Flusso (matematica)

In matematica, in particolare nello studio delle equazioni differenziali ordinarie, un flusso generalizza il concetto di funzione iterata n volte in modo che il numero di iterazioni n diventi un parametro continuo. Più rigorosamente, un flusso è un'azione di gruppo di un gruppo ad un parametro.

È utilizzato in ingegneria e fisica per formalizzare le soluzioni dell'equazione che descrive un sistema dinamico.

L'idea di un vettore di flusso, cioè il flusso di un campo vettoriale, è utilizzata nei più disparati ambiti, come la topologia differenziale, la geometria di Riemann e i gruppi di Lie. Alcuni esempi di vettori di flusso sono il flusso geodetico, il campo vettoriale hamiltoniano, il flusso di Ricci e il flusso di Anosov.

Definizione

Un flusso definito su un insieme $X$ è un'azione di gruppo di $(\mathbb {R} ,+)$ su $X$ . Più esplicitamente, un flusso è una funzione $\varphi :X\times \mathbb {R} \rightarrow X$ con $\varphi (x,0)=x$ e tale da essere coerente con la struttura di un gruppo ad un parametro:

\varphi (\varphi (x,t),s)=\varphi (x,t+s)

per ogni $s,t$ in $\mathbb {R}$ e con $x\in X$ .

L'insieme ${\mathcal {O}}(x,\varphi )=\{\varphi (x,t):t\in \mathbb {R} \}$ è chiamato orbita di $x$ attraverso $\varphi$ .

Normalmente è richiesto che un flusso sia compatibile con le strutture definite su $X$ , ad esempio se $X$ è uno spazio topologico si richiede solitamente che il flusso sia una funzione continua (in questo modo il flusso forma un sottogruppo ad un parametro di omeomorfismi). In molti casi $X=\mathbb {R} ^{n}$ , oppure è una varietà differenziabile con $\varphi$ una funzione differenziabile (che forma un sottogruppo ad un parametro di diffeomorfismi).

Un flusso locale è un flusso definito su un sottoinsieme:

\mathrm {dom} (\varphi )=\{(x,t)\ |\ t\in [a_{x},b_{x}],\ a_{x}<0<b_{x},\ x\in X\}\subset X\times \mathbb {R}

e si introduce in genere quando si trattano flussi di campi vettoriali.

In molti campi, come in ingegneria, in fisica e nello studio delle equazioni differenziali, è diffusa una particolare notazione in cui il flusso è scritto implicitamente come $x(t,x_{0})\equiv \phi _{t}(x_{0})$ , intendendo che la variabile $x$ dipende dal tempo $t$ e dal punto iniziale $x_{0}$ .

Sistemi dinamici

Un comune esempio di flusso in fisica matematica sono le soluzioni di un'equazione differenziale ordinaria autonoma, usata per descrivere i sistemi dinamici:

y'=f(y)\qquad y(t=0)=x_{0}

dove il flusso $\psi _{t}(x_{0})$ corrispondente all'orbita (evoluzione del sistema nello spazio delle fasi) per il punto iniziale $x_{0}$ è l'unica soluzione al problema ai valori iniziali dato.

Bibliografia

(EN) I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, Ergodic theory, Springer (1982)
(EN) P.R. Halmos, Lectures on ergodic theory, Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302
(EN) E. Hopf, Ergodentheorie, Springer (1970) MR0024581 Zbl 0185.29001
(EN) A.M. Vershik, Measurable realization of continuous automorphism groups of a unitary ring Izv. Akad. Nauk. SSSR Ser. Mat., 29: 1 (1965) pp. 127–136 Zbl 0194.16302
(EN) G.W. Mackey, Point realizations of transformation groups Illinois J. Math.

Voci correlate

Collegamenti esterni

(EN) D.V. Anosov, Continuous flow, in Encyclopaedia of Mathematics, Springer e European Mathematical Society, 2002.
(EN) D.V. Anosov, Measurable flow, in Encyclopaedia of Mathematics, Springer e European Mathematical Society, 2002.
(EN) D.V. Anosov, Special flow, in Encyclopaedia of Mathematics, Springer e European Mathematical Society, 2002.
(EN) flow, in PlanetMath.

Portale Controlli automatici

Portale Matematica

Read other articles:

Beverly of Graustark

Beverly of GraustarkKartu lobiSutradaraSidney FranklinDitulis olehAgnes Christine Johnston (skenario)Joseph W. Farnham (judul)BerdasarkanBeverly of Graustark olehGeorge Barr McCutcheonPemeranMarion DaviesAntonio MorenoCreighton HaleSinematograferPercy Hilburn (*French)PenyuntingFrank E. HullDistributorMetro-Goldwyn-MayerTanggal rilis 19 April 1926 (1926-04-19) (Amerika Serikat) Durasi70 menitNegaraAmerika SerikatBahasaBisu (intertitel Inggris)Anggaran$399.000[1] Beverly ...

Kim Pyong-il

Ini adalah nama Korea; marganya adalah Kim. Kim Pyong-ilLahir10 Agustus 1954 (umur 69)Pyongyang, Korea Utara (?)AlmamaterUniversitas Kim Il-sungPekerjaanKomandan batalion DPRKSekarang Duta Besar DPRK untuk PolandiaPartai politikPartai Pekerja KoreaSuami/istritidak diketahuiAnakKim Eun-song (putri)Kim In-kang (putra)Nama KoreaJosŏn-gŭl김평일 Hanja金平一 Alih AksaraGim Pyeong-ilMcCune–ReischauerKim P'yŏng'il Kim Pyong-il (kelahiran 10 Agustus 1954 – ) adalah adik seayah dari p...

العلاقات اليمنية الفانواتية

العلاقات اليمنية الفانواتية اليمن فانواتو اليمن فانواتو تعديل مصدري - تعديل العلاقات اليمنية الفانواتية هي العلاقات الثنائية التي تجمع بين اليمن وفانواتو.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجعية للدولتين: وجه المقارنة ا�...

Sabuk asteroid

Sabuk asteroid utama (warna putih) berlokasi antara orbit Mars dan Jupiter. Sabuk asteroid adalah bagian Tata Surya terletak kira-kira antara orbit planet Mars dan Jupiter. Daerah ini dipenuhi oleh sejumlah objek tak beraturan yang disebut asteroid atau planet kerdil. Sabuk asteroid disebut juga sebagai sabuk utama (main belt) untuk membedakan dari konsentrasi planet kerdil lainnya di dalam sistem tata surya, seperti Sabuk Kuiper dan scattered disc Lebih dari separuh massa sabuk utama terdapa...

イスラームにおける結婚

イスラームにおける結婚（イスラームにおけるけっこん）とは、二者の間で行われる法的な契約である。新郎新婦は自身の自由な意思で結婚に同意する。口頭または紙面での規則に従った拘束的な契約は、イスラームの結婚で不可欠だと考えられており、新郎と新婦の権利と責任の概要を示している[1]。イスラームにおける離婚は様々な形をとることができ、個�...

Allégation antisémite

L'allégation antisémite est une accusation délibérément fabriquée, présentée comme vraie, dans le seul but d'inciter à l'antisémitisme. Bien qu'elles aient été largement réfutées, les calomnies antisémites font souvent partie d'une théorie plus vaste de complot juif. Selon l'historien Kenneth Stern : « Historiquement, les Juifs ont beaucoup souffert des théories du complot. Les différents mythes, que les Juifs ont tué le Christ, ou qu'ils empoisonnent les pui...

Portugal au Concours Eurovision de la chanson 2007

Cet article est une ébauche concernant le Concours Eurovision de la chanson et le Portugal. Vous pouvez partager vos connaissances en l’améliorant (comment ?) ; pour plus d’indications, visitez le projet Eurovision. Portugalau Concours Eurovision 2007 Données clés Pays Portugal Chanson Dança comigo (Vem ser feliz) Interprète Sabrina (en) Compositeur Emanuel Parolier Emanuel, Tó Maria Vinhas Langue Portugais, anglais, espagnol, français Sélection nationale Ra...

Elections in the Sahrawi Arab Democratic Republic

Political elections for public offices in the Sahrawi Arab Democratic Republic Elections in Western Sahara redirects here. For elections in the Moroccan-administered part of Western Sahara, see Elections in Morocco. Politics of the SADR Constitution Nationality law Executive President Brahim Gali Prime Minister Bouchraya Hammoudi Bayoun Legislature Sahrawi National Council President: Hamma Salama Judiciary Constitutional Council Supreme Court Court of Appeal Judicial system Politics Parties: ...

乌克兰总理

烏克蘭總理Прем'єр-міністр України烏克蘭國徽現任杰尼斯·什米加尔自2020年3月4日任命者烏克蘭總統任期總統任命首任維托爾德·福金设立1991年11月后继职位無网站www.kmu.gov.ua/control/en/（英文）乌克兰乌克兰政府与政治系列条目宪法政府总统弗拉基米尔·泽连斯基總統辦公室国家安全与国防事务委员会总统代表（英语：Representatives of the President of Ukraine）总...

Virginia rail

Species of bird This article is about the bird. For the high speed rail line serving Northern Virginia and Washington, D.C., see Virginia Railway Express. For rail travel in the whole state, see List of railroads in Virginia. Virginia rail In Morro Bay, California, USA Conservation status Least Concern (IUCN 3.1)[1] Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Chordata Class: Aves Order: Gruiformes Family: Rallidae Genus: Rallus Species: R. limicola...

2022 Washington, D.C., mayoral election

2022 Washington, D.C., mayoral election ← 2018 November 8, 2022 2026 → Nominee Muriel Bowser Rodney Red Grant Stacia Hall Party Democratic Independent Republican Popular vote 147,433 29,531 11,510 Percentage 74.62% 14.95% 5.83% Ward results Precinct resultsBowser: 50–60% 60–70% 70–80% 80–90% Mayor before election Muriel Bowser Democ...

French forestry ordinance of 1669

17th century forest law The Forestry Ordinance of 1669 (“sur le fait des Eaux et Forêts) proclaimed by Louis XIV of France sought to protect and restore France’s timber resources as well as its considerable forestry heritage. The Ordinance consolidated and clarified older, more fragmented regulations. It was drafted after serious investigation of forestry exploitation practices by Jean-Baptiste Colbert, who successfully championed the idea of systematic reform. The Ordinance specified th...

Guy Park

Historic house in New York, United States For the Governor of Missouri, see Guy Brasfield Park. United States historic placeGuy Park ManorU.S. National Register of Historic Places View from the Mohawk River side (2020)Show map of New YorkShow map of the United StatesLocationW. Main St.,Amsterdam, New YorkCoordinates42°56′49″N 74°12′36″W / 42.94694°N 74.21000°W / 42.94694; -74.21000Built1774Architectural styleGeorgianNRHP reference No.73001206...

2011 European Athletics U23 Championships – Women's triple jump

2011 European Athletics U23 ChampionshipsTrack events100 mmenwomen200 mmenwomen400 mmenwomen800 mmenwomen1500 mmenwomen5000 mmenwomen10,000 mmenwomen100 m hurdleswomen110 m hurdlesmen400 m hurdlesmenwomen3000 msteeplechasemenwomen4 × 100 m relaymenwomen4 × 400 m relaymenwomenRoad events20 km walkmenwomenField eventsHigh jumpmenwomenPole vaultmenwomenLong jumpmenwomenTriple jumpmenwomenShot putmenwomenDiscus throwmenwomenHammer throwmenwomenJavelin throwmenwomenCombined eventsHeptathlonwome...

University of the West of England

University in South Gloucestershire, UK UWE redirects here. For other uses, see Uwe (disambiguation). University of the West of EnglandCoat of armsOther namesUWE Bristol, UWEMottoLight Liberty Learning.TypePublic research universityEstablished1595 - Merchant Venturers Navigation School[1]1853 - Gloucester and Bristol Diocesan Training Institution for School Mistresses1894 - Merchant Venturers Technical College[2] 1969 – Bristol Polytechnic1992 - University StatusEndowment£2...

Tommy Docherty

Scottish association footballer and manager (1928–2020) The Doc redirects here. Not to be confused with The D.O.C.. For other people with similar names, see Thomas Docherty (disambiguation). Tommy Docherty Docherty in 2017Personal informationFull name Thomas Henderson Docherty[1]Date of birth (1928-04-24)24 April 1928Place of birth Glasgow, ScotlandDate of death 31 December 2020(2020-12-31) (aged 92)Place of death Marple, EnglandPosition(s) Right halfSenior career*Years Team Ap...

Prince regent

Prince who rules in place of a monarch due to incapacity or absence For other uses, see Prince regent (disambiguation). This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Prince regent – news · newspapers · books · scholar · JSTOR (February 2023) (Learn how and when to remove this message) Part of a series onIm...

Montauban

Pour les articles homonymes, voir Montauban (homonymie). Montauban Blason Administration Pays France Région Occitanie Département Tarn-et-Garonne(préfecture) Arrondissement Montauban(chef-lieu) Intercommunalité Grand Montauban(siège) Maire Mandat Marie-Claude Berly 2024-2026 Code postal 82000 Code commune 82121 Démographie Gentilé Montalbanais Populationmunicipale 61 919 hab. (2021 ) Densité 458 hab./km2 Population agglomération 80 960 hab. (2021) Géographi...

القاهرة (جورجيا)

لمعانٍ أخرى، طالع القاهرة (توضيح). القاهرة الإحداثيات 30°53′00″N 84°13′00″W / 30.8833°N 84.2167°W / 30.8833; -84.2167 [1] تقسيم إداري البلد الولايات المتحدة[2][3] التقسيم الأعلى مقاطعة غرادي عاصمة لـ مقاطعة غرادي خصائص جغرافية المساحة 25.5...

2016 New Orleans Saints season

NFL team season 2016 New Orleans Saints seasonOwnerTom BensonGeneral managerMickey LoomisHead coachSean PaytonOffensive coordinatorPete Carmichael Jr.Defensive coordinatorDennis AllenHome fieldMercedes-Benz SuperdomeResultsRecord7–9Division place3rd NFC SouthPlayoff finishDid not qualifyPro BowlersDrew Brees (QB)Uniform ← 2015 Saints seasons 2017 → The 2016 season was the New Orleans Saints' 50th in the National Football League (NFL), their 41st playing home games...