Mechanics of Solids (about journal) Mechanics of Solids
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V. F. Chub, "On the possibility of application of one system of hypercomplex numbers in inertial navigation," Mech. Solids. 37 (6), 1-17 (2002)
Year 2002 Volume 37 Number 6 Pages 1-17
Title On the possibility of application of one system of hypercomplex numbers in inertial navigation
Author(s) V. F. Chub (Moscow)
Abstract A hypercomplex number system (generalized quaternions) is defined to extend the concept of Hamilton's and Clifford's biquaternions. Characteristic properties of numbers of this system, as well as their potentials for representing space and time translations, rotations, and boosts, are analyzed. Special attention is given to group properties of space-time transformations. The developed formalism is utilized to derive the relativistic equations of inertial navigation in the gravitation-free space.
References
1.  I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers [in Russian], Nauka, Moscow, 1973.
2.  Mathematics Encyclopedia [in Russian], Vol. 1, Sovetskaya Entsiklopediya, Moscow, 1977.
3.  V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in Problems of Orientation of a Rigid Body [in Russian], Nauka, Moscow, 1973.
4.  M. B. Balk, G. D. Balk, and A. A. Polukhin, Real Applications of Imaginary Numbers [in Russian], Radyan'ska Shkola, Kiev, 1988.
5.  A. T. Grigor'yan and B. A. Rozenfel'd, "History of non-Euclidean mechanics," in Investigations on History of Physics and Mechanics [in Russian], pp. 161-178, Nauka, Moscow, 1988.
6.  F. Klein, Elementary Mathematics from the Viewpoint of Higher Mathematics. Volume 1. Arithmetic, Algebra, and Analysis [Russian translation], Nauka, Moscow, 1987.
7.  C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Volume 3 [Russian translation], Mir, Moscow, 1977.
8.  F. M. Dimentberg, Screw Theory and its Applications [in Russian], Nauka, Moscow, 1978.
9.  V. N. Branets and I. P. Shmyglevskii, Introduction to the Theory of Strapdown Inertial Navigation Systems [in Russian], Nauka, Moscow, 1992.
10.  T. V. Putyata, B. L. Laptev, B. A. Rozenfel'd, and B. N. Fradlin, Aleksandr Petrovich Kotel'nikov (1865-1944) [in Russian], Nauka, Moscow, 1968.
11.  P. A. Lebedev, Kinematics of Spatial Mechanisms Mashinostroenie, Leningrad, 1966.
12.  Physics Encyclopedic Dictionary. Volume 3 [in Russian], Sovetskaya Entsiklopediya, Moscow, 1963.
13.  A. A. Logunov, Lectures on Relativity and Gravitation: Modern Analysis of the Problem [in Russian], Nauka, Moscow, 1987.
14.  A. V. Vasil'ev, "Mathematics during the recent 50 years," in Matematicheskoe Obrazovanie, No. 1, pp. 3-9, 1928; No. 2, pp. 49-58, 1928; No. 3, 1997.
15.  A. A. Logunov, To the Work "On the Electron Dynamics" by Henri Poincaré [in Russian], IYaI AN SSSR, Moscow, 1984.
16.  V. P. Vizgin, The Erlangen Program and Physics [in Russian], Nauka, Moscow, 1975.
17.  G. A. Zaitsev, "On the relation of the relativity theory to the group theory," in M.-A. Tonnela, Fundamentals of Electromagnetism and Relativity [Russian translation], pp. 447-475, IL, Moscow, 1962.
18.  Yu. V. Novozhilov, Introduction to the Theory of Elementary Particles [in Russian], Nauka, Moscow, 1972.
19.  V. Ph. Zhuravlev, Fundamentals of Theoretical Mechanics [in Russian], Nauka, Moscow, 1997.
20.  S. V. Zelenkov, I. A. Mikhalev, P. I. Saidov, and G. N. Senilov, "January 18: the 30th anniversary of the paper "To the theory of spatial orientation during blind flight by means of pendulous gyro systems" by L. I. Tkachev (1943)," in From History of Aviation and Astronautics [in Russian], Vol. 19 (Memorable Dates for 1973), pp. 8-14, Moscow, 1973.
21.  A. P. Panov, Mathematical Fundamentals of Inertial Orientation [in Russian], Naukova Dumka, Kiev, 1995.
22.  L. I. Sedov, "On the inertial navigation equations accounting for relativistic effects," Doklady AN SSSR, Vol. 231, No. 6, pp. 1131-1314, 1976. (For more detail see L. I. Sedov and A. G. Tsypkin, Fundamentals of Macroscopic Theories of Gravitation and Electromagnetism [in Russian], Nauka, Moscow, 1989.)
23.  K. Mueller, Relativity [Russian translation], Atomizdat, Moscow, 1975.
24.  E. F. Taylor and J. A. Wheeler, Spacetime Physics [Russian translation], Mir, Moscow, 1969.
25.  A. N. Krylov, "On motions of a rigid body," in Collected Works by Academician A. N. Krylov. Volume 8. Mechanics, pp. 333-350, Izd-vo AN SSSR, Moscow, Leningrad, 1950.
26.  A. P. Kotel'nikov, "Projective theory of vectors," Izv. Fiz.-mat. Ob-va pri Imperatorskom Kazan. Un-te, Vol. 9, No. 3, pp. 241-317, 1899.
27.  A. I. Kostrikin, Introduction to Algebra [in Russian], Nauka, Moscow, 1977.
28.  I. R. Shafarevich, Basic Concepts of Algebra [in Russian], Izhevsk. Respubl. Tipografiya, Izhevsk, 1999.
29.  N. N. Belova and A. N. Danilov, Algebra and Arithmetic of Quaternions [in Russian], Izd-vo ChGPI, Cherepovets, 1995.
30.  S. M. Onishchenko, Application of Hypercomplex Numbers in Inertial Navigation: Autonomous Systems [in Russian], Naukova Dumka, Kiev, 1983.
31.  B. N. Petrov, I. I Gol'denblat, G. M. Ulanov, and S. V. Ul'yanov, Problems of Control of Relativistic and Quantum Dynamical Systems: Physics and Information Aspects [in Russian], Nauka, Moscow, 1982.
32.  F. I. Tkachenko, "On relativistic inertial navigation equations," in Space Research in Ukraine [in Russian], No. 15, pp. 93-97, Kiev, 1981.
33.  A. V. Berezin, Yu. A. Kurochkin, and E. A. Tolkachev, Quaternions in Relativistic Physics [in Russian], Nauka i Tekhnika, Minsk, 1989.
34.  G. Casanova, Vector Algebra [Russian translation], Mir, Moscow, 1979.
35.  Yu. M. Lomsadze, Group-theoretic Introduction to the Theory of Elementary Particles [in Russian], Vysshaya Shkola, Moscow, 1962.
36.  D. Hestenes, Space-time Algebra, Gordon and Breach, New York, 1966.
37.  B. V. Medvedev, Fundamentals of Theoretical Physics [in Russian], Nauka, Moscow, 1977.
38.  V. A. Fok, Theory of Space, Time, and Gravitation [in Russian], Fizmatgiz, Moscow, 1961.
39.  A. Leitmann, W. H. Press, R. Price, and S. A. Teukolsky, Problems on Relativity and Gravitation [Russian translation], Mir, Moscow, 1979.
40.  K. N. Bystrov and V. D. Zakharov, "Hypercomplex structures in spaces of general relativity and field theory," in Achievements in Science and Technology [in Russian], Ser. Classical Theory of Field and Gravitation. Volume 1. Gravitation and Cosmology, pp. 111-158, VINITI, Moscow, 1991.
41.  V. V. Kassandrov, Space-time Algebraic Structure and Algebraic Dynamics [in Russian], Izd-vo RUDN, Moscow, 1992.
42.  V. Ya. Fridman, Theory of "Centaurs" and the Structure of the Real World [in Russian], Nauka, Moscow, 1996.
43.  A. L. Smolin, Hypercomplex Lorentz Transformations, Ether, and the Other Physics [in Russian], pp. 93-104, Dialog-MGU, Moscow, 1999.
44.  D. F. Kurdgellaidze, Introduction to the Non-associative Classical Theory of Field [in Russian], Metsniereba, Tbilisi, 1987.
45.  Ya. Lykhmus, E. Paal, and L. Sorgsepp, "Nonassociativity in Mathematics and Physics," in Transactions of the Institute of Physics of the Academy of Sciences of Estonia. Volume 66. Quasi-groups and Non-associative Algebras in Physics [in Russian], pp. 8-22, Tartu, 1990.
46.  G. Dixon, "Algebraic unification," Phys. Rev. D., Vol. 28, No. 4, pp. 833-843, 1983.
47.  R. Penrose and W. Rindler, Spinors and Space-time. Two-spinor Calculus and Relativistic Fields [Russian translation], Mir, Moscow, 1987.
48.  I. M. Yaglom, Complex Numbers and their Application in Geometry [in Russian], Fizmatgiz, Moscow, 1963.
49.  V. I. Arnol'd, Geometry of Complex Numbers, Quaternions, and Spins [in Russian], Izd-vo MTsNMO, Moscow, 2002.
50.  B. A. Rozenfel'd, History of Non-Euclidean Geometry: The Development of the Concept of the Geometric Space [in Russian], Nauka, Moscow, 1976.
51.  M. V. Sin'kov and N. M. Gubareni, Non-positional Representations in Multi-dimensional Number Systems [in Russian], Naukova Dumka, Kiev, 1979.
52.  N. V. Aleksandrova, "Maxwell: vectors and quaternions," in L. S. Polak (Editor), Maxwell and the Development of Physics of XIX-XX Centuries: Collected Papers [in Russian], pp. 72-76, Nauka, Moscow, 1985.
53.  W. R. Hamilton, Selected Works: Optics, Dynamics, and Quaternions [Russian translation], Nauka, Moscow, 1994.
Received 1 March 2001
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