# Integrability

From Encyclopedia of Mathematics

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A vaguely defined yet very popular notion which may mean one of the following:

- For functions - existence of the integral in some sense (Riemann integrability, Lebesgue integrability, improper integrals etc.);
- For geometric structures and partial differential equations - conditions guaranteeing existence of solutions (Frobenius integrability condition

for distributions);

- For differential equations (both ordinary and partial) and dynamical systems:

- a possibility to find solution in a given class of functions (Darbouxian integrability, Liouville integrability, integrability in quadratures etc.) or just in some closed form;
- existence of one or more first integrals, functions which remain constant along solutions;
- preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
*complete integrability*for Hamiltonian systems means existence of the maximal possible number of first integrals in*involution*.

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**How to Cite This Entry:**

Integrability.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25868