References
M. A. Balitinov, On instability of the position of equilibrium of a Hamiltonian system, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 42 (1978), 582–587.
S. V. Bolotin & V. V. Kozlov, on the asymptotic solutions of the equations of dynamics, Moscow Univ. Math. Bull. 35 (1980), 82–88.
S. D. Furta, On asymptotic solutions of the equations of motion of mechanical sytems, Prikl. Mat. Mekh. 50 (1986), 726–730.
P. Hagedorn, Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal. 42 (1971), 281–316.
P. Hagedorn, Eine zusätzliche Bemerkung zu meiner Arbeit: Die Umkehrung der Stabilitätssätze von Lagrange-Dirichlet und Routh, Arch. Rational Mech. Anal. 47 (1972), 395.
A. V. Karapetjan, On the inversion of the Routh theorem, Vestnik Moskov Univ. Ser. 1 Mat. Meh. 28 (1973), 65–69.
V. V. Kozlov, Instability of equilibrium in a potential field, Russian Math. Surveys 36 (1981), 238–239.
V. V. Kozlov, On the instability of equilibrium in a potential field, Russian Math. Surveys 36 (1981), 256–257.
V. V. Kozlov, Asymptotic solutions of equations of classical mechanics, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 46 (1982), 454–457.
V. V. Kozlov, Calculus of variations in the large and classical mechanics, Russian Mathematics Survey 40 (1985), 37–71.
V. V. Kozlov, Asymptotic motions and the inversion of the Lagrange-Dirichlet Theorem, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 50 (1987), 719–725.
V. V. Kozlov & V. V. Palamodov, On asymptotic solutions of the equations of classical mechanics, Soviet Math. Dokl. 25 (1982), 335–339.
J. L. Lagrange, Mécanique analytique, Paris, 1788.
M. Laloy, On the inversion of Lagrange-Dirichlet Theorem in the case of an analytical potential, Report 107 SMAM, Univ. Catholique de Louvain, 1977.
M. Laloy & P. Habets, On the instability of a two-dimensional conservative mechanical system with analytical potential, in Equadiff 78, Firenze, 1978, 149–164.
M. Laloy & K. Peiffer, On the instability of equilibrium when the potential has a non-strict local minimum, Arch. Rational Mech. Anal. 78 (1982), 213–222.
G. Lejeune Dirichlet, Ueber die Stabilität des Gleichgewichts, J. reine angew. Math. 32 (1846), 85–88.
E. A. Liubushin, On instability of equilibrium when the force function is not a maximum, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 44 (1980), 158–162.
J. Mawhin & M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
C. Maffei, V. Moauro & P. Negrini, On the inversion of the Lagrange-Dirichlet Theorem in a case of nonhomogeneous potential, Diff. Integral Eqs. 4 (1991), 767–782.
V. Moauro & P. Negrini, On the inversion of the Lagrange-Dirichlet Theorem, Diff. Integral Eqs. 2 (1989), 471–478.
P. Painlevé, Sur la stabilité de l'équilibre, C.R. Acad. Sci. Paris 138 (1904), 1170–1174.
V. V. Palamodov, Stability of equilibrium in a potential field, Functional Anal. Appl. 11 (1978), 227–289.
K. Peiffer, An example of non isolated equilibrium with maximum potential, stabilized by dissipative forces, Z. angew. Math. Phys. 30 (1979), 835–837.
K. Peiffer, On the inversion of Lagrange-Dirichlet Theorem, RSM UCL 162, 1989, to appear in Prikl. Mat. Mekh. (shorter version in The Lyapunov Functions, Method and Applications, Borne & Matrosov eds., Baltzer AG, 1990, 9–13).
K. Peiffer & P. Carlier, A remark on the inversion of Lagrange-Dirichlet's theorem, Qualitative theory of Differential Equations, Szeged, 1988; Coll. Math. Soc. J. Bolyai 53, Noordhoff, 1990, 473–484.
N. Rouche, P. Habets & M. Laloy, Stability Theory by Liapunov's Direct Method, Springer, New York, 1977.
V. V. Rumiantsev, On stability of motions of conservative systems, in Qualitative Theory of Differential Equations, Szeged, 1979, Coll. Math. Soc. J. Bolyai, 30 vol. II, North Holland, Amsterdam, 1981, 865–901.
M. Sofer, On the inversion of the Lagrange-Dirichlet stability Theorem — mechanical and generalized systems, Z. angew. Math. Phys. 34 (1983), 1–12.
S. D. Taliaferro, An inversion of the Lagrange-Dirichlet stability Theorem, Arch. Rational Mech. Anal. 73 (1980), 183–190.
S. D. Taliaferro, Stability for two-dimensional analytic potentials, J. Diff. Eqs. 35 (1980), 248–265.
S. D. Taliaferro, Instability of an equilibrium in a potential field, Arch. Rational Mech. Anal. (1990), 183–194.
E. W. C. Van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1–12.
E. W. C. Van Groesen, Hamiltonian flow on an energy integral: 240 years after the Euler-Maupertuis principle, Proc. Sixth Scheveningen Conference, 1984.
V. A. Vladimirov & V. V. Rumiantsev, On the inversion of Lagrange's Theorem for a rigid body with a cavity containing an ideal liquid, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 53 (1989), 608–612.
V. A. Vladimirov & V. V. Rumiantsev, Inversion of Lagrange's Theorem for a rigid body with a cavity containing a viscous liquid, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 54 (1990), 154–163.
A. Winter, The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press, Princeton, 1941.
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Hagedorn, P., Mawhin, J. A simple variational approach to a converse of the Lagrange-Dirichlet theorem. Arch. Rational Mech. Anal. 120, 327–335 (1992). https://doi.org/10.1007/BF00380318
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DOI: https://doi.org/10.1007/BF00380318