Nonlinear oscillations and boundary value problems for Hamiltonian systems

Clarke, Frank H.; Ekeland, Ivar

Clarke, Frank H.; Ekeland, Ivar (1982), Nonlinear oscillations and boundary value problems for Hamiltonian systems, Archive for Rational Mechanics and Analysis, 78, 4, p. 315-333. http://dx.doi.org/10.1007/BF00249584

Type

Article accepté pour publication ou publié

Date

1982

Journal name

Archive for Rational Mechanics and Analysis

Volume

Number

Publisher

Springer

Pages

315-333

Abstract (EN)

We prove the existence of solutions of various boundary-value problems for nonautonomous Hamiltonian systems with forcing terms x˙(t)=H′p(t,x(t),p(t))+g(t),p˙(t)=−H′x(t,x(t),p(t))−f(t). 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.

Subjects / Keywords

Hamiltonian systems