×

Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. I. (English) Zbl 0646.70013

Summary: The so-called problem of the realization of the holonomic constraints of classical mechanics is here revisited, in the light of Nekhoroshev-like classical perturbation theory. Precisely, if constraints are physically represented by very steep potential wells, with associated high frequency transversal vibrations, then one shows that (within suitable assumptions) the vibrational energy and the energy associated to the constrained motion are separately almost constant, for a very long time scale growing exponentially with the frequency (i.e., with the rigidity of the constraint one aims to realize). This result can also be applied to microscropic physics, providing a possible entirely classical mechanism for the “freezing” of the high-frequency degrees of freedom, in terms of non-equilibrium statistical mechanics, according to some ideas expressed by Boltzmann and Jeans at the turn of the century. In this Part I we introduce the problem and prove a first theorem concerning the realization of a single constraint (within a system of any number of degrees of freedom). The problem of the realization of many constraints will be considered in a forthcoming Part II.

MSC:

70H05 Hamilton’s equations
70F20 Holonomic systems related to the dynamics of a system of particles
70J99 Linear vibration theory
82B05 Classical equilibrium statistical mechanics (general)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rubin, H. Hungar, P.: Commun. Pure Appl. Math.10, 65 (1957) · Zbl 0077.17401 · doi:10.1002/cpa.3160100103
[2] Arnold, V. I.: Méthodes mathématiques de la mécanique classique Moscow: MIR 1976, p. 97
[3] Gallavotti, G.: Meccanica elementare Boringhieri, Torino: 1980 [English edition: The elements of mechanics. Berlin, Heidelberg, New York: Spring 1983], Chap. 3. · Zbl 0473.60085
[4] Takens, F.: Motion under the influence of a strong constraining force. In: Global theory of dynamical systems. Nitecki Z. Robinson C. (eds.), Lecture Notes in Mathematics, Vol. 819. Berlin, Heidelberg, New York: Springer 1979
[5] Van Kampen, N. G.: Elimination of fast variables. Phys. Rep.124, 69 (1985) · doi:10.1016/0370-1573(85)90002-X
[6] Boltzmann, L.: Nature51, 413 (1895) · JFM 26.1059.08 · doi:10.1038/051413b0
[7] Jeans, J. H.: On the vibrations set up in molecules by collisions. Phil. Mag.6, 279 (1903) · JFM 34.0833.02
[8] Jeans, J. H.: On the partition of energy between matter and aether. Phil. Mag.10, 91 (1905) · JFM 36.0842.01
[9] Nekhoroshev, N. N.: Usp. Mat. Nauk.32, 5 (1977) [Russ. Math. Surv.32, 1 (1977); Tr. Sem. Petrows. No. 5, 5 (1979)]
[10] Benettin, G. Galgani, L. Giorgilli, A.: Celest. Mech.37, 1 (1985) · Zbl 0602.58022 · doi:10.1007/BF01230338
[11] Gallavotti, G.: Quasi integrable mechanical systems. In: Critical phenomena, random systems, gauge theories. Osterwalder, K., Stora, R. (eds.). Amsterdam, New York, Oxford, Tokyo: North-Holland 1986 · Zbl 0662.70022
[12] Giorgilli, A. Galgani, L.: Celest. Mech.37, 95 (1985) · doi:10.1007/BF01230921
[13] Benettin, G. Gallavotti, G.: Stability of motions near resonances in quasi-integrable Hamiltonian systems. Stat. Phys.44, 293 (1986) · Zbl 0636.70018 · doi:10.1007/BF01011301
[14] Benettin, G.: Examples of long time scales in Hamiltonian dynamical systems. In: Proceedings of the L’Aquila Conference ?Recent Advances in Hamiltonian Dynamics,? June 1986 (to appear) · Zbl 0689.34040
[15] Benettin, G. Galgani, L. Giorgilli, A.: Boltzmann’s ultraviolet cutoff and Nekhoroshev’s theorem on Arnold diffusion. Nature311, 444 (1985) · Zbl 0602.58022 · doi:10.1038/311444a0
[16] Benettin, G. Galgani, L. Giorgilli, A.: Exponential law for the equipartition times among translational and vibrational degrees of freedom. Phys. Lett.A 120, 23 (1987) · Zbl 0646.70013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.