Abstract
For classical Hamiltonian systems containing an harmonic oscillator of high frequency, one has the problem of controlling the energy exchange between the oscillator and the remaining “slow” degrees of freedom; under very general conditions, such an exchange turns out to be exponentially small with the frequency of the oscillator. In the Jeans-Landau-Teller method, one aims to prove the exponential dependence, and to estimate the coefficient of the exponential, by exploiting the analyticity properties of the solution of the differential equations describing the motion of the system. However, in practice, since the exact solution is not known, such properties are inferred from those of an approximate solution, with no control of the difference; this fact might a priori even invalidate the exponential dependence itself. In the present paper a rigorous treatment is given, for a particular model of interest in the domain of atomic collisions, by keeping control of the difference between the exact and the approximate solution.
Similar content being viewed by others
References
Neishtadt, A.I.: Prikl. Matem. Mekan.45, 80 (1981) [PMM U.S.S.R.45, 58 (1982)]
Benettin, G., Galgani, L., Giorgilli, A.: Commun. Math. Phys.113, 87 (1987)
Benettin, G., Galgani, L., Giorgilli, A.: Commun. Math. Phys.121, 557 (1989)
Nekhoroshev, N.N.: Usp. Mat. Nauk32, (1977) [Russ. Math. Surv.32, 1 (1977)]
Nekhoroshev, N.N.: Trudy Sem Petrows. No. 5, 5 (1979) [Topics in Modern Mathematics, Petrovskii Sem. no. 5, Oleinik, O.A. (ed.) New York: Consultant Bureau 1985]
Benettin, G., Carati, A., Sempio, P.: paper in preparation
Rapp, D., Kassal, T.: Chem. Rev.65, 61 (1969)
Rapp, D.: J. Chem. Phys.32, 735 (1960)
O'Neil, T.M., Hjorth, P.G.: Phys. Fluids28, 3241 (1985); O'Neil, T.M., Hjorth, P.G., Beck, B., Fajans, J., Malmberg, J.H.: Collisional Relaxation of Strongly Magnetized Pure Electron Plasma (Theory and Experiment) In: Strongly Coupled Plasma Physics. Proceedings of the Yamada Conference No. 24, Japan, p. 313, North-Holland: Amsterdam 1990
Jeans, J.H.: Phil. Mag.6, 279 (1903)
Jeans, J.H.: Phil. Mag.10, 91 (1905)
Landau, L., Teller, E.: Physik. Z. Sowjetunion10, 34 (1936), In: Ter Haar D. (ed.) Collected Papers of L.D. Landau, p. 147, Oxford: Pergamon Press 1965
Author information
Authors and Affiliations
Additional information
Communicated by J.-P. Eckmann
Rights and permissions
About this article
Cite this article
Carati, A., Benettin, G. & Galgani, L. Towards a rigorous treatment of the Jeans-Landau-Teller method for the energy exchanges of harmonic oscillators. Commun.Math. Phys. 150, 321–336 (1992). https://doi.org/10.1007/BF02096663
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02096663