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Difference between revisions of "Integrability"

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A vaguely defined yet very popular notion which may mean one of the following:
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* For functions - existence of the integral in some sense ([[Riemann integral|Riemann integrability]], [[Lebesgue integral|Lebesgue integrability]], [[improper integral]]s etc.);
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* For geometric structures and partial differential equations - conditions guaranteeing existence of solutions ([[Frobenius integrability condition]]
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for [[distribution]]s);
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* For differential equations (both ordinary and partial) and dynamical systems:
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# a possibility to find solution in a given class of functions ([[Darboux integral|Darbouxian integrability]], [[Liouville integrability]], [[integrability in quadratures]] etc.) or just in some closed form;
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# existence of one or more [[first integral]]s, functions which remain constant along solutions;
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# preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
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# ''complete integrability'' for Hamiltonian systems means existence of the maximal possible number of first integrals in ''involution''.
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Follow the links for more details.

Latest revision as of 07:42, 3 May 2012


This page is deficient and requires revision. Please see Talk:Integrability for further comments.

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

for distributions);

  • For differential equations (both ordinary and partial) and dynamical systems:
  1. 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;
  2. existence of one or more first integrals, functions which remain constant along solutions;
  3. preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
  4. complete integrability for Hamiltonian systems means existence of the maximal possible number of first integrals in involution.

Follow the links for more details.

How to Cite This Entry:
Integrability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25842