an:00477933
Zbl 0785.34032
Ambrosetti, Antonio; Coti Zelati, Vittorio
Periodic solutions of singular Lagrangian systems
EN
Progress in Nonlinear Differential Equations and their Applications. 10. Boston, MA: Birkh??user. xii, 157 p. (1993).
00401448
1993
b
34C25 70H03 37J99 37G99 34-02 70-02
Lagrangian systems; Kepler and the \(N\)-body problems; periodic solutions; singular potential; Hamiltonian systems
The monograph deals with the existence of periodic solutions of system \(\ddot q+ \text{grad}_ qV(t,q)=0\), where \(V\) is a singular potential, i.e. a real valued function, defined on an open unlimited subset \(\Omega \subset \mathbb{R}^ n\), such that \(V\) diverges as \(q\) approaches the boundary \(\partial \Omega\) of \(\Omega\). On the whole there is considered the case when \(\Omega=\mathbb{R}^ n \backslash \{0\}\). \(V\) is assumed periodical with respect to \(t\). Several physical problems are governed by Lagrangians which are not regular. The major purpose of the monograph is to present methods and tools which have been used in research of this topic, that is still in evolution. The offered approach is more on the lines of nonlinear functional analysis. There is given a functional frame for systems with singular potential, including the Kepler and the \(N\)- body problems as particular cases. There is used critical point theory to obtain existence results for broad classes of potentials. However the variational methods, which have been employed to obtain important advances in the study of regular Hamiltonian systems, can be succesfully used to handle singular potentials as well.
N.Medvedeva (Chelyabinsk)