From Newton to chaos and modern physics - I

Abstract

In this paper on chaos and its impact on modern science, we first present the Newtonian, Lagrangian and Hamiltonian formalism of dynamics and their recent development. In doing this, the language of differential geometry is favoured so as to gain deeper insights into the nature of symmetry, integrability and chaos. The classical examples such as the Newtonian and relativistic central field problems are discussed in the light of differential geometry. In addition to the Toda lattice problem, more examples such as the 2-centre and the Kerr metric problems are included to show how the approximation method can lead to loss of integrals and the occurrence of chaos in a large class of problems. Also discussed are the reduction of the Newtonian and post-Newtonian N-body problems. Central to this first part are symmetry and integral constant, since a critical understanding of these concepts greatly facilitates a comprehensive understanding on chaos. Although the early assumption on the integrability feature of mechanical systems was completely changed by the work of Poincare, this concept is having a renewed interests being not only related to the integrability of infinite dimensional dynamical systems described by partial differential equations, but also important in the study of semi-classical quantisation.