Positivity 10 (2006), 681–692
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040681-12, published online July 11, 2006
The Brake Orbits of Hamiltonian Systems
on Positive-type Hypersurfaces
Abstract. This paper deals with the brake orbits of Hamiltonian system
˙x(t)=J ∇H(x(t)) (HS)
on given energy hypersurfaces Σ = H
(1). We introduce a class of contact
type but not necessarily star-shaped hypersurfaces in R
and call them nor-
malized positive-type hypersurfaces. By using of the critical point theory, we
prove that if Σ is a partially symmetric normalized positive-type hypersurface,
it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity
results under certain pinching conditions. Our results include the earlier works
on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case.
An example of partially symmetric normalized positive-type hypersurface in
that is not star-shaped is also presented.
Mathematics Subject Classiﬁcation (2000). 34C25; 58E05; 58F05; 58F10.
Keywords. brake orbit, critical point, Hamiltonian system, multiplicity, positive-
type hypersurface, Z
1. Introduction and Results
Consider the periodic orbits of Hamiltonian system
on energy hypersurface Σ = H
(1), where H ∈ C
is the standard symplectic matrix. In 1978, Rabinowitz and Weinstein proved the
existence of at least one periodic orbit on Σ provided Σ is star-shaped (or convex).
Their works stimulated great interest and much research on this problem. Many
remarkable results for existence and multiplicity of periodic orbits were obtained
by several authors. See for example, the references Viterbo (1987); Ekeland and
Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.