×

On a class of dynamical systems with singular potential. (English) Zbl 0744.34039

The authors consider the nonlinear differential system (*) \(-\ddot x=\nabla V(x)\), where \(x(t)\in C^ 2(R,R^ n)\), \(V\in C^ 1(R/\{0\},R)\) and \(V\) behaves like \(-k/| x|^ \alpha \) near the origin.
Their aim is to search for \(T\)-periodic solutions (with \(T\) prescribed) of (*). In literature this problem has been studied mainly by applying variational methods and considering two different situations: (i) \(1\leq \alpha < 2\), (ii) \(\alpha \geq 2\). In the present paper the particular case when \(\alpha=1\) is examined (assuming moreover that \(n=2\)); this case, as far as the authors know, was considered in one paper only [A. Ambrosetti and V. Coti Zelati, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 287-295 (1988; Zbl 0667.58055)]. Two new theorems concerning the existence and the properties of periodic solutions of (*) are proved.

MSC:

34C25 Periodic solutions to ordinary differential equations

Citations:

Zbl 0667.58055
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrosetti, A.; Coti Zelati, V., Periodic solutions of singular dynamical systems, (Rabinowitz, P. H., Periodic Solutions of Hamiltonian Systems and Related Topics, V209 (1987), D. Reidel), 1-10, NATO ASI series · Zbl 0757.70007
[2] Ambrosetti, A.; Coti Zelati, V., Solutions periodiques sans collision pour une classe de potentiels de type Keplerian, C.r. Acad. Sci. Paris, 305, 813-815 (1987) · Zbl 0639.34038
[3] Ambrosetti, A.; Coti Zelati, V., Noncollision orbits for a class of Keplerian-like potentials, Ann. Inst. H. Poincaré Analyse non Linéaire, 5, 287-295 (1988) · Zbl 0667.58055
[6] Benci, V.; Giannoni, F., Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials,, J. diff. Eqns, 82, 60-70 (1989) · Zbl 0689.34034
[7] Capozzi, A.; Salvatore, A.; Singh, Periodic solutions of Hamiltonian systems: the case of the singular potential, Proc. NATO-ASI, 207-216 (1986) · Zbl 0601.58031
[8] Capozzi, A.; Greco, C.; Salvatore, A., Lagrangian systems in presence of singularities, Proc. Am. math. Soc., 102, 125-130 (1988) · Zbl 0664.34054
[9] Coti Zelati, V., Remarks on dynamical systems with weak forces, Manuscripta math., 57, 417-424 (1987) · Zbl 0606.58039
[10] Coti Zelati, V., Dynamical systems with effective-like potential, Nonlinear Analysis, 12, 209-222 (1988) · Zbl 0648.34050
[11] Degiovanni, M.; Giannoni, F., Periodic solutions of dynamical systems with Newtonian type potentials, Ann. Scu. norm. sup. Pisa, 15, 467-494 (1988) · Zbl 0692.34050
[12] Degiovanni, M.; Giannoni, F.; Marino, A., Periodic solutions of dynamical systems with Newtonian type potentials, Atti Accad. naz. Lincei Rc., LXXXI, 271-278 (1987) · Zbl 0667.70010
[13] Degiovanni, M.; Giannoni, F., Nonautonomous perturbations of Newtonian potential, Proc. Nonlinear Analysis and Variational Problems (1986), Isola d’Elba · Zbl 0682.34031
[14] Gordon, W., A minimizing property of Keplerian orbits, Am. J. Math., 99, 961-971 (1975) · Zbl 0378.58006
[15] Gordon, W., Conservative dynamical systems involving strong forces, Trans. Am. math. Soc., 204, 113-135 (1975) · Zbl 0276.58005
[16] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis, 12, 259-269 (1988) · Zbl 0648.34048
[17] Solimi, S., On forced dynamical systems with a singularity of repulsive type, 14, 489-500 (1990) · Zbl 0708.34041
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.