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In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.
Three features are often referred to as characterizing integrable systems:
Integrable systems may be seen as very different in qualitative character from more generic dynamical systems,which are more typically chaotic systems. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over sufficiently large time.
Complete integrability is thus a nongeneric property of dynamical systems. Nevertheless, many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center or two. Other elementary examples include the motion of a rigid body about its center of mass and the motion of an axially symmetric rigid body about a point in its axis of symmetry.