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Matematiska biljarder och KAM teori.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2014 (Swedish)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [sv]

En matematisk biljard är en idealisering av det fysiska spelet vari

man betraktar kollisionerna av en punktmassa i rörelse innanför ett

slutet område. Studiet av matematiska biljarder motiveras delvis av

mekaniska samt optiska system men det ges även exempel på rent matematiska

sådana. Denna något lekfulla inkörsport till det generellare

studiet av dynamiska system visar sig vara en slagkraftig sådan, och

rapporten tar upp några viktiga begrepp inom studiet av matematiska

biljarder såsom biljardavbildingen samt kaustikor. Denna rapport ämnar

även tillhandage en kort introduktion till KAM teori, en teori först

introducerad av Andrej Kolmogorov för att delvis lösa problem inom

celest mekanik. Det bevisas två satser inom denna teori, båda starkt

beroende av tidigare resultat i rapporten samt av det som introduceras

som det diofantiska villkoret.

Abstract [en]

A mathematical billiard is an idealization of the real, physical, game

where one studies the collisions of a point mass within a closed area.

The study of mathematical billiards is partly motivated by both mechanical

and optical systems, but also by pure mathematical ones. This

approach to the more general study of dynamical systems is not to be

dismissed by its playfulness, and this report highlights some important

concepts of mathematical billiards such as the billiard map and

the notion of a caustic. Moreover, the aim of this rapport is also to

give a short introduction to KAM theory; a theory first developed by

Andrej Kolmogorov to partly solve problems in celestial mechanics.

This rapport proves two KAM theorems, both heavily dependent on

previews results in the rapport and also on, what will be introduced

as, the Diophantine condition.

Place, publisher, year, edition, pages
2014. , 54 p.
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:kth:diva-147665OAI: oai:DiVA.org:kth-147665DiVA: diva2:731401
Supervisors
Available from: 2014-07-01 Created: 2014-07-01 Last updated: 2014-10-21Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
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