Book: New Methods of Celestial Mechanics

Introduction
Published beginning in 1892 as the multi-volume Les méthodes nouvelles de la mécanique céleste, Henri Poincaré's New Methods of Celestial Mechanics reconceived the mathematical foundations of planetary theory. Addressing classical problems of perturbation theory and the long-term stability of the solar system, the work bridges rigorous analytical calculation and a new geometric, topological viewpoint on dynamical motion. The book grew out of investigations for a prize on the stability of the solar system and transformed an applied problem into a source of far-reaching mathematical ideas.
Poincaré approached celestial mechanics not merely as a collection of series to be computed but as a study of the qualitative structure of trajectories in phase space. He emphasized periodic solutions, the behavior of nearby trajectories, and the role of singularities and resonances, thereby converting questions of astronomy into a general theory of dynamical systems.

Central Results
A striking outcome was the demonstration that the three-body problem is not integrable in the sense of admitting a general closed-form solution expressed by first integrals. Poincaré showed that series methods used by earlier investigators can diverge and that formal power-series expansions may fail to capture global behavior. He proved the existence of complicated intersection structures now identified as homoclinic points and tangles, revealing that trajectories could display extreme sensitivity to initial conditions.
Poincaré developed the concept of periodic orbits as organizing centers for motion and used them to classify nearby behavior. He articulated the distinction between stable and unstable motions, established criteria for the multiplicity and persistence of periodic solutions under perturbation, and identified secular effects that accumulate over long times. Together these results undermined the expectation of simple, closed-form predictability for many-body systems and replaced it with a nuanced portrait of possible behaviors.

Methods and Techniques
The book introduced fundamental tools that became staples of modern dynamics. The Poincaré map or return map provided a discrete snapshot of a continuous flow and made periodic orbits and their stability accessible through iteration. Normal forms, canonical transformations, and careful treatment of small divisors refined perturbation theory, while asymptotic expansions and the recognition of divergent series clarified when and how series approximations are meaningful. Topological reasoning, classification of singularities, invariant manifolds, and the study of intersections of stable and unstable manifolds, brought geometry to the forefront of analysis.
Poincaré combined rigorous estimates with imaginative geometric pictures. Rather than relying solely on explicit integrals, he pursued invariant structures in phase space and leveraged continuity and compactness arguments to deduce existence and multiplicity results. This mixture of analytic and qualitative techniques allowed him to navigate the complexities produced by resonances and near-collisions among bodies.

Legacy and Influence
New Methods laid the groundwork for the modern theory of dynamical systems and had immediate and lasting impact across mathematics and physics. The ideas of recurrence, sensitivity to initial conditions, invariant manifolds, and homoclinic intersections evolved into the language of chaos theory in the twentieth century. Subsequent developments such as KAM theory, ergodic theory, and the global qualitative study of differential equations trace intellectual lineage to Poincaré's synthesis.
Beyond pure mathematics, the work reframed how scientists think about predictability and stability in celestial mechanics, influencing astronomical practice and the philosophy of determinism. Poincaré's blend of conceptual clarity, technical ingenuity, and geometric intuition remains a model for approaching complex nonlinear phenomena.