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KAM Theory, or Kolmogorov-Arnold-Moser theory, explains the persistence of quasi-periodic orbits in non-linear, near-integrable Hamiltonian systems when subjected to small perturbations.

In the context of KAM Theory, small perturbations refer to minor changes or disturbances in the system parameters used to slightly alter the Hamiltonian's integrability.

Action-angle variables are a set of canonical coordinates used in Hamiltonian mechanics where the action is preserved under Hamiltonian flow and the angle variables evolve linearly in time.

Quasi-periodic orbits are motions that repeat over time with a set of frequencies that are incommensurate, meaning that the ratio of frequencies is irrational.

This theorem, a precursor to KAM Theory, initially explored the conditions under which the n-torus is preserved in a Hamiltonian system and led to insights about the stability of motions in many dimensions.

A Hamiltonian system is a dynamical system governed by Hamilton's equations, describing the evolution of a system in terms of coordinates and conjugate momenta.

A nearly integrable Hamiltonian refers to a system that is close to being completely integrable, where the motions can typically be described by action-angle variables and small perturbations are considered.

In dynamical systems, resonance occurs when a system is able to absorb more energy from its surroundings periodically due to the matching of certain natural frequencies.

KAM Theory contributes to the understanding of the long-term stability of the solar system by explaining why the planetary orbits don’t deviate significantly over time despite small perturbations.

In KAM Theory, it is shown that most of the invariant tori of the integrable system will survive under small perturbations, although they may be deformed.

Arnold diffusion is a phenomenon in dynamical systems where there is a very slow change in the energies or actions in a system resulting from the interaction of various resonances.