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Pseudo-orbits of contact forms. (English) Zbl 0677.58002
Pitman Research Notes in Mathematics Series, 173. Harlow (UK): Longman Scientific & Technical; New York: Wiley. i. 47, 296 p. £20.00 (1988).
Since around 1960 the modern approach to the geometric theory of mechanics has been flourishing. Various schools put different emphasis on the subject like abstract dynamical systems, explicit problems from mechanics or, as in this book, variational problems related to contact geometry.
Consider a contact vector field on a compact orientable $$(2n+1)$$- dimensional manifold. The Weinstein conjecture states that the vector field has a periodic orbit under certain weak conditions. Primarily Weinstein, but later also Rabinowitz and Ekeland, proved a number of results for Hamiltonian systems. Motivated by the general conjecture one studies in this book variational problems on a submanifold of the loopspace M; this is non-compact in the sense that it does not satisfy the Palais-Smale condition. The problem gives rise to a new variational problem at infinity in which a certain pendulum equation turns out to play a crucial role.
After introducing the necessary functionals with corresponding critical points the pendulum equation is derived. Topological aspects and convergence results are tied in with as different techniques as singular perturbations and Morse theory.
The text is rather technical but has been well written. In the future one would expect to have in addition some explicit nontrivial examples. In the mean time this is a nice addition to the geometric theory of mechanics.
Reviewer: F.Verhulst

MSC:
 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 37G99 Local and nonlocal bifurcation theory for dynamical systems