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Motion of Charged Particles in a Line

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Mathematical Models with Singularities

Part of the book series: Atlantis Briefs in Differential Equations ((ABDE,volume 1))

Abstract

Let us consider a fixed electrical charge \(Q_1\) placed at the origin of the real line and a point particle with charge \(Q_2\) moving on \({\mathbb R}^+\) and subjected to an external \(T\)-periodic excitation \(h(t)\). The main objective of this chapter is to unveil the main dynamical aspects of this model.

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References

  1. Blackmore, D., Rosato, A., Tricoche, X., Urban, K., Ratnaswamy, V.: Tapping dynamics for a column of particles and beyond. J. Mech. Mater. Struct. 6, 71–86 (2011)

    Article  Google Scholar 

  2. Blackmore, D., Rosato, A., Tricoche, X., Urban, K., Zou, L.: Analysis, simulation and visualization of 1D tapping via reduced dynamical models. Phys. D 273–274, 14–27 (2014)

    Article  MathSciNet  Google Scholar 

  3. Brown, M.: Homeomorphisms of two-dimensional manifolds. Houst. J. Math. 11, 455–469 (1985)

    MATH  Google Scholar 

  4. Campos, J., Torres, P.J.: On the structure of the set of bounded solutions on a periodic Lienard equation. Proc. Am. Math. Soc. 127, 1453–1462 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Caraballo, T., Cheban, D.: Almost periodic and asymptotically almost periodic solutions of Liénard equation. Discret. Contin. Dyn. Syst. Ser. B 16(3), 703–717 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cesari, L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  7. Cieutat, P.: On the structure of the set of bounded solutions on an almost periodic Liénard equation. Nonlinear Anal. 58, 885–898 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cieutat, P., Fatajou, S., N’Guerekata, G.M.: Bounded and almost automorphic solutions of Lienard equation with a singular nonlinearity. Electron. J. Qual. Theory Differ. Equ. 21, 1–15 (2008)

    MathSciNet  Google Scholar 

  9. Corbera, M., Llibre, J.: Periodic orbits of a collinear restricted three-body problem. Celest. Mech. Dynam. Astron. 86, 163–183 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dietrich, S.: Wetting phenomena. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena. Academic Press, London (1988)

    Google Scholar 

  11. Gazzola, F.: Periodic motions of a lattice of particles with singular forces. Differ. Integral Equ. 10, 245–264 (1997)

    MATH  MathSciNet  Google Scholar 

  12. Gordon, W.B.: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113–135 (1975)

    Article  MATH  Google Scholar 

  13. Fonda, A., Manásevich, R., Zanolin, F.: Subharmonic solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24, 1294–1311 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Forbat, N., Huaux, A.: Détermination approchée et stabilité locale de la solution périodique d’une équation différentielle non linéaire. Mém. et Public. Soc. Sci. Artts Letters du Hainaut 76, 3–13 (1962)

    MathSciNet  Google Scholar 

  15. Habets, P., Sanchez, L.: Periodic solutions of some lienard equations with singularities. Proc. Am. Math. Soc. 109, 1035–1044 (1990)

    MATH  MathSciNet  Google Scholar 

  16. Halk, R., Torres, P.J.: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248, 111–126 (2010)

    Article  Google Scholar 

  17. Hakl, R., Zamora, M.: On the open problems connected to the results of Lazer and Solimini. Proc. R. Soc. Edinb. Sect. A 144(1), 109–118 (2014)

    Article  MathSciNet  Google Scholar 

  18. Huaux, A.: Sur l’existence d’une solution périodique de l’équation différentielle non linéaire \(x^{\prime \prime }+0,2x^{\prime }+x/(1-x)=(0,5)\cos \omega t\). Bull. Cl. Sci. Acad. R. Belgique 5(48), 494–504 (1962)

    MathSciNet  Google Scholar 

  19. Lazer, A.C., Solimini, S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Martínez-Amores, P., Torres, P.J.: Dynamics of a periodic differential equation with a singular nonlinearity of attractive type. J. Math. Anal. Appl. 202, 1027–1039 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nagumo, M.: On the periodic solution of an ordinary differential equation of second order, Zenkoku Shijou Suugaku Danwakai, 54–61(1944) (in Japanese). English translation in Mitio Nagumo collected papers, Springer, Berlin (1993)

    Google Scholar 

  22. Njoku, F.I., Omari, P.: Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions. Appl. Math. Comput. 135, 471–490 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ortega, R.: Linear motions in a periodically forced Kepler problem. Port. Math. (N.S.) 68, 149–176 (2011)

    Article  MATH  Google Scholar 

  24. Parry, A.O., Rascón, C., Wood, A.J.: Universality for 2D wedge wetting. Phys. Rev. Lett. 83(26), 5535–5538 (1999)

    Article  Google Scholar 

  25. Rascón, C., Parry, A.O.: Wetting on non-planar and heterogeneous substrates. J. Phys.: Condens. Matter 12, A369–A374 (2000)

    Google Scholar 

  26. Rascón, C., Parry, A.O., Sartori, A.: Wetting at nonplanar substrates: unbending and unbinding. Phys. Rev. E 59(5), 5697–5700 (1999)

    Article  Google Scholar 

  27. Ruiz-Herrera, A., Torres, P.J.: Periodic solutions and chaotic dynamics in forced impact oscillators. SIAM J. Appl. Dyn. Syst. 12(1), 383–414 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rachůnková, I., Tvrdý, M.: Vrkoc̆ I.: Existence of nonnegative and nonpositive solutions for second-order periodic boundary-value problems. J. Differ. Equ. 176, 445–469 (2001)

    Article  MATH  Google Scholar 

  29. Sperling, H.J.: The collision singularity in a perturbed two-body problem. Celest. Mech. 1, 213–221 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  30. Torres, P.J.: Bounded solutions in singular equations of repulsive type. Nonlinear Anal. T.M.A. 32, 117–125 (1998)

    Article  MATH  Google Scholar 

  31. Torres, P.J.: Periodic motions of forced infinite lattices with nearest neighbor interaction. Z. Angew. Math. Phys. 51, 333–345 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Torres, P.J.: Necessary and sufficient conditions for existence of periodic motions of forced systems of particles. Z. Angew. Math. Phys. 52, 535–540 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Torres, P.J.: Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle. Mediterr. J. Math. 1(4), 479–486 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Torres, P.J., Zanolin, F.: Periodic motion of a system of two or three charged particles. J. Math. Anal. Appl. 250, 375–386 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Torres, P.J., Zhang, M.: Twist periodic solutions of repulsive singular equations. Nonlinear Anal. 56, 591–599 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Granada, Granada, Spain

    Pedro J. Torres

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  1. Pedro J. Torres
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Correspondence to Pedro J. Torres .

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Torres, P.J. (2015). Motion of Charged Particles in a Line. In: Mathematical Models with Singularities. Atlantis Briefs in Differential Equations, vol 1. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-106-2_1

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