In the 1960s, Arnol'd first predicted (Ar65) the existence of Lagrangian intersection theory (on the cotangent bundle) as the intersection-theoretic version of theMorse theory and posed Arnol'd's conjecture: the geometric intersection number of the zero section of T∗N for a compact manifold N is bounded from below by the one given by the number of critical points provided by the Morse theory on N. This original version of the conjecture is still open due to the lack of understanding of the latter Morse-theoretic invariants. However, its cohomological version was proven by Hofer (H85) using the direct approach of the classical variational theory of the action functional. This was inspired by Conley and Zehnder's earlier proof (CZ83) of Arnol'd's conjecture on the number of fixed points of Hamiltonian diffeomorphisms. Around the same time Chaperon (Ch84) and Laudenbach and Sikorav (LS85) used the broken geodesic approximation of the action functional and the method of generating functions in their proof of the same result. This replaced Hofer's complicated technical analytic details by simple more or less standard Morse theory.

The proof published by Chaperon and by Laudenbach and Sikorav is reminiscent of Conley and Zehnder's proof (CZ83) in that both proofs reduce the infinite-dimensional problem to a finite-dimensional one. (Laudenbach and Sikorav's method of generating functions was further developed by Sikorav (Sik87) and then culminated in Viterbo's theory of generating functions quadratic at infinity (Vi92).)

In the meantime, Floer introduced in (Fl88b) a general infinite-dimensional homology theory, now called the Floer homology, which is based on the study of the moduli space of an elliptic equation of the Cauchy–Riemann type that occurs as the L2-gradient flow of the action integral associated with the variational problem. In particular Hofer's theorem mentioned above is a special case of Floer's (Fl88a) (at least up to the orientation problem, which was solved later in (Oh97b)), if we set L0 = φ(oN), L1 = oN in the cotangent bundle. (Floer's construction is applicable not only to the action functional in symplectic geometry but also to the various first-order elliptic systems that appear in low-dimensional topology, e.g., the anti-self-dual Yang–Mills equation and the Seiberg–Witten monopole equation, and has been a fundamental ingredient in recent developments in low-dimensional topology as well as in symplectic topology.)