In the middle of the nineteenth century, Michel Chasles strongly advocated that geometry and analysis be considered complementary disciplines, not to be separated if one wanted a complete picture of a mathematical theory. To put it simply, analysis provides shortcuts and routine in proofs, while geometry gives insight, showing the meaning of the results. Henri Poincaré’s dissertation advisor, Gaston Darboux, was a student of Chasles. Poincaré was 12 years younger than Darboux, but he underwent a similar influence by studying the writings of Chasles. It shows in his early treatment of ODEs, where he introduced the geometry of the flow near critical points (equilibria) and used projection methods to clarify the structure of solution space. His geometric ideas helped him to handle automorphic functions, where he proposed the relationship between singularities of linear differential equations, Riemann sheets, and non-Euclidean geometry. That influence also came out abundantly in his analysis of dynamical systems, including conservative systems. His concept of “consequents” (Poincaré map) led him to formulate fixed-point theorems to obtain periodic solutions; see Section 9.5. The dynamics of high-dimensional dynamical systems with homoclinic and heteroclinic solutions together with their doubly asymptotic manifolds required subtle analysis in combination with geometric visualization. Also, in his work on the Laplace and Poisson equations, Poincaré’s balayage method clearly pictures the analytic tool of shifting mass distributions in a convenient way.
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