Abstract
We develop a symplectic method of finding the adiabatic invariants of nonlinear dynamic systems with small parameter. We show that a necessary and sufficient condition for the existence of quasi-Hamiltonian adiabatic invariants of nonlinear dynamic systems with regular dependence on a small parameter is that the Cauchy problem be well-posed for an equation of Lax type in the class of nongradient local functionals on the cotangent manifold of the phase space. It is established that scalar nonlinear dynamic systems always have a priori complete evolution invariants, not only adiabatic invariants. We also consider typical applications in hydrodynamics and oscillatory systems of mathematical physics.
Similar content being viewed by others
Literature cited
A. S. Bakai and Yu. P. Stepanovskii,Adiabatic Invariants [in Russian], Naukova Dumka, Kiev (1981).
B. A. Dubrovin, S. P. Novikov, and A. G. Fomenko,Modern Geometry, Springer-Verlag, New York (1984).
M. Kruskal,Adiabatic Invariants [Russian translation], Izdatel'stvo Inostrannoi Literatury, Moscow (1962).
Yu. A. Mitropol'skii, “On adiabatic invariants in problems of nonlinear mechanics,” Preprint, Institute of Mathematics of the Ukrainian Academy of Sciences, No. 83.37 (1983).
Yu. A. Mitropol'skii et al.,Integrable Dynamic Systems [in Russian], Naukova Dumka, Kiev (1987).
Yu. O. Mitropol's'kii, A. K. Prikarpats'kii, and B. M. Fil', “Some aspects of a gradient-holonomic algorithm in the theory of integrability of nonlinear dynamic systems and problems of computer algebra,”Ukr. Mat. Zh.,43, No. 1, 78–91 (1991).
Yu. A. Mitropol'skii, A. K. Prikarpatskii, and V. G. Samoilenko, “Asymptotic methods of constructing the implectic and recursion operators of nonlinear dynamic systems,”Dokl. Akad. Nauk SSSR,287, No. 6, 1312–1317 (1986).
S. P. Novikov, ed.,Theory of Solitons, Consultants Bureau, New York (1984).
V. G. Samoilenko, “Jet-analysis on smooth functional manifolds and its applications for the study of integrable nonlinear dynamic systems,” Preprint, Institute of Mathematics of the Ukrainian Academy of Sciences, Kiev (1988).
Additional information
Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 179–185.
Rights and permissions
About this article
Cite this article
Mitropol's'kii, Y.O., Antonishin, I.O. & Prikarpats'kii, A.K. Adiabatic invariants of nonlinear dynamic systems with small parameter. J Math Sci 67, 2993–2998 (1993). https://doi.org/10.1007/BF01095884
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01095884