Periodic solutions in the singular logarithmic potential


  • Claudio Vidal Universidad del Bío-Bío.



Periodic solutions, Logarithmic potential, Symmetry.


We consider the singular logarithmic potential periodic.JPG, a potential which plays an important role in the modelling of triaxial systems, such as elliptical galaxies or bars in the centres of galaxy discs. Using properties of the central field in the axis-symmetric case we obtain periodic solutions which are symmetric with respect to the origin for weak anisotropies. Also we generalize our result in order to include more general perturbations of the logarithmic potential.

Author Biography

Claudio Vidal, Universidad del Bío-Bío.

Departamento de Matemática, Facultad de Ciencias.


[1] Arnold, V. Mathematical Methods of Classical Mechanics, Springer Verlag, Berlin Heidelberg, New York, (1978).

[2] Azevêdo, C. Dinâmica do problema do fio circular homogñeo, Thesis for Degree of Doctor in Mathematics, Universidade Federal de Pernambuco. Brazil, (2002)

[3] Azevêdo, C. and Ontaneda, P. Continuous symmetric perturbations of planar power law forces, J. Diff. Eq., 211, pp. 20-37, (2005).

[4] Birkhoff, G. ‘The restricted problem of three bodies. Rend. Circ. Mat. Palermo, 30, pp. 1-70, (1915).

[5] Boccaletti, D. and Pucaco, G. Theory of orbits, Springer-Verlag, Berlin Heidelberg, New York, (1996).

[6] Cabral, H. and Vidal, C. Periodic solutions of symmetric perturbations of the Kepler problem. J. Diff. Eq., 163, pp. 76-88, (2000).

[7] Caranicolas, N., and Barbanis, B. ‘Periodic orbits in nearly axisymmetric stellar systems’. Astron. and Astroph., 114, pp. 360-366, (1982).

[8] Caranicolas, N., and Vozikis, CH. ‘Orbital characterizations of dynamical models of elliptical galaxies’, Celest. Mech., 39, pp. 85-102, (1986).

[9] Casasayas, J. and Llibre, J.: ‘Qualitative analysis of the Anisotropic Kepler problem’, Memoirs Amer. Math. Soc., 314, (1984).

[10] Piccinini, L., Stampacchia, G. and Vidossich, G. Ordinary Differential Equations in IRn, Aplied Mathematical Sciences, 39, Springer-Verlag, (1978).

[11] Santoprete, M. ‘Symmetric periodic solutions of the anisotropic Manev problem’, Journal of Mathematical Physics, 43, pp. 3207-3219, (2002).

[12] Stoica, C. and Font, A. ‘Global dynamics in the singular logarithmic potential’, J. Phys. A: Math. Gen., 36, pp. 7693-7714, (2003).

[13] Schwarzschild, M. Astrophys. J, 232, 236, (1979).

[14] Vidal, C. ‘Periodic solutions for any planar symmetric perturbation of the Kepler problem’, Celest. Mech., 80, pp. 119-132, (2001).

[15] Vidal, C.: ‘Periodic solutions of symmetric perturbations of Gravitational potentials, J. Dyn. Diff. Eqs., 17, 1, pp. 85-114, (2005).



How to Cite

C. Vidal, “Periodic solutions in the singular logarithmic potential”, Proyecciones (Antofagasta, On line), vol. 26, no. 2, pp. 189-206, Apr. 2017.