Difference between revisions of "Integrability"
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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 integral|Riemann integrability]], [[Lebesgue integral|Lebesgue integrability]], [[improper integral]]s etc.); | ||
| + | * For geometric structures and partial differential equations - conditions guaranteeing existence of solutions ([[Frobenius integrability condition]] | ||
| + | for [[distribution]]s); | ||
| + | * For differential equations (both ordinary and partial) and dynamical systems: | ||
| + | # 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; | ||
| + | # existence of one or more [[first integral]]s, 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''. | ||
| + | |||
| + | 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 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.
Follow the links for more details.
How to Cite This Entry:
Integrability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25868
Integrability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25868