Integrable system

An integrable system is a dynamical system whose evolution can be reduced to explicit solutions using conserved quantities or equivalent mathematical structure. In Hamiltonian mechanics, integrability requires enough independent, mutually commuting conserved quantities.

Hamiltonian mechanics: A formulation of classical mechanics in which states evolve on phase space according to a Hamiltonian function. Hamiltonian integrability is defined through conserved functions on phase space.
Hamiltonian system: A dynamical system whose evolution is generated by a Hamiltonian function on a symplectic phase space. Hamiltonian integrability is usually stated as a condition on such systems.
Chaos theory: The study of deterministic systems whose dynamics can be highly sensitive to initial conditions. Chaotic behavior often emerges where integrable structure breaks down.
Kepler problem: The two-body gravitational problem in which one body moves under an inverse-square central force. Its conserved energy, angular momentum, and Runge–Lenz vector make it integrable.
Joseph Liouville: A French mathematician whose work included the theory of integrable Hamiltonian motion. His name is attached to the classical criterion for Hamiltonian integrability.
First integral: A function of a system’s state that remains constant along its trajectories. Each independent first integral constrains the motion to a smaller subset of phase space.
Phase space: A space whose points represent the complete states of a dynamical system. Integrals and trajectories are functions and curves on this state space.
Nonintegrable system: A dynamical system that lacks enough structure to be solved by the standard criteria for integrability. It marks the failure of the conserved quantities that make integrable systems tractable.
Euler top: The torque-free rotation of a rigid body about a fixed point with principal moments of inertia. Its conserved energy and angular momentum allow its motion to be solved exactly.
Henri Poincare: A French mathematician who founded major parts of modern qualitative analysis of differential equations and celestial mechanics. His work on the three-body problem exposed limits of classical integrability.

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