Abstract
A technique is developed to search for differential equations that describe any dynamic processes for which only their dependence on some parameters is known. A set of new differential equations are found; they posses interesting properties and are promising to be used in various branches of engineering, especially in flight dynamics. The extremal theory of dimensions was used and was shown to allow finding the form of differential equations for any complex dynamic processes and determining physical laws for them.
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References
E. R. Smoliakov, “Special extremals in dimension analysis,” Dokl. RAN, 421, No. 5, 602–606 (2008).
E. R. Smoliakov, “Variational equations of electrodynamics,” Differents. Uravn., 43, No. 4, 475–480 (2007).
E. Buckingham, “On physically similar systems; illustrations of the use of dimensional equations,” Phys. Rev., 4, 345–376 (1914).
P. V. Bridgeman, Dimensional Analysis, Yale Univ. Press (1937).
L. A. Sena, Units of Physical Quantities and Their Dimensions [in Russian], Mir, Moscow (1977).
A. E. Bryson and Yu Chi Ho, Applied Optimal Control: Optimization, Estimation and Control, Hemisphere, Washington, DC (1975).
E. R. Smoliakov, Theory of Conflict Equilibria [in Russian], Editorial URSS, Moscow (2005).
E. R. Smol’yakov, “Using singular extremals to obtain new equations of motion and unknown constants,” Cybern. Syst. Analysis, 45, No. 4, 605–612 (2009).
V. A. Gelovani and E. R. Smoliakov, “The hypothesis about the influence of higher derivatives on the motion of the center of mass,” Dokl. RAN, 375, No. 2, 159–162 (2000).
E. R. Smoliakov, “Nonlinear motion laws and substantiation of reactionless drive,” Dokl. RAN, 393, No. 6, 770–775 (2003).
E. R. Smoliakov, Theoretical Substantiation of Interstellar Flights [in Russian], KomKniga, Moscow (2005).
E. R. Smoliakov, “Dynamics and energetics of transitions between dual spaces,” Dokl. RAN, 406, No. 6, 734–737 (2006).
E. R. Smoliakov, “Motion integrals in a dual space,” Dokl. RAN, 414, No. 4, 459–463 (2007).
E. R. Smoliakov, “Theory of motion of electrically charged solid bodies in Minkowsky space and the space dual to it,” in: Dynamics of Inhomogeneous Systems, Tr. Inst. Syst. Analysis of the Russian Academy of Science, 29, Issue 11, 85–117 (2007).
E. R. Smoliakov, “Extremality principle in dimensional theory and new fundamental physical constants,” in: Dynamics of Inhomogeneous Systems, Tr. Inst. Syst. Analysis of the Russian Academy of Science, 33, Issue 12, 7–95 (2008).
G. N. Duboshin, Celestial Mechanics. Basic Problems and Methods [in Russian], Nauka, Moscow (1968).
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1The study was carried out in line with the program of the Russian Academy of Sciences “The fundamentals of information technologies and systems,” project No. 1–3, and was supported by the Russian Fund for Fundamental Research, project 09-01-00655-a.
Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 72–82, September–October 2011.
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Smoliakov, E.R. Searching for unknown motion laws based on the extremal theory of dimensions1 . Cybern Syst Anal 47, 731–740 (2011). https://doi.org/10.1007/s10559-011-9352-0
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DOI: https://doi.org/10.1007/s10559-011-9352-0