×

Dissipation-induced instabilities in finite dimensions. (English) Zbl 1205.70002

Summary: The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson (Lord Kelvin), it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin’s theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetaev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader’s intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given.

MathOverflow Questions:

Consequences of lack of rigour

MSC:

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70K20 Stability for nonlinear problems in mechanics
37C99 Smooth dynamical systems: general theory
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0167-6911(86)90095-2 · Zbl 0587.93049
[2] DOI: 10.1023/A:1020570204883
[3] DOI: 10.1073/pnas.0408383102
[4] Arnold, V. I., Sov. Math. Dokl. 6 pp 773– (1965) ISSN: http://id.crossref.org/issn/0197-6788
[5] Arnold, V. I., Am. Math. Soc. Transl. 19 pp 267– (1969) ISSN: http://id.crossref.org/issn/0065-9290
[6] DOI: 10.1070/RM1971v026n02ABEH003827 · Zbl 0259.15011
[7] Arnold, V. I., in: Ordinary Differential Equations (1973)
[8] Arnold, V. I., in: Mathematical Methods of Classical Mechanics (1978) · Zbl 0386.70001
[9] Arnold, V. I., in: Dynamical Systems III (1993)
[10] Arnold, V. I., in: Ergodic Problems of Classical Mechanics (1968)
[11] DOI: 10.1098/rspa.1996.0062
[12] Bloch, A., Ann. Inst. Henri Poincare, Anal. Non Lineaire 11 pp 37– (1994) ISSN: http://id.crossref.org/issn/0294-1449
[13] DOI: 10.1007/BF02101622 · Zbl 0846.58048
[14] DOI: 10.1023/B:JOSS.0000022367.36305.d3 · Zbl 1066.70014
[15] Bloch, A. M., Proc. CDC 36 pp 2356– (1997)
[16] DOI: 10.1119/1.1519232
[17] Bolotin, V. V., in: Nonconservative Problems of the Theory of Elastic Stability (1963)
[18] DOI: 10.1023/A:1013039024380
[19] DOI: 10.1070/PU2003v046n04ABEH001306
[20] DOI: 10.1038/182760a0
[21] DOI: 10.1007/s002050050141 · Zbl 0965.76015
[22] Buttazzo, G., in: One-Dimensional Variational Problems (1998)
[23] DOI: 10.1111/j.1749-6632.1998.tb11252.x
[24] DOI: 10.1051/cocv:2002045 · Zbl 1070.70013
[25] Cherry, T. M., Trans. Cambridge Philos. Soc. 23 pp 165– (1925) ISSN: http://id.crossref.org/issn/0371-5779
[26] Chetayev, N. G., in: The Stability of Motion (1961)
[27] DOI: 10.1103/PhysRevE.64.067603
[28] DOI: 10.1119/1.10926
[29] Dacorogna, B., in: Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001
[30] Daleckii, J. L., in: Stability of Solutions of Differential Equations in Banach Space (1974)
[31] DOI: 10.1088/0951-7715/5/4/008 · Zbl 0783.58024
[32] DOI: 10.1088/0951-7715/15/3/301 · Zbl 1026.37048
[33] Dirichlet, G. L., Crelle 32 pp 85– (1846) · ERAM 032.0907cj
[34] Earnshaw, S., Trans. Cambridge Philos. Soc. 7 pp 97– (1842) ISSN: http://id.crossref.org/issn/0371-5779
[35] DOI: 10.1006/aphy.1995.1097 · Zbl 0836.70018
[36] Ekeland, I., in: Convexity Methods in Hamiltonian Mechanics (1990) · Zbl 0707.70003
[37] Evans, L. C., in: Partial Differential Equations (1998)
[38] Friedlander, S., Not. Am. Math. Soc. 46 pp 1358– (1999) ISSN: http://id.crossref.org/issn/0002-9920
[39] Galin, D. M., Am. Math. Soc. Transl. 118 pp 1– (1975) ISSN: http://id.crossref.org/issn/0065-9290 · Zbl 0495.58019
[40] Gantmacher, F. R., in: Lectures on Analytical Mechanics (1966)
[41] Gantmacher, F. R., in: The Theory of Matrices (1977)
[42] Goldstein, H., in: Classical Mechanics (1956)
[43] DOI: 10.1016/0167-2789(87)90087-X · Zbl 0612.58013
[44] DOI: 10.1103/PhysRevLett.68.2257 · Zbl 1050.37516
[45] Hagerty, P., in: Proceedings of the 38th CDC (1999)
[46] Hagerty, P., SIAM J. Appl. Math. 64 pp 484– (2003) ISSN: http://id.crossref.org/issn/0036-1399
[47] Herrman, G., Appl. Mech. Rev. 20 pp 103– (1967) ISSN: http://id.crossref.org/issn/0003-6900
[48] DOI: 10.1016/0370-1573(85)90028-6 · Zbl 0717.76051
[49] Hopf, E., Akad. Wiss. (Leipzig) 94 pp 3– (1942)
[50] DOI: 10.1002/1521-4001(200008)80:8<507::AID-ZAMM507>3.0.CO;2-5
[51] DOI: 10.1109/9.871756 · Zbl 0985.70015
[52] Kane, T. R., J. Appl. Mech. 45 pp 903– (1978) ISSN: http://id.crossref.org/issn/0021-8936
[53] Kapitsa, P. L., Zh. Tekh. Fiz. 9 pp 124– (1939) ISSN: http://id.crossref.org/issn/0044-4642
[54] Khalil, H. K., in: Nonlinear Systems (2001)
[55] DOI: 10.1016/0167-2789(94)90225-9 · Zbl 0814.76048
[56] DOI: 10.1134/1.163851
[57] Krasovskii, N. N., in: Stability of Motion (1963)
[58] DOI: 10.1016/j.physd.2005.12.003 · Zbl 1159.70357
[59] DOI: 10.1006/jsvi.2000.3137
[60] DOI: 10.1007/BF01048153 · Zbl 0747.58030
[61] DOI: 10.1109/9.59804 · Zbl 0709.93603
[62] DOI: 10.1016/0005-1098(95)00096-8 · Zbl 0847.93049
[63] Lin, C. C., in: The Theory of Hydrodynamic Stability (1955)
[64] Luo, Z.-H., in: Stability and Stabilization of Infinite Dimensional Systems with Applications (1999)
[65] DOI: 10.1016/0375-9601(91)90480-V
[66] Marsden, J. E., in: Lectures on Mechanics (1992) · Zbl 0744.70004
[67] DOI: 10.1007/BF00914351 · Zbl 0778.70016
[68] Marsden, J. E., Contemp. Math. 97 pp 297– (1989) ISSN: http://id.crossref.org/issn/0271-4132
[69] Mawhin, J., in: Critical Point Theory and Hamiltonian Systems (1989) · Zbl 0676.58017
[70] Merkin, D. R., in: Gyroscopic Systems (1974) · Zbl 0308.70008
[71] Merkin, D. R., in: Introduction to the Theory of Stability (1997)
[72] DOI: 10.1016/0167-2789(86)90209-5 · Zbl 0661.70025
[73] DOI: 10.1103/RevModPhys.70.467 · Zbl 1205.37093
[74] DOI: 10.1006/icar.1994.1198
[75] DOI: 10.1098/rspa.1998.0174 · Zbl 0913.70008
[76] Nijmeijer, H., in: Nonlinear Dynamical Control Systems (1996)
[77] Nikolai, E. L., in: Theoretical Mechanics (1939)
[78] O’Brien, S., Proc. R. Ir. Acad., Sect. A 56 pp 23– (1953) ISSN: http://id.crossref.org/issn/0035-8975
[79] DOI: 10.1137/S0036139992235123 · Zbl 0805.70008
[80] Pedlosky, J., in: Geophysical Fluid Dynamics (1987) · Zbl 0713.76005
[81] Phillips, N. A., J. Meteorol. 8 pp 381– (1951) ISSN: http://id.crossref.org/issn/0095-9634
[82] Pollard, H., in: Mathematical Introduction to Celestial Mechanics (1966) · Zbl 0141.23803
[83] DOI: 10.1002/cpa.3160310203
[84] DOI: 10.1175/1520-0469(1977)034<1689:TEOFAO>2.0.CO;2
[85] Routh, E. J., in: Elementary Rigid Dynamics (1913)
[86] Rumyantsev, V. V., J. Appl. Math. Mech. 57 pp 1101– (1994) ISSN: http://id.crossref.org/issn/0021-8928 · Zbl 0808.70018
[87] DOI: 10.1007/BF01291002 · Zbl 0030.22103
[88] Seyranian, A. P., in: Multiparameter Stability Theory with Mechanical Applications (2003) · Zbl 1047.34063
[89] Siegel, C. L., in: Lectures on Celestial Mechanics (1971) · Zbl 0312.70017
[90] DOI: 10.1007/BF01881678 · Zbl 0738.70010
[91] DOI: 10.1119/1.18488
[92] DOI: 10.1007/BF01418778 · Zbl 0202.23201
[93] DOI: 10.1007/s002220050303 · Zbl 0910.35107
[94] Struwe, M., in: Variational Methods (1990)
[95] Szebehely, V. G., in: Theory of Orbits: The Restricted Problem of Three Bodies (1967)
[96] Thomson, W., in: Treatise on Natural Philosophy: Part 1 (1879)
[97] DOI: 10.1088/0951-7715/3/4/005 · Zbl 0719.58027
[98] DOI: 10.1016/0362-546X(86)90086-6 · Zbl 0613.93049
[99] Walker, J., Sci. Am. 241 pp 144– (1979) ISSN: http://id.crossref.org/issn/0036-8733
[100] DOI: 10.1007/BF01209527 · Zbl 0800.93563
[101] Wang, L.-S., Celest. Mech. Dyn. Astron. 50 pp 349– (1991) ISSN: http://id.crossref.org/issn/0923-2958 · Zbl 0737.70003
[102] DOI: 10.2307/1971185 · Zbl 0403.58001
[103] DOI: 10.1215/S0012-7094-43-01016-6 · Zbl 0061.37207
[104] Whittaker, E. T., in: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (1917) · Zbl 0665.70002
[105] Wiggins, S., in: Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003) · Zbl 1027.37002
[106] Wintner, A., in: The Analytical Foundations of Celestial Mechanics (1947) · Zbl 0041.59006
[107] Yudovich, V. I., in: The Linearization Method in Hydrodynamic Stability Theory (1989) · Zbl 0727.76039
[108] Zajac, E. E., J. Astronaut. Sci. 11 pp 46– (1964) ISSN: http://id.crossref.org/issn/0021-9142
[109] DOI: 10.1007/BF00536796 · Zbl 0047.42606
[110] DOI: 10.1007/BF02067575 · Zbl 0051.16304
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.