Publication result detail

Straight Lines in Three-Dimensional Space and the Ultrahyperbolic Equation.

CHRASTINOVÁ, V.

Original Title

Straight Lines in Three-Dimensional Space and the Ultrahyperbolic Equation.

English Title

Straight Lines in Three-Dimensional Space and the Ultrahyperbolic Equation.

Type

Peer-reviewed article not indexed in WoS or Scopus

Original Abstract

The straight lines in three-dimensional vector space realize the shortest distance for various metrics. This property is reformulated in terms of the inverse problem of the calculus of variations and closely related to the ultrahyperbolic equation with four independent variables. The interrelation is useful in both directions. For instance, polynomial solutions of the ultrahyperbolic equation provide all polynomial metrics with extremals the straight lines and conversely, a~slight generalization of the Hilbert metrics leads to rather nontrivial (multi-valued or focusing) solutions of the ultrahyperbolic equation. In general, the article clarifies some well-known achievements concerning the 4th Hilbert Problem.

English abstract

The straight lines in three-dimensional vector space realize the shortest distance for various metrics. This property is reformulated in terms of the inverse problem of the calculus of variations and closely related to the ultrahyperbolic equation with four independent variables. The interrelation is useful in both directions. For instance, polynomial solutions of the ultrahyperbolic equation provide all polynomial metrics with extremals the straight lines and conversely, a~slight generalization of the Hilbert metrics leads to rather nontrivial (multi-valued or focusing) solutions of the ultrahyperbolic equation. In general, the article clarifies some well-known achievements concerning the 4th Hilbert Problem.

Keywords

Inverse problem of the calculus of variations, Poincar\'{e}-Cartan form, ultrahyperbolic equation, Hilbert projective metrics, 4th Hilbert Problem.

Key words in English

Inverse problem of the calculus of variations, Poincar\'{e}-Cartan form, ultrahyperbolic equation, Hilbert projective metrics, 4th Hilbert Problem.

Authors

CHRASTINOVÁ, V.

RIV year

2011

Released

31.07.2010

Publisher

Krymské vědecké centrum národní akademie věd

Location

Kiev

ISBN

1729-3901

Periodical

Tavricheskiy vestnik informatiki i matematiki

Volume

2010

Number

1

State

Ukraine

Pages from

35

Pages to

49

Pages count

15

BibTex

@article{BUT50311,
  author="Veronika {Chrastinová}",
  title="Straight Lines in Three-Dimensional Space and the Ultrahyperbolic Equation.",
  journal="Tavricheskiy vestnik informatiki i matematiki",
  year="2010",
  volume="2010",
  number="1",
  pages="35--49",
  issn="1729-3901"
}