The original breakthrough to this problem was given by Andrey Kolmogorov in 1954.

Springer 1997. Rafael de la Llave (2001) A tutorial on KAM theory. KAM theory: the legacy of Kolmogorov’s 1954 paper Kolmogorov-Arnold-Moser theory from Scholarpedia H Scott Dumas.

This was rigorously proved and extended by Jürgen Moser in 1962 (for smooth twist maps) and Vladimir Arnold in 1963 (for analytic Hamiltonian systems), and the general result is known as the KAM theorem. Arnold originally thought that this theorem could apply to the motions of the solar system or other instances of the -body problem, but it turned out to work only for the three-body problem because of a degeneracy in his formulation of the problem for larger numbers of bodies.

This was rigorously proved and extended by Jürgen Moser in 1962 (for smooth twist maps) and Vladimir Arnold in 1963 (for analytic Hamiltonian systems), and the general result is known as the KAM theorem. Arnold originally thought that this theorem could apply to the motions of the solar system or other instances of the -body problem, but it turned out to work only for the three-body problem because of a degeneracy in his formulation of the problem for larger numbers of bodies.

Percival in 1979. The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom.

Springer 1997. Rafael de la Llave (2001) A tutorial on KAM theory. KAM theory: the legacy of Kolmogorov’s 1954 paper Kolmogorov-Arnold-Moser theory from Scholarpedia H Scott Dumas.

The KAM Story – A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory, 2014, World Scientific Publishing, .

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