  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
Print ISSN 00256544 Online ISSN 19347936 
Archive of Issues
Total articles in the database:   3667 
In Russian (Èçâ. ÐÀÍ. ÌÒÒ):   2068

In English (Mech. Solids):   1599 

<< Previous article  Volume 37, Issue 6 / 2002  Next article >> 
V. F. Chub, "On the possibility of application of one system of hypercomplex numbers in inertial navigation," Mech. Solids. 37 (6), 117 (2002) 
Year 
2002 
Volume 
37 
Number 
6 
Pages 
117 
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 spacetime
transformations. The developed formalism is utilized to
derive the relativistic equations of inertial navigation
in the gravitationfree 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 nonEuclidean mechanics,"
in Investigations on History of Physics and Mechanics
[in Russian], pp. 161178, 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 (18651944)
[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. 39, 1928; No. 2, pp. 4958,
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. 447475, 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. 814, 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. 11311314, 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. 333350, Izdvo AN SSSR, Moscow, Leningrad, 1950. 
26.  A. P. Kotel'nikov, "Projective theory of vectors,"
Izv. Fiz.mat. Obva pri Imperatorskom
Kazan. Unte, Vol. 9, No. 3, pp. 241317, 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], Izdvo 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. 9397, 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, Grouptheoretic Introduction to the Theory
of Elementary Particles [in Russian], Vysshaya Shkola, Moscow, 1962. 
36.  D. Hestenes, Spacetime 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. 111158, VINITI,
Moscow, 1991. 
41.  V. V. Kassandrov, Spacetime Algebraic Structure
and Algebraic Dynamics [in Russian], Izdvo 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. 93104, DialogMGU, Moscow, 1999. 
44.  D. F. Kurdgellaidze, Introduction to the Nonassociative
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. Quasigroups
and Nonassociative Algebras in Physics [in Russian], pp. 822, Tartu,
1990. 
46.  G. Dixon, "Algebraic unification," Phys. Rev.
D., Vol. 28, No. 4, pp. 833843, 1983. 
47.  R. Penrose and W. Rindler, Spinors and Spacetime.
Twospinor 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], Izdvo MTsNMO, Moscow, 2002. 
50.  B. A. Rozenfel'd, History of NonEuclidean Geometry:
The Development of the Concept of the Geometric Space [in Russian],
Nauka, Moscow, 1976. 
51.  M. V. Sin'kov and N. M. Gubareni, Nonpositional
Representations in Multidimensional 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 XIXXX
Centuries: Collected Papers [in Russian], pp. 7276, Nauka, Moscow, 1985. 
53.  W. R. Hamilton, Selected Works: Optics, Dynamics,
and Quaternions [Russian translation], Nauka, Moscow, 1994. 

Received 
1 March 2001 
<< Previous article  Volume 37, Issue 6 / 2002  Next article >> 

If you find a misprint on a webpage, please help us correct it promptly  just highlight and press Ctrl+Enter

