Newton's theory of gravity was a triumph of human thought and of mathematics. Newton's theory allows the exact mathematical solution of the interrelated motions of two heavenly bodies.

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. (Wikipedia)

If the sun and earth are considered point masses, then Newton's gravitational equation can be used to determine exactly the attraction and orbit, etc., of the earth around the sun. More precisely, they both go around each other.

Assume that the bodies are collapsed to points.

The theory works well for sun and earth.

Likewise, if the earth and moon earth are considered point masses, then Newton's gravitational equation can be used to determine exactly the attraction and orbit, etc., of the moon around the earth. More precisely, they both go around each other.

The theory works well for earth and moon.

The above are called two body problems.

However, there is a problem with more than two bodies. No one has ever been able to mathematically solve the three body problem, such as the sun, earth, and moon.

How can a problem such as weather prediction be solved exactly? It can't.

• Weather is not really "predicted" (exactly)
• Weather is "forecast" (approximately)

Why do we not just approximate it over time to get the solution? We can't.

Any attempt to approximate solutions result in round-off errors that accumulate and can cause the same model to vary widely based in the initial conditions.