Integrable system

In mathematics, integrability is a property of certain dynamical systems, that means very roughly that the solutions of the system are "simple" enough that they can be written down exactly (in principle). More precisely, while there are several distinct formal definitions, an integrable system can be thought of as a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Integrable systems are in this sense the opposite of chaotic systems, where small changes in initial conditions create wild deviations in the phase space behavior of solutions. Three features are often referred to as characterizing integrable systems: the existence of a maximal set of conserved quantities (the usual defining property of complete integrability) the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) 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 a sufficiently large time. 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 (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top)....