The existence of homoclinic motions. (English. Russian original) Zbl 0549.58019

Mosc. Univ. Math. Bull. 38, No. 6, 117-123 (1983); translation from Vestn. Mosk. Univ., Ser. I 1983, No. 6, 98-103 (1983).
A motion of a Hamiltonian system is said to be homoclinic, or doubly- periodic, if it tends asymptotically to a periodic motion as \(t\to\pm \infty\). The main result of the concisely written paper is presented in the form of the following theorem: Let \(\gamma\) be a minimal periodic motion with non-vanishing characteristic determinants. Then a homoclinic motion to \(\gamma\) exists. Morse theory is employed to prove the theorem. To understand the paper the reader should be acquainted with two previous contributions by the author [Vestn. Mosk. Univ., Ser. I 1978, No.6, 72-77 (1978; Zbl 0403.34053)] and by V. V. Kozlov and the author [ibid. 1980, No.4, 84-89 (1980; Zbl 0439.70020)].
Reviewer: W.R.Bielski


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H99 Hamiltonian and Lagrangian mechanics
49S05 Variational principles of physics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces