What is the formula for vector sum of all the gravitational forces acting on the ith body?

The two-body problem is a classical mechanics problem that predicts the motion of two massive objects. The n-body problem is a classical physics problem that predicts the motion of a group of celestial objects that interact gravitationally.  The two-body problem is a fundamental problem that has an explicit analytic solution. It's the only case of the n-body problem that can be solved completely.  The n-body problem can be stated as follows: Given the quasi-steady orbital properties of a group of celestial bodies Predict their interactive forces Predict their true orbital motions for all... Show more

The two-body problem is a classical mechanics problem that predicts the motion of two massive objects. The n-body problem is a classical physics problem that predicts the motion of a group of celestial objects that interact gravitationally. 

The two-body problem is a fundamental problem that has an explicit analytic solution. It's the only case of the n-body problem that can be solved completely. 

The n-body problem can be stated as follows:
Given the quasi-steady orbital properties of a group of celestial bodies
Predict their interactive forces
Predict their true orbital motions for all future times 

The motion of the planets of the solar system over time scales approaching its 4.6-billion-year age is a classic n-body problem. 

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