Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions. Accordingly, an integrable system is a system of differential equations whose behavior is determined by initial conditions and which can be integrated from those initial conditions. In the context of differential equations to integrate an equation means to solve it from initial conditions. Accordingly, an integrable system is a system of differential equations whose behavior is determined by initial conditions and which can be integrated from those initial conditions. Many systems of differential equations arising in physics are integrable. A standard example is the motion of a rigid body about its center of mass. This system gives rise to a number of conserved quantities: the angular momenta. Conserved quantities such as these are also known as the first integrals of the system. Roughly speaking, if there are enough first integrals to give a coordinate system on the set of solutions, then it is possible to reduce the original system of differential equations to an equation that can be solved by computing an explicit integral. Other examples giving rise to integrable systems in physics are some models of shallow water waves (Korteweg–de Vries equation), the nonlinear Schrödinger equation, and the Toda lattice in statistical mechanics. While the presence of many conserved quantities is generally a fairly clear-cut criterion for integrability, there are other ways in which integrability can appear. It is famously difficult to be precise about what the term means. Nigel Hitchin (Hitchin, Segal & Ward, p. 1) draws a comparison to a quotation by jazz musician Louis Armstrong, who when asked what jazz was, is rumored to have replied 'If you gotta ask, you'll never know.' Hitchin identifies three generally recognizable features of integrable systems: An example of a non-integrable system is a multi-jointed robot arm: for a given initial and final position, there are many possible paths that the robot arm can take to get from here to there; even specifying the initial and final velocities is insufficient to constrain the problem. An even simpler example is that of a rolling ball on a flat surface: one can roll it anywhere, but the final orientation of the ball depends on the path taken. Such systems are constrained (the lengths of the robot arm are fixed; the ball is constrained to roll without slipping), but the constraints are not enough to determine a unique solution. A more concise, worked example of a non-integrable system is given in the article on integrability conditions for differential systems. Some of the primary tools for studying non-integrable systems are sub-Riemannian geometry and contact geometry. A foundational result for integrable systems is the Frobenius theorem, which effectively states that a system is integrable only if it has a foliation; it is completely integrable if it has a foliation by maximal integral manifolds. In the context of differentiable dynamical systems, the notion of integrability refers to the existence of invariant, regular foliations; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context. An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations. The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form. In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense, see the Liouville-Arnold theorem. Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fieldsassociated to the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of Poisson commuting invariants (i.e., functions on the phase space whose Poisson brackets with the Hamiltonian of the system,and with each other, vanish).