Convexity Methods in Hamiltonian Mechanics

The book delves into the complexities of integrable and nonintegrable systems in celestial mechanics, particularly focusing on periodic solutions. It discusses Poincaré's contributions, including his geometric theorem about area-preserving maps and the significance of the least action principle in Hamiltonian mechanics. The text highlights the historical context of these concepts, from Fermat to Maupertuis, and emphasizes the limited impact of the least action principle until recently, despite its aesthetic appeal and the connection established by Emmy Noether's theorem between symmetries and motion integrals.