Integrable system
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, that its motion is confined to a submanifold
of much smaller dimensionality than that of its phase space.
Three features are often referred to as characterizing integrable systems:
- the existence of a maximal set of conserved quantities
- the existence of algebraic invariants, having a basis in algebraic geometry
- the explicit determination of solutions in an explicit functional form
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 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.
In the late 1960s, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves, the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice. The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967.
In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets, and if the flows are complete and the energy level set is compact, this implies the Liouville–Arnold theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.
A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable if, locally, it has a foliation by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.
Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of special functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.
General dynamical systems
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, 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 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 an exact form.
Hamiltonian systems and Liouville integrability
In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense. Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of functionally independent Poisson commuting invariants.In finite dimensions, if the phase space is symplectic, it must have even dimension and the maximal number of independent Poisson commuting invariants is. The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomous Hamiltonian systems have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical -form are called the action variables, and the resulting canonical coordinates are called action-angle variables.
There is also a distinction between complete integrability, in the Liouville sense, and partial integrability, as well as a notion of superintegrability and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal, we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with one-dimensional leaves, this is called maximally superintegrable.
Action-angle variables
When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense,and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables,
such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow, and the angle variables are the natural periodic coordinates on the tori. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
The Hamilton–Jacobi approach
In canonical transformation theory, there is the Hamilton-Jacobi method, in which solutions to Hamilton's equations are sought by first finding a complete solution of the associated Hamilton-Jacobi equation. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of completely ignorable variables; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities. In the case of compact energy level sets, this is the first step towards determining the action-angle variables. In the general theory of partial differential equations of Hamilton-Jacobi type, a complete solution, exists in very general cases, but only in the local sense. Therefore, the existence of a complete solution of the Hamilton-Jacobi equation is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a complete separation of variables, in which the separation constants provide the complete set of integration constants that are required. Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.Solitons and inverse spectral methods
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg–de Vries equation, could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the inverse scattering transform and more general inverse spectral methods,which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.
The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" is invariant under the evolution, cf. Lax pair. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.