this is true for the equations we believe are good predictors of the world. The three body problem, and indeed the n-body problem all fit into this regime. It can be highly chaotic, but the time evolution function is indeed an injective function.

There is one corner case which cuts away at this certainty, metastability. consider the case of a ball on the top of a hill which could roll to the left or the right. It's just perfectly balanced at the top. Rolling one way or the other are both perfectly legitimate results. We would call that state nondeterministic, because the current state does not say which way it falls. It could even stay up there as long as it "pleases."

The more formal version of this argument would be made in Lagrangian Mechanics. In this formulation of mechanics, we define a "Lagrangian" in terms of positions and velocities, and try to find a path through time which minimizes the action, which is the integral of the Lagrangian over time. This path describes the evolution of the system in an injective manner. Well... almost. Actually, in Lagrangian Mechanics one seeks stationary points, which are minima, maxima, or inflection/saddle points, not just minima. It turns out that you can prove that minimizing action is insufficient to describe things we see in the real world, but finding stationary points is sufficient.

So consider the case of a frictionless ball rolling back and forth in a valley, with a hill on each side. If we give the ball a large velocity, the only valid solution to these equations is for the ball to go over one hill and continue on its way, never to return. It had enough energy to go over the hill, and that was that. If we give the ball a small velocity, then the only valid solution to these equations is indeed a minima, and we can watch the ball roll back and forth in the valley forever, never having enough energy to escape.

But what about the perfect velocity, right in the middle? There is a velocity with exactly the same amount of kinetic energy at the bottom as potential energy at the top. When it gets to the top, it has zero velocity -- a metastable state.

As it turns out, this is a saddle point in the Lagrangian formulation. There are three possibilities: it can stay at the top forever, it can roll back into the valley, or it can roll out of the valley. It turns out all are valid stationary points for the action. Indeed, it turns out that it can also fall off of the top at any arbitrary time, and still be a stationary point. We have a truly non-deterministic system there.

These points are infinitely rare in a larger system. You and I know that in practice, something will eventually perturb the ball off of the infinitely perfect metatstable point. Indeed this is a foundation of the concept of entropy. However, in theory, the entire universe could be in one of these saddle points. Such a universe would be nondeterministic.

But if I may bring in a level of empiricism, it would be impossible to deduce the difference between such a nondeterministic universe and a deterministic universe where the key state variables simply have not been measured to sufficient precision yet.

For a concrete example of this, one might look at Norton's Dome. It defines a construct which generates a non-deterministic result using Newtonian Physics, without having to resort to the complexities of Lagrangian Mechanics. It's interesting because there's a lot of disagreement as to whether or not the argument is valid. It violates some principles(Lipschitz continuity), but there is not a consensus as to whether or not that makes the construct "non-physical" or not.

• Norton's dome is not an example of metastability. Metastability means a state that is stable in time but collapses under any small perturbation - this is the case, as you pointed out, for many ordinary balls resting on top of hills. But the point of Norton's dome is that for the special shape of the dome it is consistent with the equations of motions that the resting ball at the top begins rolling at any point in time without any perturbation, i.e. it is not a metastable state since it can change without any perturbation. Nov 15, 2020 at 11:03
• @ACuriousMind: But the Norton dome cannot exist, neither in a truly classical world (where it cannot be constructed), nor in a quantum world (where there are no infinitely indivisible particles), so it is a useless imaginary concept. Nov 15, 2020 at 11:55
• And Cort, your answer is false for classically possible hills, even if they have infinite mass (which is physically impossible). If a ball has just enough energy to get to the top of a hill, it will stay there, and the fact that it is not a stable equilibrium has absolutely nothing to do with determinism. Please fix your error....... Nov 15, 2020 at 12:20

You don't need a three body system. All that is required is nonlinearity. Even if the evolution in time is completely fixed by initial conditions, there is no way to predict the future beyond a certain time if the system is nonlinear. In a nonlinear system any error in an initial condition will exponentially grow until the initial finger print is eventually erased.

• Error doesn't always grow in nonlinear systems - a slight error in the phase of a sine wave, for example, will not compound over time. Either way, chaotic behavior does not suggest non-determinism - identical initial conditions still result in identical outcomes. The fact that initial conditions cannot be measured or replicated with infinite precision is beside the point. An inability to generate predictions of arbitrary precision at arbitrary time scales does not imply non-determinism. Nov 13, 2020 at 16:19
• @NuclearHoagie The fact that it is fundamentally impossible to predict the future beyond a characteristic time for me means no determinism. In my opinion there is no difference between fundamentally inaccessible information and no information at all. Nov 13, 2020 at 16:23
• But it's not fundamentally impossible - if you want a better prediction, you just need a better measurement. You can predict the behavior of a chaotic system with arbitrary precision, given sufficient knowledge of the initial state. You're talking about practical limitations, not fundamental ones. A pseudo-random number generator, for example, yields a sequence of numbers that cannot be predicted a priori, despite the fact that its behavior is completely deterministic. If I write a number down on a piece of paper and have you guess it, the outcome is also unpredictable yet deterministic. Nov 13, 2020 at 16:37