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Nonlinear oscillations and boundary value problems for Hamiltonian systems

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Abstract

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms

$$\begin{gathered} \dot x(t) = H'_p (t, x(t), p(t)) + g(t), \hfill \\ \dot p(t) = - H'_x (t, x(t), p(t)) - f(t). \hfill \\ \end{gathered} $$

Among these problems is the existence of T-periodic solutions, namely those satisfying x(t+T)=x(t) and p(t+T)+p(t). A special study is made of the classical case, where H(x, p)=1/2 |p|2+V(x). In the case of parametric oscillations, where (f, g)=(0, 0) and t ↦ H(t, x, p) is T-periodic, we give a lower bound for the true (minimal) period of the T-periodic solution (x, p) produced by our method, and we prove the existence of an infinite number of subharmonics.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver

    Frank H. Clarke & I. Ekeland

  2. Ceremade Université de Paris IX-Dauphine, Paris

    Frank H. Clarke & I. Ekeland

Authors
  1. Frank H. Clarke
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  2. I. Ekeland
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Additional information

This work was supported in part by the Canadian NSERC under Grant A 9082.

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Clarke, F.H., Ekeland, I. Nonlinear oscillations and boundary value problems for Hamiltonian systems. Arch. Rational Mech. Anal. 78, 315–333 (1982). https://doi.org/10.1007/BF00249584

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  • DOI: https://doi.org/10.1007/BF00249584

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