1.1 Introduction

Mechanics is the study of how things move; from the Earth orbiting the Sun, to a ball rolling down a hill, to an electron in a cathode ray tube. The modern formulation of classical mechanics was first developed in the 17^th and 18^th centuries by European natural philosophers like Galileo and Newton, who based their ideas on previous theories like those from ancient Greece. It was then reformulated in the 19^th century by French mathematicians such as Lagrange, Poisson and Liouville, as well as Hamilton, whose developments paved the way for the innovations of the 20^th century, when it was realised that classical mechanics cannot accurately describe objects travelling close to the speed of light or objects that are extremely small (on an atomic scale). These discoveries led to the development of relativistic mechanics and quantum mechanics, for which the advanced Lagrangian and Hamiltonian formalisms serve as a mathematical basis.

In this text we will be studying the Newtonian formulation of classical mechanics, which is still used today not only as a teaching method, but also as a tool in research. The Lagrangian and Hamiltonian formalisms come into their own for advanced problems, but can be quite unwieldy for the simple systems that we will be describing the motion of. One may wonder why we still study classical mechanics if it has been proven to be obsolete in some areas. The answer is that there are still many real-life systems which are best described using a classical description. It is also a great opportunity to become familiar with the language of vector calculus while studying examples which are relevant to real life. Without further ado, let us dive in to how we describe the world in mechanics.