Complete Systems of Galileo Invariants of a Motion of Parametric Figure in the Three Dimensional Euclidean Space

Yıl 2022, Cilt: 15 Sayı: 2, 334 - 342, 31.10.2022

Djavvat Khadjiev , İdris Ören , Gayrat Beshimov

https://doi.org/10.36890/iejg.1091348

Öz

Let $E^{3}$ be the 3-dimensional Euclidean space and $S$ be a set with at least two elements. The notions of an $S$-parametric figure and the motion of an $S$-parametric figure in $E^{3}$ are defined. Complete systems of invariants of an $S$-parametric figure in $E^{3}$ for the orthogonal group $O(3,R)$ , the special orthogonal group $SO(3,R)$, Euclidean group $MO(3,R)$, the special Euclidean group $MSO(3,R)$ and Galileo groups $Gal_{1}(3,R)$ , $Gal^{+}_{1}(3,R)$ are obtained.

Anahtar Kelimeler

Galilean group, invariant, figure, Euclidean geometry

Kaynakça

• [1] Aripov R. and Khadjiev D.:The complete system of global differential and integral invariants of a curve in Euclidean geometry, Izvestiya Vuzov, Ser. Mathematics, 542, 1-14 (2007).(Russian). (English translation) Aripov R. and Khadzhiev D., Russian Mathematics (Iz VUZ), 2007, Vol. 51, No. 7, pp.1-14.
• [2] BergerM.:Geometry I, Springer-Verlag,Berlin,Heidelberg,1987.
• [3] ElleryB.G.:Foundations of Euclidean and Non-Euclidean Geometry,HoltRinehartand WinstonINC,NewYork,1968.
• [4] Marvin,Y.G.:EuclideanandNon-EuclideanGeometries,W.H.FreemanandCompany,NewYork,1993.
• [5] Ho ̋ferR.:m-pointinvariantsofrealgeometries,BeitrageAlgebraGeom,40,261-266(1999).
• [6]O ̋ren I.,KhadjievD.and Peksen O.:Identificationsofpathsandcurvesundertheplanesimilaritytransformationsandtheirapplicationsto mechanics, Journal of Geometry and Physics, 151, (2020).
• [7]KhadjievD.:CompletesystemsofdifferentialinvariantsofvectorfieldsinaEuclideanspace,TurkishJournalofMathematics,34,543-559(2010).
• [8] KhadjievD.,O ̋renI.andPeks ̧enO ̋.:Globalinvariantsofpathsandcurvesforthegroupofalllinearsimilaritiesinthetwo-dimensionalEuclidean space, International Journal of Geometric Methods in Modern Physics, 15, No. 6, (2018).
• [9] Khadjiev D.:Ayupov Sh. and Beshimov G.:Complete systems of invariants of m-tuples for fundamental groups of the two-dimensional Euclidean space, Uzbek Mathematical Journal, 1,71-98 (2020,).
• [10] KhadjievD.,BekbaevU.,AripovR.:Onequivalenceofvector valuedmaps,arXiv:2005.08707v1[mathGM]13May2020.
• [11] PatricJ.R.:EuclideanandNon-EclideanGeometry,AnAnalyticApproach,CambridgeUniversityPress,Cambridge,2008. [12]Mattila,P.:GeometryofSetandMeasuresinEuclideanSpaces,CambridgeUniversityPress,Cambridge,1995.
• [13] RogerA.J.:AdvancedEuclideanGeometry,DoverPublications,INC,NewYork,1960.
• [14]Musielak,Z.E.,Fry,J.L.:GeneraldynamicalequationsforfreeparticlesandtheirGalileaninvariance,IntJTheorPhys,48,1194-1202(2009).
• [15] Godunov,S.K.,Gordienko,V.M.:ComplicatedstructuresofGalilean-invariantconservationlaws,JournalofAppliedMechanicsandTechnical Physics, 43, No. 2, 175-189,(2002).
• [16] W.W,H.,Havas,P.:GeneralizedGalilei-invariantclassicalmechanics,InternationalJournalofModernPhysicsA20,No.18.,4259-4289(2005).
• [17]Karger,A.:KinematicgeometryinndimensionalEuclideanandsphericalspace,CzechoslovakMathematicalJournal,22,No.1,83–107(1972).