Integrable system

A differential system of dimension $p$ (cf. Involutive distribution) on an $n$-dimensional differentiable manifold $M^n$ that has, in a neighbourhood of each point $x\in M^n$, an $(n-p)$-parameter family of $p$-dimensional integral manifolds (cf. 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 $x\in M^n$ a subspace $D(x)$ of dimension $p$ in the tangent space $T_x(M^n)$ has been distinguished, such that a differential system, or distribution, $D$ of class $C^r$, $r\geq1$, of dimension $p$ is given on $M^n$. The system $D$ is called totally integrable if for each point $a\in M^n$ there is a coordinate system $(U,\phi)$, $x\in U$, $\phi(x)=\phi(x_1,\dots,x_n)$, such that for any constants $c^j$, $p<j\leq n$, the manifold $U_c=\{x\in U:x^j=c^j\}$ is an integral submanifold, i.e. its tangent space at an arbitrary point coincides with $D(x)$. For analytic conditions that are necessary and sufficient for this, see Involutive distribution.

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Cf. also 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 $2n$-dimensional phase space which has $n$ (including the Hamiltonian itself) integrals in involution, cf. Hamiltonian system.