Notes on Hamiltonian Dynamical Systems

The theory of Hamiltonian dynamical systems has a rich history, going back to at least the nineteenth century. Celestial mechanics provided a good deal of the original motivation, but the scope has expanded considerably since then. In this book the author provides a very complete development of the theory that is accessible to newcomers and still maintains clear connections to its historical context. He begins with early work in the nineteenth century and proceeds to the major developments of the twentieth century. The book is based on the author’s notes collected over years of presenting aspects of the subject to advanced undergraduates and graduate students. His intention is to provide a path from the basics of Hamiltonian formalism to the most recent results in dynamical systems theory.

 

The discussion is divided into essentially three parts. The first part describes the basic ideas behind Hamiltonian systems, how they arose, and what had been achieved by the second half of the nineteenth century when Poincaré appeared. The second part takes the reader through Poincaré’s realization that not all the equations of mechanics were integrable and the consequences of that. The final part describes developments in the last half of the twentieth century. The author incorporates all the major elements of the theory.

 

The treatment begins with a description of Hamiltonian formalism, Hamilton’s equations, and phase space with a discussion of dynamical variables and the concept of first integrals (quantities like energy or momentum that are preserved by the flow associated with the Hamiltonian). The author then introduces canonical transformations on the phase space – those changes of coordinates that preserve the form of Hamilton’s equations – and briefly discusses the associated symplectic formulation.

 

 Then the author turns to the work of Liouville and his result that Hamiltonian systems in n dimensions are integrable if there are n first integrals. When Poincaré appears we learn that many Hamiltonians do not have n first integrals – and indeed that generically most Hamiltonian systems are not integrable. The emphasis then changes to analysis of perturbations of integrable Hamiltonian systems. In particular, in becomes important to understand how the dynamics behaves in a neighborhood of equilibrium.

 

This leads in turn to the results of the twentieth century: Kolmorgorov’s theorem on the persistence of invariant tori and the complementary exponential stability result of Nekhoroshev that insures long term stability for orbits in an open set. Nonetheless, the shadow of Poincaré and the notions of chaos and homoclinic orbits that he discovered have continuing repercussions in current research. We don’t even know if the solar system is stable in the long term; numerical simulations suggest that the effects of chaos eventually appear.

 

The author provides a treatment that is accessible to newcomers, complete, and attentive to the complexities of the subject’s historical development. One of the appeals of the book is the way it mixes the technical details of formal statements and proofs with numerical experiments and historical digression. The use of theorems to explain the results of numerical experiments, which are then used to aid interpretation of the theorems, is particularly effective. The book covers a great deal of material, but it does so gracefully and thoroughly. 

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.