Abstract
We have elaborated the methods for the qualitative study of dissipative systems and systems with anti-dissipation that allow us, for example, to obtain conditions for bifurcation of birth of stable and unstable auto-oscillations. We succeeded in generalizing the method for studying plane topographical Poincaré systems to higher dimensions. In three-dimensional rigid body dynamics, we have discovered complete lists of first integrals of dissipative systems and systems with anti-dissipation that are transcendental (in the sense of classification of their singularities) functions that are expressed through elementary functions in a number of cases. We have discovered new qualitative analogs between the properties of motion of free bodies in a resisting medium that is fixed at infinity and bodies in an overrun medium flow.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Chaplygin, S.A.: On motion of heavy bodies in an incompressible fluid. In: A Complete Collection of Works [in Russian]. Vol. 1, Izd. Akad. Nauk SSSR, Leningrad 133–135 (1933)
Chaplygin, S.A.: Selected Works [in Russian]. Nauka, Moscow (1976)
Gurevich, M.I.: Jet Theory of Ideal Fluid [in Russian]. Nauka, Moscow (1979)
Samsonov, V.A., Shamolin, M.V.: Body motion in a resisting medium. Moscow Univ. Mech. Bull. 44(3), 16–20 (1989)
Sedov, L.I.: Continuous Medium Mechanics [in Russian]. vol. 1, 2. Nauka, Moscow (1983, 1984)
Shamolin, M.V.: Applications of Poincaré topographical system methods and comparison systems in some concrete systems of differential equations. Vestn. MGU, Ser. 1, Mat., Mekh. 2, 66–70 (1993)
Shamolin, M.V.: Spatial Poincaré topographical systems and comparison systems. Usp. Mat. Nauk. 52(3), 177–178 (1997)
Shamolin, M.V.: On integrability in transcendental functions. Usp. Mat.Nauk. 53(3), 209–210 (1998)
Shamolin, M.V.: New Jacobi integrable cases in dynamics of a rigid body interacting with a medium. Dokl. Ross. Akad. Nauk. 364(5), 627–629 (1999)
Shamolin, M.V.: A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium. Dokl. Ross. Akad. Nauk. 371(4), 480–483 (2000)
Shamolin, M.V.: New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium. J. Math. Sci. 114(1), 919–975 (2003)
Shamolin, M.V.: Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow around. Prikl. Mat. Mekh. 69(6), 1003–1010 (2005)
Shamolin, M.V.: Integrability of Some Classes of Dynamic Systems in Terms of Elementary Functions. Vestn MGU, Set. 1, Mat., Mekh. 3, 43–49 (2008)
Shamolin, M.V.: On integrability in elementary functions of certain classes of nonconservative dynamical systems. Contemporary Mathematics and Its Applications. Geometry Mech. 62, 131–171 (2009)
Trofimov, V.V., Shamolin, M.V.: Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems. Fund. Prikl. Mat. 16(4), 3–229 (2010)
Acknowledgements
This work was supported by Russian Fund of Basic Research, project 12-01-00020-a.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Shamolin, M.V. (2014). Dynamical Pendulum-Like Nonconservative Systems. In: Awrejcewicz, J. (eds) Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-319-08266-0_38
Download citation
DOI: https://doi.org/10.1007/978-3-319-08266-0_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08265-3
Online ISBN: 978-3-319-08266-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)