Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface

Abstract

Two theorems about the existence of periodic solutions with prescribed energy for second order Hamiltonian systems are obtained. One gives existence for almost all energies under very natural conditions. The other yields existence for all energies under a further condition.

Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems

“…Then with the help of Lemma 2.10 (the Egorov Theorem), for any given ε > 0, there exists a closed set B ε ⊂ B such that q n (t) → c, as n → +∞ unif ormlly in B ε (10) with m(B − B ε ) < ε. From q n (t + 1) = q n (t) and (10), we see that q n (t + 1/2) → c, as n → +∞ unif ormlly in B ε (11) with m(B ε − B ε ) < ε. Since q n (t + 1/2) = −q n (t) inM h , then c = 0.…”

“…For nonsingular second-order Hamiltonian 1618 LIANG DING, RONGRONG TIAN AND JINLONG WEI system with fixed energy, there were some references: Benci [5] obtained one periodic solution for C 2 nonsingular potential; for nonconstant periodic solution with C 1 nonsingular potential, Zhang [25] gained one nonconstant periodic solution with positive fixed energy. Later, Xue [10] and Wu, Li and Yuan [23] extended the result of [25], and it is noted that, in [10], [23] and [25], there are some constraints on the fixed energy h. For example, in [10] and [25], the authors presume that h > 0; in [23], h > V (0). For singular second-order Hamiltonian systems with fixed energy, there are also many study works on the existence of periodic solutions [1,2,7,9,18,21,24] etc.…”

“…In 1978, by imposing some constraints on the energy sphere, Rabinowitz [19] used variational methods to prove the existence of periodic solutions for a class of firstorder Hamiltonian systems with fixed energy. Since then, the existence of periodic solutions of Hamiltonian system with fixed energy is extensively investigated [1,2,5,7,9,10,18,21,22,23,24,25] etc. For nonsingular second-order Hamiltonian 1618 LIANG DING, RONGRONG TIAN AND JINLONG WEI system with fixed energy, there were some references: Benci [5] obtained one periodic solution for C 2 nonsingular potential; for nonconstant periodic solution with C 1 nonsingular potential, Zhang [25] gained one nonconstant periodic solution with positive fixed energy.…”