Periodic solutions of Lagrangian systems with bounded potential

Multiple forced rotational solutions for the planar N-pendulum equation

“…6) In (1.6), if v = 0, solutions are oscillatory. Existences and multiplicities of oscillations for planar double, triple and N-pendulum have been studied in [3,8,4,17,15] and references therein. In [14], Rabinowitz showed the existence of at least N +1 2π-periodic forced oscillations for more general Lagrangian functions.…”

Remarks on Forced Equations of the Double Pendulum Type

“…This situation has attracted the attention of several authors (e.g., [2,3,4,8]) and multiple forced oscillations have been obtained for (0.2) provided the forcing term / = f(t) has mean value zero (i.e., J0 f(t) dt = 0 ). Here we investigate the problem without this restriction on the mean value of /.…”

Lagrangian Systems in the Presence of Singularities

“…In this paper we look for T-periodic solutions of the Lagrangian system of ordinary differential equations: ÍGR, q,£eRN, and dij(t,q), bi(t,q), c(t,q), V(t,q) are C1 real-valued functions, T-periodic in t. Moreover we suppose that the "potential" V(t,q) is defined in R x fi, where fi is an open subset of Rw, and V{t,q) -► -oo as?-» dQ. Many authors have studied this problem in the case when fi = R^ (so dQ = 0) under various assumptions on the growth of V(t,q) as \q\ -► oo: cf., for instance, [2,3,5,9,10]. W. B. Gordon was the first to study our case by means of variational methods, and we refer to [6,7] for the physical motivation of the problem (cf.…”

“…Since fi can be homotopically trivial (for example if fi = R3\{0}), we shall restrict the functional / to the subspace (cf. [3]) E = {qeHl\q(t + T/2) = -q(t)}.…”

“…Then it is easy to check that qh(t) -q{t/k) is still a critical point of /, and that f{qk) < f{q) (cf. [3]). Therefore T is the minimal period of the solutions given by Theorems 0.1 and 0.2.…”