Why is the 3 body problem unsolvable? And what will happen if someone solves it?

[physicistwithaheart]

I feel it should be solvable we just need to find a perfect pattern, and there will be a general pattern since the forces acting are based on a single function, so..... you can't actually say it is unsolvable right? Cause imaging 3 bodies actually existed somwhere in this universe then nature isn't gonna wait till we predict it! And yea I have checked in many places that tiny changes cause large changes so it becomes chaos........ but still I just can't accept that it is impossible to solve, its just that we are not good enough right? I have also checked that you can predict the forces active if you know some special cases but still..... How can you prove my logic wrong. I feel the 3 body problem is a problem that needs to be revived in physics. "Only then Classical Mechanics can be solved completely and Sir Isaac Newton can rest in peace!"

 

[Dale]

What makes you think it is not solvable?

Are you possibly confusing "solvable" with "has an analytical solution" or "has a closed form solution"?

 

[Filip Larsen]

Please give https://en.wikipedia.org/wiki/Three-body_problem a read.

 

[PeroK]

Even the two body problem only has an analytical solution in terms of the shape. There's no closed form solution for the position as a function of time in an elliptical orbit.

https://physics.stackexchange.com/q...n-as-a-function-of-time-for-elliptical-orbits

Analytic solutions in general are the rare exception in physics.

 

[kuruman]

Cause imaging 3 bodies actually existed somwhere in this universe then nature isn't gonna wait till we predict it!

And what do you think Nature will do once she gets tired of waiting?

Spoiler

Wait some more.

 

[PeroK]

And what do you think Nature will do once she gets tired of waiting?

Spoiler

Wait some more.

Nature laughs at the difficulties of integration!

 

[physicistwithaheart]

it helped thanks
Are you possibly confusing "solvable" with "has an analytical solution" or "has a closed form solution"?

 

[physicistwithaheart]

Please give https://en.wikipedia.org/wiki/Three-body_problem a read.

it helped thanks. but it still feels intutive to think of analytical form

 

[kuruman]

it helped thanks. but it still feels intutive to think of analytical form

And for good reason. The first four years of undergraduate physics education train students in methods for finding analytical solutions to problems. At that level, models of nature are idealized, e.g. no air resistance in projectile motion, massless strings and pulleys, etc. As a result, students falsely inform their intuition that all problems have analytical solutions without realizing that such solutions are based on idealizations. Nature is much more complicated than its description in physics textbooks and that's why she patiently waits and waits to be mathematically described, one step at a time, in a process known as research.

 

[PeroK]

And for good reason. The first four years of undergraduate physics education trains students in methods for finding analytical solutions to problems. At that level, models of nature are idealized, e.g. no air resistance in projectile motion, massless strings and pulleys, etc. As a result, students falsely inform their intuition that all problems have analytical solutions without realizing that such solutions are based on idealizations. Nature is much more complicated than its description in physics textbooks and that's why she patiently waits and waits to be mathematically described, one step at a time, in a process known as research.

I once heard the analogy that studying systems with analytic solutions is a bit like studying a single animal, like the elephant. And everything else (systems without analytic solutions) is studying non-elephant animals!

 

[physicistwithaheart]

And for good reason. The first four years of undergraduate physics education train students in methods for finding analytical solutions to problems. At that level, models of nature are idealized, e.g. no air resistance in projectile motion, massless strings and pulleys, etc. As a result, students falsely inform their intuition that all problems have analytical solutions without realizing that such solutions are based on idealizations. Nature is much more complicated than its description in physics textbooks and that's why she patiently waits and waits to be mathematically described, one step at a time, in a process known as research.

lets just hope someday it gets solved! or better i hope i can solve it someday

 

[Dale]

lets just hope someday it gets solved! or better i hope i can solve it someday

It is already solved. The solution is just not analytic.

 

[physicistwithaheart]

I once heard the analogy that studying systems with analytic solutions is a bit like studying a single animal, like the elephant. And everything else (systems without analytic solutions) is studying non-elephant animals!

so If we assume, standard cases like energy conservation and no collision, then can we solve it?

 

[physicistwithaheart]

It is already solved. The solution is just not analytic.

the purpose of physics, is to predict we still can't predict ther degree's of freedom. so you cannot technically say it is solved. yea sure it is not analytical, But isn't that what we want?

 

[Dale]

so If we assume, standard cases like energy conservation and no collision, then can we solve it?

Again, it is already solved. And assuming energy conservation and no collision will not make the solution analytic.

 

[Dale]

the purpose of physics, is to predict we still can't predict ther degree's of freedom. so you cannot technically say it is solved. yea sure it is not analytical, But isn't that what we want?

I can definitely say it is solved because it is.

Prediction in no way requires an analytical solution. Quite the opposite. To make a prediction with an analytical solution we need to evaluate it numerically.

 

[PeroK]

the purpose of physics, is to predict we still can't predict ther degree's of freedom. so you cannot technically say it is solved. yea sure it is not analytical, But isn't that what we want?

The problem is that there are hardly any standard functions: polynomials, trig functions, exponential and logs. That's about it. There are a couple more, like the Lambert W-function and the Gamma function and some Error functions etc. Those are defined specifically because they cover some common cases. After that you are looking at specific power series, say, that don't have a name.

What you really want to prove, following my analogy, is that all animals are actually elephants. Or, that every differential equation has a trignometric solution.

 

[PeroK]

PS the ancient Greeks were obsessed by rational numbers and didn't like it when irrational numbers starting popping up. We know that no rational number can be squared to give a whole number. There's no point in hoping that one day you'll find whole numbers $m, n$ where $(\frac m n)^2 = 2$. We know there is a square root of 2, and we know it's not rational.

Likewise, we know the classes of differential equation that do not have solutions in terms of elementary functions. The solutions exist in some cases and not in others.

These things have been proven mathematically. There is no point in hoping otherwise.

 

[Dale]

It is interesting to know that the first computers (e.g. the ENIAC) were built specifically to conveniently calculate non-analytical solutions to physics problems (e.g. projectile motion with drag)

 

[kuruman]

Again, it is already solved. And assuming energy conservation and no collision will not make the solution analytic.

I think that the first manned Moon landing in 1969 was a spectacular solution to the 3-body problem. I scrimped and saved for three months to buy a cheap black & white TV set to watch it live.

 

[Nugatory]

that studying systems with analytic solutions is a bit like studying a single animal, like the elephant

One particular animal, the spherical cow. Unfortunately it’s mythical.

 

[James Demers]

I feel it should be solvable we just need to find a perfect pattern, and there will be a general pattern since the forces acting are based on a single function, so..... you can't actually say it is unsolvable right?

This is a bit of an over-simplification, but essentially you have 18 variables: 3 bodies x (3 velocity vectors + 3 coordinates). Newton's equations of motion give you ten integrals, so you have more variables than you have equations. No amount of monkeying around with coordinate systems and transformations gets you past this basic problem. There is a solution in the form of an infinite series (look up Karl Sundman), which can give you arbitrary, but still-not-perfect, precision.
(You could say that Nature solves the problem with an analog computer: the three bodies. Problem with this computer is that it's too slow to ever predict the future.)

 

[kuruman]

You could say that Nature solves the problem with an analog computer: the three bodies.

I don't think you could say that. There is no such thing as three bodies in Nature that can be studied as a separate and isolated system. Asserting that there is, is an approximation and a human creation. Nature solved the problem of putting the entire universe together. We're just trying to figure out how that was done.

 

[Herman Trivilino]

I feel it should be solvable we just need to find a perfect pattern, and there will be a general pattern since the forces acting are based on a single function, so..... you can't actually say it is unsolvable right?

Oh, it's definitely solvable, and we know how to solve it. So your conclusion is correct, but I don't see how it follows from your premises.

........ but still I just can't accept that it is impossible to solve, its just that we are not good enough right?

Oh, I think we're very good at it. An eclipse is a three body problem, and we can predict with any desired precision when and where they will occur.

I think what you're confused about is what's referred to as closed form solutions. Those solutions are exact, but rare. It's just that in school we spend most of our time studying only those rare situations, possibly leading us to believe that they are the rule rather than the exception.

By the way, it's not just newtonian physics where the three body problem is encountered. We see it in quantum physics, where only a one-electron atom has a closed form solution.

Don't be fooled by the notion that a lack of an exact solution implies a lack of a precise solution. Numerical methods are used to find approximate solutions to any desired amount of accuracy. There is no limit on how close we can get to the exact solution.

 

[wrobel]

The 3 body problem was the first example on which people became aware of what dynamical chaos is. The effect of separatrices splitting was discovered in this system by Poincare . This was the first chaotic effect.

 

[Lincon Ribeiro]

This is one of the most interesting problems in the history of physics, and, adding to other replies, there are several steps between the basic question you posed and the answer, starting from modeling it using classical physics framework, until Poincaré's solution.
For instance (without proper formalism), just to give an idea on how complex it is:
Let $m_1,m_2,m_3 > 0$ be the masses of the objects in the system and $r_i (t) \in \mathbb{R^3}$ their respective position vectors. Newton's equations could be writen as :

$$
m_i \ddot{r}_{i(t)} = \sum_{j \ne i} Gm_i m_j \frac{r_j-r_i}{||r_j-r_i||^3}, \quad i=1,2,3
$$

We're trying to solve these equations for each $r_i(t)$, you'll have to integrate all of them, given the constraints and dependencies. You might try, as a first attempt, to rewrite them using Hamiltonian framework:

$$
H(\vec{r}, \vec{p})=\sum_{i=1}^{3} \frac{||p_i||^2}{2m_i}-G\sum_{1\leq i<j\leq 3} \frac{m_i m_j}{||r_i-r_j||}
$$

where $\vec{p}_i=m_i \dot{r}_i$. Solving for canonical equations will result in Newton's equations.

Now, how to proceed from this? This is where things start to get complicated. Take a look at June Barrow-Green's book "Poincaré and the Three-Body Problem" : https://en.wikipedia.org/wiki/Poincaré_and_the_Three-Body_Problem or Alain Chenciner paper, with the same name. There are several topics in mathematical-physics to grasp, way beyond an undergraduate course in Physics, such that:

• Kepler's Problem
• Lagrange's and Laplace's secular systems approximations
• Periodic solutions
• Perturbation theory
• Non-linear differential equations
• Quasi-periodic solutions: Lindstedt series
• Divergence series
• Bohlin series
• Poisson stability

After all of that, when reading Poincaré's paper, you'll figure out that the solution, using series, is divergent, meaning, there is no analytical result for the 3-body problem, as pointed out before. You can only find approximate solutions for it.

 

[sophiecentaur]

I feel it should be solvable

"Feelings" are not to be trusted where Science is concerned. I remember having been confused by the difference between Randomness and Chaos; they are totally different things. Many systems can be treated by using an 'ideal' model with straight lines, smooth (continuous) curves etc.. That's what we all grew up on. Chaotic behaviour has been known about for a long time but the idea of fractals (look up Mandelbrot Sets) was new to the public only a few decades ago. There's plenty written about chaos higher up the thread so no need to re-visit it in this post. It's very much not just adding random noise to a conventional relationship between variables. New thinking and acceptance is needed here.