Foreword

Summary

In variance principles and integrability (or lack of it) are the organizing principles of this text. Chaos, fractals, and strange attractors occur in different nonintegrable Newtonian dynamical systems. We lead the reader systematically into modern nonlinear dynamics via standard examples from mechanics. Goldstein and Landau and Lifshitzpresume integrability implicitly without defining it. Arnol'd's inspiring and informative book on classical mechanics discusses some ideas of deterministic chaos at a level aimed at advanced readers rather than at beginners, is short on driven dissipative systems, and his treatment of Hamiltonian systems relies on Cartan's formalism of exterior differential forms, requiring a degree of mathematical preparation that is neither typical nor necessary for physics and engineering graduate students.

The old Lie-Jacobi idea of complete integrability (‘integrability’) is the reduction of the solution of a dynamical system to a complete set of independent integrations via a coordinate transformation (‘reduction to quadratures’). The related organizing principle, invariance and symmetry, is also expressed by using coordinate transformations. Coordinate transformations and geometry therefore determine the method of this text. For the mathematically inclined reader, the language of this text is not ‘coordinatefree’, but the method is effectively coordinate-free and can be classified as qualitative methods in classical mechanics combined with a Lie–Jacobi approach to integrability. We use coordinates explicitly in the physics tradition rather than straining to achieve the most abstract (and thereby also most unreadable and least useful) presentation possible.