Hamiltonian systems arise naturally in classical mechanics as models of physical motion, describing how particles evolve over time. Their motions correspond to critical points of the Hamiltonian action functional, which solves the Euler–Lagrange equation. Through the study of this action functional, Hamiltonian systems have been shown to differ from general dynamical systems in several fundamental ways. In this talk, we highlight these differences by comparing the orbit structures of Hamiltonian systems with those of more general dynamical systems, tracing developments from Poincaré to Floer. In particular, we focus on how these results can be understood using modern symplectic geometry, especially through the framework of Hamiltonian Floer homology. Building on this framework, I will also present my recent result on a rigidity phenomenon in the topological entropy of Hamiltonian systems, which I call effective robustness.
