Periodic solutions in the singular logarithmic potential
DOI:
https://doi.org/10.4067/S0716-09172007000200003Keywords:
Periodic solutions, Logarithmic potential, Symmetry.Abstract
We consider the singular logarithmic potential 
, 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.
References
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[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).
[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).
Published
2017-04-18
How to Cite
[1]
C. Vidal, “Periodic solutions in the singular logarithmic potential”, Proyecciones (Antofagasta, On line), vol. 26, no. 2, pp. 189-206, Apr. 2017.
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