zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Variational methods for Hamiltonian systems. (English) Zbl 1048.37055
Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 1091-1127 (2002).

Variational methods have been used very successfully to find periodic, homoclinic, or heteroclinic orbits of Hamiltonian systems. The author, a most influential contributor to this area for three decades, surveys some of the representative problems, methods and results. He treats first-order Hamiltonian systems (HS) Jz ˙+H z (t,z)=0 as well as second-order systems (HS2) q ¨+V q (t,q)=0, both autonomous or nonautonomous. In the nonautonomous case it is required that H,V depend periodically on t.

The paper consists of two parts: Part 1 is concerned with periodic solutions, Part 2 with homoclinic and heteroclinic orbits. After formulating without details a technical framework for periodic solutions, the following topics are discussed: superquadratic autonomous Hamiltonian systems, fixed energy results, brake orbits, time dependent superquadratic fixed period problems, perturbations from symmetry, subquadratic Hamiltonian systems, asymptotically quadratic Hamiltonians, singular potentials. Part 2 starts with the variational formulation for homoclinics to 0, and contains some results for homoclinics, basic heteroclinic results, multibump solutions in the time dependent case, and multibump solutions in the autonomous case.

The author states a number of selected theorems precisely and gives sometimes ideas of proofs or of essential ingredients of the proofs. Other results are discussed informally. For all results mentioned, references to the literature are given.

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C25Periodic solutions of ODE
34C37Homoclinic and heteroclinic solutions of ODE
58E05Abstract critical point theory
37-02Research exposition (Dynamical systems and ergodic theory)