Quaternions and Cauchy Classical Theory of Elasticity

• Autorzy: Danielewski Marek , Sapa Lucjan

Identyfikatory

DOI

10.2478/amst-2019-0014

• Języki publikacji: EN

Abstrakty

EN

Developed by French mathematician Augustin-Louis Cauchy, the classical theory of elasticity is the starting point to show the value and the physical reality of quaternions. The classical balance equations for the isotropic, elastic crystal, demonstrate the usefulness of quaternions. The family of wave equations and the diffusion equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic solid. Using the quaternion algebra, we present the derivation of the quaternion form of the multiple wave equations. The fundamental consequences of all derived equations and relations for physics, chemistry, and future prospects are presented.

Słowa kluczowe

EN

wave mechanics; quaternions; Cauchy model; elastic solid; Schrödinger equation

PL

mechanika falowa; kwaterniony; model Cauchy'ego; ciało sprężyste; równanie Schrödingera

• Wydawca: Komitet Budowy Maszyn PAN ; De Gruyter
• Czasopismo: Advances in Manufacturing Science and Technology
• Rocznik: 2020
• Tom: Vol. 44, nr 2
• Strony: 67--70
• Opis fizyczny: Bibliogr. 16 poz., rys., tab. wykr.

Twórcy

autor

Danielewski Marek

• Faculty of Mat. Sci. and Ceramics, AGH UST, Mickiewicza 30, 30-059 Kraków, Poland

autor

Sapa Lucjan

• Faculty of Applied Mathematics, AGH UST, Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

• [1] http://en.wikipedia.org/wiki/Quaternion.
• [2] A.E.H. LOVE: Mathematical theory of elasticity, 4th Ed. Dover Publications Inc., New York 1944, p. 8.
• [3] S.D. POISSON: Mémoires Académie Science Paris, 8(1829)356, 623.
• [4] J.C. MAXWELL: Remarks on the mathematical classification of physical quantities. Proc. London Math. Soc., 3(1869), 224-233.
• [5] F. NEUMANN: Vorlesungen über die Theorie der Elasticität der festen Körper und des Lichtäthers. B.G. Teubner, Leipzig 1885.
• [6] P. DUHEM: Mém. Soc. Sci. Bordeaux, Ser. V, 3(1898)316.
• [7] V.V. KRAVCHENKO: Applied quaternionic analysis. Heldermann Verlag, Lemgo 2003.
• [8] H. KLEINERT: Gravity as theory of defects in a crystal with only second-gradient elasticity. Annalen der Physik, 44(1987), 117- 119.
• [9] M. DANIELEWSKI: The Planck-Kleinert Crystal. Z. Naturforsch., 62a(2007), 564-568.
• [10] M. DANIELEWSKI, L. SAPA: Diffusion in Cauchy elastic solid. Diffus. Fundam., 33(2020), 1-14; http://diffusion.uni-leipzig.de/ contents_vol33.php.
• [11] M. DANIELEWSKI, L. SAPA: Nonlinear Klein-Gordon equation in Cauchy-Navier elastic solid. Cherkasy Univ. Bull. Phys. Math. Sci., 1(2017), 22-29.
• [12] A.L. CAUCHY: Récherches sur l’équilibre et le movement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bull. Sot. Philomath., 9(1823), 300-304.
• [13] A.L. CAUCHY: De la pression ou tension dans un corps solide. Ex. Math., 2(1827), 42.
• [14] S. FLÜGGE (ed.): Mechanics of solids. ii Encyclopedia of Physics, vol. VIa/2, Springer, Berlin 1972, p. 208.
• [15] K. GÜRLEBECK, W. SPRÖßIG: Quaternionic analysis and elliptic boundary value problems. Akademie-Verlag, Berlin 1989.
• [16] S. ULRYCH: Higher spin quaternion waves in the Klein-Gordon theory. Int. J. Theor. Phys., 52(2013), 279-292.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).