Prof. Daniel Goroff: Contextual Area History of the Mathematical Understanding of Dynamics

1) Describe a few of the most important developments in the history of dynamical systems theory. Understanding dynamical systems has required the development of many tools that radically altered our corresponding understanding of the physical world and mathematics itself. The tools of infinitesimal calculus, the principle of least action, and topological descriptions of the qualitative behavior of dynamical systems are three of the most important contributions that dynamical systems theory has provided. These have greatly increased our insight into the causes and forces that give rise to motion. Infinitesimal Calculus Probably the most widely applied notion developed by those trying to understand mechanics is that of infinitesimal calculus. While many problems in statics could be understood without the aid of calculus, the dynamic problems of celestial mechanics and the computation of accurate ephemerides demanded more complex tools. The roots of infinitesimal calculus can be traced to the work of the ancient Greek geometers. The notion of quadrature of a shape, that is finding a square whose area equaled the shape, was o