##
**Critical point theory and Hamiltonian systems.**
*(English)*
Zbl 0676.58017

Applied Mathematical Sciences, 74. New York etc.: Springer-Verlag. xiv, 277 p. DM 108.00 (1989).

In the past two decades, one of the main endeavours in nonlinear functional analysis was focused on minimax methods for semilinear differential equations. The roots of this machinery are the Lyusternik- Schnirelman and the Morse theory. To the R. Palais and S. Smale extensions for mappings on Banach manifolds, given in the 60s, there have been added in the 70s a dual least action principle by D. C. Clark, the A. Ambrosetti and P. H. Rabinowitz dual variational methods and later I. Ekeland’s variational principle. The subject received a tremendous impetus in the last decade from the development of a general theory of periodic solutions of Hamiltonian systems \(J\dot u(t)+\nabla H(t,u(t))=0\). The book is an outgrowth of the authors’ outstanding contributions and is organized in ten chapters.

After prerequisites of the convex analysis and periodic solutions, the Fenchel transform and Clarke duality in the first two chapters, the authors consider the circumstances where \(H(t,\cdot)\) is convex. In this case the dual least action principle of Chapter 3 seems to provide the best results in the simplest way. In the absence of convexity, the existence of critical point of the saddle point type can be proved by using minimax techniques.

For indefinite functionals, Chapter 4 describes the mountain pass lemma, which is derived from Ekeland’s variational principle. Based on the Borsuk-Ulam theorem, the index is introduced as a “size” measure of some sets, symmetric with respect to the origin. The Lyusternik- Schnirelman theory and corresponding equivariant deformations are presented in Chapter 6. This allows a local description of periodic orbits with prescribed energy. In Chapter 7 one finds the Morse-Ekeland index for linear positive definite and convex asymptotically linear autonomous Hamiltonian systems. A more difficult part of the critical point theory is its connection to algebraic topology concepts. The properties of the critical points of a smooth function \(f\) on a manifold \(M\) are related to the topology of \(M\). The Morse theory derives homotopy properties of \(M\) from an analysis of the second derivative of \(f\) at its critical points.

Chapter 8 deals with critical groups at a point of mountain pass type or a saddle point and connections with bifurcation phenomena. Applications of the Morse theory are made to asymptotically linear nonautonomous systems. The final chapter is devoted to nondegenerate critical manifolds and periodic solutions of forced superlinear equations.

The volume concludes with a rich bibliography of more than 450 titles, most of them from the 80s. Each chapter begins with a heuristic presentation of the basic ideas in the topic and is followed by reference notes and well chosen problems that are invitations to deeper investigations. Although, there are several books in the subject as those of P. H. Rabinowitz [Minimax methods in critical point theory with applications to differential equations. Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] or K. C. Chang [Infinite dimensional Morse theory and its applications. NATO Advanced Study Institute, 97. Département de Mathématiques et de Statistique, Université de Montréal. Montréal (Québec), Canada: Les Presses de l’Université de Montréal (1985; Zbl 0609.58001)], the clear and progressive exposition of the book under review should be fully appreciated. The tools can provide valuable insight that will stimulate our understanding of analytic mechanics. The book constitutes a self-contained and comprehensive introduction in a top research field and it can also be an excellent text for an up-to-date graduate course.

After prerequisites of the convex analysis and periodic solutions, the Fenchel transform and Clarke duality in the first two chapters, the authors consider the circumstances where \(H(t,\cdot)\) is convex. In this case the dual least action principle of Chapter 3 seems to provide the best results in the simplest way. In the absence of convexity, the existence of critical point of the saddle point type can be proved by using minimax techniques.

For indefinite functionals, Chapter 4 describes the mountain pass lemma, which is derived from Ekeland’s variational principle. Based on the Borsuk-Ulam theorem, the index is introduced as a “size” measure of some sets, symmetric with respect to the origin. The Lyusternik- Schnirelman theory and corresponding equivariant deformations are presented in Chapter 6. This allows a local description of periodic orbits with prescribed energy. In Chapter 7 one finds the Morse-Ekeland index for linear positive definite and convex asymptotically linear autonomous Hamiltonian systems. A more difficult part of the critical point theory is its connection to algebraic topology concepts. The properties of the critical points of a smooth function \(f\) on a manifold \(M\) are related to the topology of \(M\). The Morse theory derives homotopy properties of \(M\) from an analysis of the second derivative of \(f\) at its critical points.

Chapter 8 deals with critical groups at a point of mountain pass type or a saddle point and connections with bifurcation phenomena. Applications of the Morse theory are made to asymptotically linear nonautonomous systems. The final chapter is devoted to nondegenerate critical manifolds and periodic solutions of forced superlinear equations.

The volume concludes with a rich bibliography of more than 450 titles, most of them from the 80s. Each chapter begins with a heuristic presentation of the basic ideas in the topic and is followed by reference notes and well chosen problems that are invitations to deeper investigations. Although, there are several books in the subject as those of P. H. Rabinowitz [Minimax methods in critical point theory with applications to differential equations. Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] or K. C. Chang [Infinite dimensional Morse theory and its applications. NATO Advanced Study Institute, 97. Département de Mathématiques et de Statistique, Université de Montréal. Montréal (Québec), Canada: Les Presses de l’Université de Montréal (1985; Zbl 0609.58001)], the clear and progressive exposition of the book under review should be fully appreciated. The tools can provide valuable insight that will stimulate our understanding of analytic mechanics. The book constitutes a self-contained and comprehensive introduction in a top research field and it can also be an excellent text for an up-to-date graduate course.

Reviewer: D.Pascali

### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

70H05 | Hamilton’s equations |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58Exx | Variational problems in infinite-dimensional spaces |