By variational methods, we study the motions of a relativistic particle under the action of an external scalar potential. We consider standard static spacetimes whose metric coefficients grow at most quadratically at infinity. Fixing suitable values for the energy, we obtain timelike trajectories.
R. Bartolo and A. M. Candela, “Quadratic Bolza problems in static spacetimes with critical asymptotic behavior,” Mediterranean Journal of Mathematics, vol. 2, no. 4, pp. 459–470, 2005.
View at: Publisher Site | Google Scholar | MathSciNetR. Bartolo, A. M. Candela, J. L. Flores, and M. Sánchez, “Geodesics in static Lorentzian manifolds with critical quadratic behavior,” Advanced Nonlinear Studies, vol. 3, no. 4, pp. 471–494, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetR. Bartolo and A. Germinario, “Timelike spatially closed trajectories under a potential on splitting Lorentzian manifolds,” Communications in Applied Analysis, vol. 9, no. 2, pp. 177–205, 2005.
View at: Google Scholar | MathSciNetR. Bartolo, A. Germinario, and M. Sánchez, “Convexity of domains of Riemannian manifolds,” Annals of Global Analysis and Geometry, vol. 21, no. 1, pp. 63–84, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian Geometry, vol. 202 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 2nd edition, 1996.
View at: Zentralblatt MATH | MathSciNetV. Benci and D. Fortunato, “On the existence of infinitely many geodesics on space-time manifolds,” Advances in Mathematics, vol. 105, no. 1, pp. 1–25, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetV. Benci, D. Fortunato, and F. Giannoni, “On the existence of multiple geodesics in static space-times,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 8, no. 1, pp. 79–102, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNetV. Benci, D. Fortunato, and F. Giannoni, “On the existence of geodesics in static Lorentz manifolds with singular boundary,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 19, no. 2, pp. 255–289, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. M. Candela, J. L. Flores, and M. Sánchez, “A quadratic Bolza-type problem in a Riemannian manifold,” Journal of Differential Equations, vol. 193, no. 1, pp. 196–211, 2003.
View at: Publisher Site | Google Scholar | MathSciNetE. Fadell and S. Husseini, “Category of loop spaces of open subsets in Euclidean space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 17, no. 12, pp. 1153–1161, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. Masiello, Variational Methods in Lorentzian Geometry, vol. 309 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1994.
View at: Zentralblatt MATH | MathSciNetB. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, 1983.
View at: Zentralblatt MATH | MathSciNetR. S. Palais, “Lusternik-Schnirelman theory on Banach manifolds,” Topology, vol. 5, pp. 115–132, 1966.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. Romero and M. Sánchez, “Completitud de las soluciones de una ecuación diferencial que extiende a la de las geodésicas: movimiento de las partículas en un campo de fuerzas,” preprint.
View at: Google ScholarM. Sánchez, “Some remarks on causality theory and variational methods in Lorenzian manifolds,” Conferenze del Seminario di Matematica dell'Università di Bari, vol. 265, pp. 1–12, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetE. W. C. van Groesen, “Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,” Journal of Mathematical Analysis and Applications, vol. 132, no. 1, pp. 1–12, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet