View source for Integrable system

A differential system of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513301.png" /> (cf. [[Involutive distribution|Involutive distribution]]) on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513302.png" />-dimensional [[Differentiable manifold|differentiable manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513303.png" /> that has, in a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513304.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513305.png" />-parameter family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513306.png" />-dimensional integral manifolds (cf. [[Integral manifold|Integral manifold]]). One often speaks of a totally-integrable system in this case; more precisely it is defined as follows. Suppose that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513307.png" /> a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513308.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i0513309.png" /> in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133010.png" /> has been distinguished, such that a differential system, or distribution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133011.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133013.png" />, of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133014.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133015.png" />. The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133016.png" /> is called totally integrable if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133017.png" /> there is a coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133020.png" />, such that for any constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133022.png" />, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133023.png" /> is an integral submanifold, i.e. its tangent space at an arbitrary point coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133024.png" />. For analytic conditions that are necessary and sufficient for this, see [[Involutive distribution|Involutive distribution]].



====Comments====
Cf. also [[Pfaffian equation|Pfaffian equation]]. The phrase integrable system is also used to refer to a completely-integrable Hamiltonian system or equation, i.e. a Hamiltonian equation (system) on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133025.png" />-dimensional phase space which has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051330/i05133026.png" /> (including the Hamiltonian itself) [[Integrals in involution|integrals in involution]], cf. [[Hamiltonian system|Hamiltonian system]].