Hyperbolic-like solutions for singular Hamiltonian systems

  • Patricio Felmer
  • Kazunaga Tanaka


We study the existence of unbounded solutions of singular Hamiltonian systems: \(\ddot q + \nabla V(q) = 0,\) where \(V(q) \sim -{1\over{|q|^\alpha}}\) is a potential with a singularity. For a class of singular potentials with a strong force \(\alpha>2\), we show the existence of at least one hyperbolic-like solutions. More precisely, for given \(H>0\) and \(\theta_+, \theta_-\in S^{N-1}\), we find a solution q(t) of (*) satisfying \({1\over 2} |\dot q|^2 + V(q) = H,\) \(|q(t)| \longrightarrow \infty \quad {as} \quad t\longrightarrow\pm\infty\) \(\lim \limits_{t\to\pm\infty} {q(t)\over |q(t)|} = \theta_\pm.\)


Hamiltonian System Strong Force Singular Potential Unbounded Solution Singular Hamiltonian System 
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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • Patricio Felmer
    • 1
  • Kazunaga Tanaka
    • 2
  1. 1.Departamento de Ingeniería Matemática F.C.F.M., Universidad de Chile, Casilla 170 Correo 3, Santiago, ChileCH
  2. 2.Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan, e-mail:

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