Inversion Problem for the Lagrange Theorem on the Stability of Equilibrium and Related Problems

  • Valery V. Kozlov
  • Stanislav D. Furta
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we consider problems that we will solve by a method that we originally used for proving instability in cases where linearized equations alone were insufficient. This first section will serve mostly as an introduction. Here we will outline a range of problems and formulate a number of theorems on stability for which converse assertions are introduced in the final sections, their proof being based on the construction of asymptotic solutions. In these theorems the principal stability condition will be the presence of an isolated minimum of some function that plays the role of the potential energy of the system.


Asymptotic Stability Equilibrium Position Asymptotic Solution Invariant Manifold Center Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 3.
    Arnol’d, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv. 18(6), 85–191 (1963). Translated from: Usp. Mat. Nauk. 18(6) (114), 91–192 (1963)Google Scholar
  2. 18.
    Bogolyubov, N.N., Mitropolsky, Y.O.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)zbMATHGoogle Scholar
  3. 25.
    Bolotin, S., Negrini, P.: Asymptotic solutions of Lagrangian systems with gyroscopic forces. Nonlinear Differ. Equ. Appl. 2(4), 417–444 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 34.
    Bulatovich, R., Kazhich, M.: Asymptotic solutions of equations of motion of nonholonomic Chaplygin systems (Russian). Mathematica Montesnigri 2, 11–20 (1993). MR 1284893Google Scholar
  5. 40.
    Chetayev, N.G.: The Stability of Motion. Pergamon, Oxford/New York/Paris (1961)Google Scholar
  6. 53.
    Furta, S.D.: On the asymptotic solutions of the equations of motion of mechanical systems. J. Appl. Math. Mech. 50, 726–731 (1986). Translated from: Prikl. Mat. Mekh. 50, 938–944 (1986)Google Scholar
  7. 54.
    Furta, S.D.: Instability of the equilibrium states of nonnatural conservative systems (Russian). In: The Method of Lyapunov Functions in the Analysis of the Dynamics of Systems, pp. 203–206. Nauka, Novosibirsk (1987)Google Scholar
  8. 55.
    Furta, S.D.: The instability of equilibrium of a gyroscopic system with two degrees of freedom (Russian). Vestn. MGU, Ser. I Mat. Mekh. (5), 100–101 (1987). MR 913275Google Scholar
  9. 57.
    Furta, S.D.: Asymptotic trajectories of mechanical systems and the problem of the inversion of the Routh theorem on stability of equilibrium (Russian). In: Differential and Integral Equations: Methods of Topological Dynamics, pp. 80–86. GGU, Gor’kiy (1989). Zbl 0800.35002Google Scholar
  10. 58.
    Furta, S.D.: Asymptotic trajectories of natural systems under the action of forces of viscous friction (Russian). In: Analytic and Numerical Methods for Studying Mechanical Systems, pp. 35–38. MAI, Moscow (1989)Google Scholar
  11. 63.
    Furta, S.D.: Instability of the equilibrium positions of restricted mechanical systems. Sov. Appl. Mech. 27(2), 204–208 (1991). Translated from: Prikl. Mekh. (Kiev) 27(2), 43–46 (1991)Google Scholar
  12. 66.
    Gantmacher, F.R.: The Theory of Matrices, vols. 1, 2. Chelsea, New York (1959)Google Scholar
  13. 67.
    Gilmore, R.: Catastrophe Theory for Scientists and Engineers. Wiley, New York (1981)zbMATHGoogle Scholar
  14. 83.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge/New York (1991)zbMATHCrossRefGoogle Scholar
  15. 96.
    Karapetyan, A.V., Rumyancev, V.V.: Stability of conservative and dissipative systems (Russian). Itogi Nauki i Tekhniki, Obshchaya Mekhanika 6, 3–128 (1983). MR 766567Google Scholar
  16. 100.
    Khazin, L.G., Shnol’, E.E.: Stability of Critical Equilibrium States. Manchester University Press, Manchester (1991). Publication source: Nauchn. Tsentr Biol. Issled/Akad. Nauk. SSSR, Pushchino (1985)Google Scholar
  17. 104.
    Kozlov, V.V.: Asymptotic solutions of equations of classical mechanics. J. Appl. Math. Mech. 46(4), 454–457 (1983). Translated from: Prikl. Mat. Mekh. 46, 573–577 (1982)Google Scholar
  18. 105.
    Kozlov, V.V.: On the stability of equilibria of nonholonomic systems. Sov. Math. Dokl. 33, 654–656 (1986). Translated from: Dokl. Akad. Nauk. SSSR 288(2), 289–291 (1986)Google Scholar
  19. 106.
    Kozlov, V.V.: Asymptotic motions and the inversions of the Lagrange-Dirichlet theorem. J. Appl. Math. Mech. 50(6), 719–725 (1987). Translated from: Prikl. Mat. Mekh. 50(6), 928–937 (1986)Google Scholar
  20. 108.
    Kozlov, V.V.: On a Kelvin problem. J. Appl. Math. Mech. 53(1), 133–135 (1989). Translated from: Prikl. Mat. Mekh. 53(1), 165–167 (1989)Google Scholar
  21. 110.
    Kozlov, V.V.: On equilibrated nonholonomic systems (Russian). Vestn. Mosk. Univ. Ser. I Mat. Mekh. (3), 74–79 (1994). MR 1315728Google Scholar
  22. 111.
    Kozlov, V.V.: The asymptotic motions of systems with dissipation. J. Appl. Math. Mech. 58(5), 787–792 (1994). Translated from: Prikl. Mat. Mekh. 58(5), 31–36 (1994)Google Scholar
  23. 112.
    Kozlov, V.V.: Symmetries, Topology and Resonances in Hamiltonian Mechanics. Springer, New York (1996)Google Scholar
  24. 114.
    Kozlov, V.V.: Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties. Dokl. Math. 77(3), 412–415 (2008). Translated from Dokl. Akad. Nauk., Ross. Akad. Nauk. 420(4), 447–450 (2008)Google Scholar
  25. 117.
    Kozlov, V.V., Palamodov, V.P.: On asymptotic solutions of equations of classical mechanics. Sov. Math. Dokl. 25, 335–339 (1982). Translated from: Dokl. Akad. Nauk. SSSR 263(2), 285–289 (1982)Google Scholar
  26. 124.
    Krasovskiy, N.N.: Certain Problems in the Theory of Stability of Motion (Russian). Gos, Mosc. Izdat. Fiz.-Mat. Lit. (1959) MR 0106313Google Scholar
  27. 127.
    Laloy, M.: On equilibrium instability for conservative and partially dissipative systems. Int. J. Non-Linear Mech. 11, 295–301 (1976)zbMATHCrossRefGoogle Scholar
  28. 129.
    Lavrent’ev, M.A.: Methods of the Theory of Functions of a Complex Variable (Russian). Nauka, Moscow (1987). MR 1087298Google Scholar
  29. 131.
    Lyapunov, A.M.: Sur l’instabilité de équilibre dans certains cas où la fonction de forces n’est pas un maximum. J. Math. V 3, 81–94 (1897)Google Scholar
  30. 133.
    Lyapunov, A.M.: The General Problem of the Stability of Motion. Taylor & Francis, London (1992)zbMATHGoogle Scholar
  31. 137.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)zbMATHCrossRefGoogle Scholar
  32. 138.
    Matveev, M.V.: Lyapunov stability of equilibrium states of reversible systems. Math. Notes 57(1), 63–72 (1995). Translated from: Mat. Zametki 57(1), 90–104 (1995)Google Scholar
  33. 141.
    Moser, J.: Lectures on Hamiltonian Systems. American Mathematical Society, Providence (1968)Google Scholar
  34. 144.
    Neymark, Y.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, vol. 33. American Mathematical Society, Providence (1972). Publication source: Moscow, Nauka (1967)Google Scholar
  35. 148.
    Palamodov, V.P.: Stability of motion and algebraic geometry. In: Kozlov V.V. (ed.) Dynamical Systems in Classical Mechanics. Translations Series 2, vol. 168, pp. 5–20. American Mathematical Society, Providence (1995)Google Scholar
  36. 149.
    Peiffer, K., Carlier, P.: A remark on the inversion of Lagrange-Dirichlet’s theorem. Coll. Math. Soc. Janos Bolyai 53, Qual. Theor. Differ. Equ. Szeged 473–484 (1988)Google Scholar
  37. 154.
    Rouche, N., Habets, P., Laloy, M.: Stability by Ljapunov’s Direct Method. Springer, New York (1977)CrossRefGoogle Scholar
  38. 155.
    Rumyantsev, V.V.: On stability of motion of nonholonomic systems. J. Appl. Math. Mech. 31, 282–293 (1967). Translated from: Prikl. Mat. Mekh. 31, 260–271 (1967)Google Scholar
  39. 156.
    Rumyantsev, V.V., Sosnitskiy, S.P.: The stability of the equilibrium of holonomic conservative systems. J. Appl. Math. Mech. 57(6), 1101–1122 (1993). Translated from: Prikl. Mat. Mekh. 57(6), 143–166 (1993)Google Scholar
  40. 157.
    Salvadori, L.: Criteri di instabilità per i moti merostatici di un sistema olonomo. Rendiconti. Accademia delle scienze fisiche e matematiche. Società nazionale di Scienze. Lettere ed Arti di Napoli 27(4), 535–542 (1960)MathSciNetzbMATHGoogle Scholar
  41. 158.
    Salvadori, L.: Sull’estensione ai sistemi dissipativi del criterio di stabilità del Routh. Ricerche di Matematica 15, 162–167 (1966)MathSciNetzbMATHGoogle Scholar
  42. 159.
    Salvadori, L.: On the conditional total stability of equilibrium for mechanical systems. Le Mat. 46, 415–427 (1991)MathSciNetzbMATHGoogle Scholar
  43. 160.
    Salvadori, L.: Stability problems for holonomic mechanical systems. In: La mechanique analitique de Lagrange et son héritage II. Atti della Accademia delle Scienze di Torino, suppl. no. 2 al vol. 126, pp. 151–168. Classe di Scienze Fisiche, Matematiche e Naturali (1992)Google Scholar
  44. 161.
    Salvadori L., Visentin F.: Sulla stabilità totale condizionata nella meccanica dei sistemi olonomi. Rendiconti di Matematica e delle sue Applicazioni 12, 475–495 (1992)MathSciNetzbMATHGoogle Scholar
  45. 174.
    Smale, S.: On gradient dynamical systems. Ann. Math. 74, 199–206 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 176.
    Sokol’skiy, A.G.: On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance. J. Appl. Math. Mech. 41(1), 20–28 (1977). Translated from Prikl. Mat. Mekh. 41(1), 24–33 (1977)Google Scholar
  47. 183.
    Tamm, I.E.: Fundamentals of the Theory of Electricity (Russian). Nauka, Moscow (1966)Google Scholar
  48. 193.
    Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Dover, New York (1944)zbMATHGoogle Scholar
  49. 196.
    Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Valery V. Kozlov
    • 1
  • Stanislav D. Furta
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Faculty for Innovative and Technological BusinessRussian Academy of National Economy and Public AdministrationMoscowRussia

Personalised recommendations