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Encyclopedia > Integrability conditions

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example. A Pfaffian system is one specified by 1-forms alone, but the theory includes other types of example of differential system. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This is a glossary of terms specific to differential geometry and differential topology. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...


Given a collection of differential 1-forms on an n-dimensional manifold M, an integral submanifold is an embedding

of a submanifold N into M such that the kernel of the restriction map on forms

is spanned by the αi at every point p of N. If in addition the αi are linearly independent, then N is (nk)-dimensional.


An integrability condition is a condition on the αi to guarantee that there will be an integral submanifold.

Contents

Example of a non-integrable system

Not every such differential system has integral manifolds, however. For example, consider the following one-form on the standard simplex :

θ = xdy + ydz + zdx

Suppose that N is an integral submanifold for θ, so that i * θ = 0. In particular, i * dθ = di * θ = 0. So dθ is also in the kernel of i * , which means that we must have for some 1-form α on M. On the other hand, by the skewness of the wedge product, this implies that

But a direct calculation verifies that

which is a nonzero multiple of the standard volume on the simplex S, and so is never zero.


Necessary and sufficient conditions

The necessary and sufficient conditions for integrability of a system generated by 1-forms are supplied by the Frobenius theorem. One form states that if the ideal algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed , then the system admits an integral manifold. In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ...


Examples

In Riemannian geometry, we may consider the problem of finding an orthogonal coframe θi (i.e., collection of 1-forms forming a basis of the cotangent space at every point with ) which are closed . By the Poincaré lemma, the θi locally will have the form dxi for some functions xi on the manifold, and thus provide an isometry of an open subset of M with an open subset of . Such a manifold is called locally flat. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms which form a basis of the cotangent bundle at every point. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β unknown. ...


This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe

.

If we had another coframe , then the two coframes would be related by an orthogonal transformation

Φ = MΘ

If the connection 1-form is ω, then we have

On the other hand,

But ω = (dM)M − 1 is the Maurer-Cartan form for the orthogonal group. Therefore it obeys the structural equation and this is just the curvature of M: After an application of the Frobenius theorem, one concludes that a manifold M is locally flat if and only if its curvature vanishes. In mathematics, the Maurer-Cartan form for a Lie group G is a distinguished differential form on G that carries within itself the basic infinitesimal information about the structure of G. It was much used by Elie Cartan, as a basic ingredient of his method of moving frames. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ...


Generalizations

Many generalizations exist to integrability conditions on differential systems which are not necessarily generated by one-forms. The most famous of which are the Cartan-Kähler theorem, which only works for real analytic differential systems, and the Cartan-Kuranishi prolongation theorem. See Further reading for details. In mathematics, the Cartan-Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I. It is named for Élie Cartan and Erich Kähler It is not true that merely having dI contained in I is... Analytic may refer to Analytic proposition or analytic philosophy, in philosophy Analytic geometry, analytic function, analytic continuation, analytic set in mathematics. ... Given an exterior differential system defined on a manifold M, the Cartan-Kuranishi prolongation theorem says that after a finite number of prolongations the system is either in involution (admits at least one large integral manifold), or is impossible. ...


Further reading

  • Bryant, Chern, Gardner, Goldschmidt, Griffiths, "Exterior Differential Systems," Mathematical Sciences Research Institute Publications, Springer-Verlag, ISBN 0-387-97411-3
  • Olver, P., "Equivalence, Invariants, and Symmetry," Cambridge, ISBN 0-521-47811-1
  • Ivey, T., Landsberg, J.M., "Cartan for Beginners: Differential Geometry via Moving Frames

and Exterior Differential Systems", American Mathematical Society, ISBN 0-8218-3375-8


  Results from FactBites:
 
Beyond Integrability (389 words)
There is one final problem of the hidden field of integrable models which I want to discuss.
Namely the question of what is the relation of an integrable model to a generic physical system which does not satisfy a Yang-Baxter equation.
43] on the basis of perturbation studies of nonintegrable models which show ever increasingly complicated singularity structures as the order of perturbation increases; a complexity which magically disappears when an integrability condition is imposed.
  More results at FactBites »

 
 

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