# Why is it Impossible to Solve the Three Body Problem Analytically?

• I
• Dario56
In summary, the 3-body problem, as well as many-body problems in general, are impossible to solve analytically due to the nonlinearity and chaotic nature of these systems. This is because nature is inherently nonlinear and the set of all possible functions is uncountably infinite. While linear models may provide approximations, they are unable to capture the interesting phenomena such as chaos and emergent behavior in these systems. Therefore, it is not a matter of lacking enough elementary functions, but rather the complexity and diversity of physical systems.

#### Dario56

Three (many) body problems where three or many bodies (particles) interact are impossible to solve analytically. First one appeared in classical mechanics where equations of motion of planets were tried to be found by applying Newton's 2nd law for system of planets and stars interacting via gravity. In quantum mechanics, problem appears in solving Schrödinger equation for molecules (finding a molecular wave function) which consist of at least 3 particles interacting via electromagnetism.

I am not sure why is solving such problems impossible to do analytically. I am guessing it has to do with the fact that we don't know enough elementary functions to be able to give a solution in closed form or simply that combination of elementary functions can't describe solution to many body problems.

Delta2
The short answer is that the general 3-body problem allows for chaotic solutions as proven by Poincaré. A result so fundamental that a recent approach rather interestingly simply models some 3-body configurations as random walks.

PeroK, DrClaude and bob012345
Dario56 said:
In physics generally analytic solutions are a rarity. Laplace is often quoted as saying "nature laughs at the difficulties of integration".

Delta2 and Dario56
PeroK said:
In physics generally analytic solutions are a rarity.
Indeed. Nature is nonlinear at almost every level, with the set of realistic linear models having (waves hand) a measure of zero relative to the set of all models. The only reason linear models have "gotten" so much theoretical attention over time is because they are simple enough to be able to conclude a lot of stuff that, while interesting, at best only works as an approximation for the real world or in situations where we (e.g. via engineering) deliberately can construct parts of the real world to stay in the linear realm under some ideal conditions. But linear models are almost never able to capture interesting phenomenons like chaos and emergent behavior.

In fairness of the OP question I would like to add that the 2-body problem is (of course) not to be considered a linear problem, but (more handwaving) more like a degenerate nonlinear problem that has sufficiently few degrees of freedom for it to be unable to exhibit chaos.

Dario56
Filip Larsen said:
The short answer is that the general 3-body problem allows for chaotic solutions as proven by Poincaré. A result so fundamental that a recent approach rather interestingly simply models some 3-body configurations as random walks.
Yes, this is certainly the case. Chaos means extreme sensitivity on initial conditions and it is impossible to express such sensitivity mathematically in closed form.

Non linear ODEs can be hard enough to solve already, imagine what we have when we have a system of tenths or hundreds of non linear PDEs, which are possibly needed to describe accurately many physical systems. Simply chaos lol (pun intended).

Dario56 said:
I am guessing it has to do with the fact that we don't know enough elementary functions to be able to give a solution in closed form or simply that combination of elementary functions can't describe solution to many body problems.
Yes well I would dare to say that no matter how many elementary functions we have we still won't be able to give closed form solutions to all the complex physical systems, because every such system requires its own elementary functions. And the set of all functions is uncountably infinite.