There is a two-body problem—and it's actually much easier to solve.

The phrase “three-body problem” refers to what happens when you have three objects gravitationally interacting with one another.

Two bodies

Imagine just the Earth and the Sun.

The Sun pulls on Earth, and Earth pulls on the Sun. Under Newton's laws, you can mathematically solve their motion exactly. They follow predictable elliptical orbits (more precisely, both orbit their common centre of mass).

So the two-body problem has a general exact solution.

Add a third body

Now add the Moon:

• Sun pulls on Earth
• Earth pulls on Sun
• Sun pulls on Moon
• Moon pulls on Sun
• Earth pulls on Moon
• Moon pulls on Earth

The difficulty is that each object's motion changes the gravitational forces acting on the other two.

For example, as the Moon moves, its gravitational pull slightly changes Earth's orbit. But Earth's changed position then alters the Moon's motion, which alters Earth's motion again, and so on.

There is no general mathematical formula that gives the exact future positions of three mutually gravitating bodies for arbitrary starting conditions.

That's the famous three-body problem.

Why does one extra object make it so much harder?

The equations themselves aren't enormously complicated. For three bodies, Newton's law gives you three coupled equations:

mi​r¨i​=Gj=i∑​mi​mj​∣rj​−ri​∣3rj​−ri​​

The problem is that the equations are coupled and nonlinear. You can't neatly separate them into independent orbital equations like you can with two bodies.

And three-body systems can exhibit chaotic behaviour: a tiny difference in the initial positions or velocities can eventually produce a dramatically different configuration.

So:

2 bodies → exact general solution exists.
3 bodies → no general exact solution; usually solved numerically or approximately.

It's worth noting that special three-body solutions do exist—for example, the Lagrange points are special configurations—but there's no single general solution for every possible three-body system.

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is a general solution a theoretical impossibility

No — not in the sense of “mathematically impossible to ever solve.” It's more subtle.

For the classical three-body problem, a general solution in terms of elementary functions (like the familiar formulas for the two-body problem) does not exist. Mathematicians have proved very strong results about this.

But there are several meanings of “solution”:

• Exact numerical solution: Yes. Given the starting positions and velocities, you can calculate the motion to essentially any desired precision.
• Special exact solutions: Yes. There are many, including the famous Lagrange and Euler solutions.
• General closed-form formula like the two-body solution: No known such formula exists, and results from Poincaré and later mathematicians show that the dynamics cannot generally be reduced to the kind of simple integrable form available for two bodies.
• Arbitrary precision mathematical description: Yes. There are sophisticated mathematical representations of the general solution, including convergent series in appropriate formulations.

The really important distinction is between “there is no solution” and “there is no simple formula.”

The three-body problem is not unsolvable. We can simulate the Earth-Moon-Sun system extremely accurately. What we lose is the beautiful situation of the two-body problem where you can write down a relatively simple equation and say:

“Given these initial conditions, here's the exact orbit for all future time.”

And there's an even deeper reason: chaos. In some three-body systems, minuscule uncertainties in the starting conditions grow over time. So even though the equations are deterministic, predicting the system arbitrarily far into the future from measurements with finite precision can become practically impossible.

So the short answer is:

A general solution isn't theoretically impossible in every mathematical sense. What is impossible is reducing the general three-body problem to the same kind of simple, globally predictive closed-form solution that exists for the two-body problem.

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what is the nature of these uncertainties

The key idea is that the uncertainty isn't necessarily coming from the laws of physics being uncertain. It comes from our knowledge of the starting state, combined with the dynamics of the system.

Imagine three bodies whose initial positions and velocities you know almost perfectly.

Suppose the true initial position of one body is:

$$x=1.000000000000$$

but your measurement is uncertain by just:

$$\Delta x=0.000000000001$$

That's an incredibly tiny error.

In a simple two-body orbit, that tiny error generally stays relatively well behaved. Your predicted orbit might be slightly wrong, but it remains recognisably the same orbit.

In a chaotic three-body system, however, the error can grow exponentially.

Very roughly:

Δ(t)≈Δ(0)eλt

where λ is related to the system's Lyapunov exponent.

So you might get something like:

10−12→10−10→10−8→10−6→10−2→1

over successive periods of time.

Eventually, two simulations that started with essentially indistinguishable initial conditions can predict completely different positions.

But where does the initial uncertainty come from?

In the real universe, we can never know the state of a system with literally infinite precision.

There are several sources:

1. Measurement limitations

You can't measure an object's position and velocity with infinite precision. Telescopes, clocks, sensors, etc. have finite accuracy.

2. Quantum mechanics

At a fundamental level, particles don't have perfectly definite classical positions and momenta simultaneously. For macroscopic celestial bodies this usually isn't the dominant practical limitation, but it establishes that "perfect initial conditions" aren't physically available.

3. Unmodelled influences

Suppose you're modelling three objects but there is actually a fourth object far away. Its gravitational influence might be tiny, but it isn't exactly zero.

Likewise, radiation pressure, other planets, asteroids, etc. can introduce small effects.

4. Numerical error

Even if you had perfect initial conditions, a computer represents numbers with finite precision and solves the differential equations approximately. Those tiny numerical errors can also be amplified by chaotic dynamics.

Here's the really interesting part

The system itself can be completely deterministic.

Imagine running the universe twice with exactly the same initial state.

You would get exactly the same future.

There's no randomness required.

The problem is that if you change the initial state by an unimaginably tiny amount:

same laws + almost identical starting state → potentially radically different future

That's what sensitive dependence on initial conditions means.

And this is what people are getting at when they say that the three-body problem is chaotic.

It doesn't mean "we don't know what will happen because nature is random."

It means:

The equations determine what will happen, but predicting it requires initial information of a precision that can become impossibly high as you look farther into the future.

That's one of the profound differences between determinism and predictability. A system can be completely deterministic while still being practically unpredictable.