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The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing Moon's position with respect to the stars).

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincare was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called ``chaos'' in this work. The work of Bruns and Poincare gave a strong indication that no solution in the form proposed by Weierstrass exists.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion. It has been estimated that ne needs about $10^{8,000,000}$ terms of Sundman's series to compute Moon's position with the accuracy required in astronomy.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164-203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to Newtonian n-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105-119.

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing the Moon's position with respect to the stars.)

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincaré was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called “chaos” in this work. The work of Bruns and Poincaré gave a strong indication that no solution in the form proposed by Weierstrass exists.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion. It has been estimated that ne needs about $10^{8{,}000{,}000}$ terms of Sundman's series to compute Moon's position with the accuracy required in astronomy.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164–203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to the Newtonian $N$-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105–119.

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing Moon's position with respect to the stars).

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincare was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called ``chaos'' in this work. The work of Bruns and Poincare gave a strong indication that no solution in the form proposed by Weierstrass exists.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164-203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to Newtonian n-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105-119.

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing Moon's position with respect to the stars).

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincare was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called ``chaos'' in this work. The work of Bruns and Poincare gave a strong indication that no solution in the form proposed by Weierstrass exists.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion. It has been estimated that ne needs about $10^{8,000,000}$ terms of Sundman's series to compute Moon's position with the accuracy required in astronomy.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164-203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to Newtonian n-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105-119.

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing Moon's position with respect to the stars).

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincare was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called ``chaos'' in this work.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164-203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to Newtonian n-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105-119.

The three-body problem is one of the most famous problems in the history of mathematics, which also has an important application in science: it was supposed to explain the Moon's motion, among other things. Enormous effort was spent on this problem by many famous mathematicians of the 18th and 19th centuries. Since Newton's time it was clear that there was no simple closed form solution. (The problem also had an important practical application in 18th century, namely to navigation. If you can predict the motion of the Moon for few years ahead with sufficient accuracy, you can determine longitude at sea without a chronometer, just by observing Moon's position with respect to the stars).

Bruns proved in 1887 that the problem is not integrable (does not have a complete system of algebraic integrals). In the end of the 19th century, an exact mathematical formulation of what was desired was achieved: to express the motions of the bodies in the form of convergent series of functions of time, valid for all times. This statement is due to Weierstrass.

Poincare was awarded a prize for his work on this problem, but he did not solve it. His results were mostly negative; he discovered what is called ``chaos'' in this work. The work of Bruns and Poincare gave a strong indication that no solution in the form proposed by Weierstrass exists.

Few people remember nowadays that in this precise form the problem was actually solved by Karl Frithiof Sundman, in 1912. This solution can be found in Siegel's book on celestial mechanics.

But by that time it was already understood that this solution was useless for practical purposes, namely for prediction of the Moon's motion over long time periods. It was also useless for understanding the qualitative features of the motion.

This does not mean that the work of Sundman was useless: the methods developed there had a substantial influence on the further development of celestial mechanics.

Here is an excellent historical exposition, with many references:

J. Barrow-Green, The dramatic episode of Sundman, Historia Mathematica, 3 (2010) 164-203.

A modern mathematical exposition of the main idea can be found in

D. Saari, A visit to Newtonian n-body problem via elementary complex variables, American Mathematical Monthly, 1990, Vol. 97, No. 2, 105-119.