This is a two-volume series of monographs. This series provides a self-contained exposition of basic Floer homology in both open and closed string contexts, and systematic applications to problems in Hamiltonian dynamics and symplectic topology. The basic objects of study in these two volumes are the geometry of Lagrangian submanifolds and the dynamics of Hamiltonian diffeomorphisms and their interplay in symplectic topology.

The classical Darboux theorem in symplectic geometry reveals the flexibility of the group of symplectic transformations. On the other hand, Gromov and Eliashberg's celebrated theorem (El87) reveals the subtle rigidity of symplectic transformations: the subgroup Symp (M, ω) consisting of symplectomorphisms is closed in Diff(M) with respect to the C0 topology. This demonstrates that the study of symplectic topology is subtle and interesting. Eliashberg's theorem relies on a version of the non-squeezing theorem, such as the one proved by Gromov (Gr85) using the machinery of pseudoholomorphic curves. Besides Eliashberg's original combinatorial proof of this non-squeezing result, there is another proof given by Ekeland and Hofer (EkH89) using the classical variational approach of Hamiltonian systems. The interplay between these two facets of symplectic geometry, namely the analysis of pseudoholomorphic curves and Hamiltonian dynamics, has been the main driving force in the development of symplectic topology since Floer's pioneering work on his semi-infinite dimensional homology theory, which we now call Floer homology theory.

Hamilton's equation ẋ = XH (t, x) arises in Hamiltonian mechanics and the study of its dynamics has been a fundamental theme of investigation in physics since the time of Lagrange, Hamilton, Jacobi, Poincaré and others. Many mathematical tools have been developed in the course of understanding its dynamics and finding explicit solutions of the equation.