Invariants and stability in classical mechanics

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Abstract
In the last 20 years there have been discoveries in classical mechanics that have made an impact on several other branches of physics. The important result from which these developments stem has become known as the KAM theorem. This relates to a situation in which a simple integrable dynamical system is subjected to perturbations. Whereas before perturbation the motion could be described as regular, after perturbation the phase space is sharply divided into regions of regular and highly irregular motion. Computer calculations have made it possible to appreciate both the complexity and the detailed structure of the situation. Examples of this behaviour have been found in plasma physics, statistical mechanics and astronomy. A qualitative review is given of the KAM theorem, the work that led up to it and its application to various branches of physics.