To understand what the Maxwell equations are telling us, it’s useful to dissect them piece by piece. The simplest piece comes from looking at stationary electric charges and how they give rise to electric fields. A consequence of this is the Coulomb force law between charges. This, and much more, will be described in this chapter.

There are two great post-Newtonian steps in classical mechanics. The first is the Lagrangian formulation and the accompanying principle of least action. The second is the Hamiltonian formulation, which is yet another way of writing Newtons equation of motion that uncovers what is really going on. This is where we start to see the deep and beautiful mathematical structure that underlies classical mechanics. It is also where we can make connections to what comes next, with quantum mechanics following very naturally from the Hamiltonian formulation.

The two body problem is the question of how two objects – say the Sun and the Earth – move under their mutual gravitational attraction. The problem is, happily, fully solvable and the purpose of this chapter is to fully solve it. We will understand how Keplers laws of planetary motion arise from the more fundamental Newtonian law of gravity. Because the electrostatic force has exactly the same form as the force of gravity, we can also use our solutions to understand how electrons scatter off atoms, a famous experiment performed by Rutherford that led to an understanding of the structure of matter.

Any education in theoretical physics begins with the laws of classical mechanics. The basics of the subject were laid down long ago by Galileo and Newton and are enshrined in the famous equation $F=ma$ that we all learn in school. But there is much more to the subject and, in the intervening centuries, the laws of classical mechanics were reformulated to emphasise deeper concepts such as energy, symmetry, and action. This textbook describes these different approaches to classical mechanics, starting with Newton’s laws before turning to subsequent developments such as the Lagrangian and Hamiltonian approaches. The book emphasises Noether’s profound insights into symmetries and conservation laws, as well as Einstein’s vision of spacetime, encapsulated in the theory of special relativity. Classical mechanics is not the last word on theoretical physics. But it is the foundation for all that follows. The purpose of this book is to provide this foundation.

Much of classical mechanics treats particles as infinitesimally small. But most of our world is not like this. Planets and cats and tennis balls are not infinitesimally small, but have an extended size and this can be important for many applications. The purpose of this chapter is to understand how to describe the complicated motion of extended objects as they tumble and turn.

Space and time are not what they seem. Their true nature only becomes clear as particles reach the speeds close to the speed of light where some of the common sense ideas start to break down. Indeed, one of major themes of twentieth century physics is that common sense is not a good guide when we look closely at the universe. In this chapter, we start to understand the true nature of space and time, as encapsulated in Einsteins theory of special relativity. We will see many wonderful things, from time slowing down to the lengths shrinking. There will be stories of twins and trains and elementary particles failing to die.

The full beauty of Maxwell equations only becomes apparent when we realise that they are consistent with Einstein’s theory of special relativity. The purpose of this chapter is to make this relationship manifest. We rewrite the Maxwell equations in relativistic notation, where the four vector calculus equations are condensed into one, simple tensor equation. Viewed through the lens of relativity and gauge theory, the Maxwell equations are forced upon us: the world can’t be any other way.

At the heart of classical mechanics sits the venerable equation F=ma. To solve this equation, we first need to specify the force at play. In this chapter, we start along this journey. We will look at various forces, including gravity, electromagnetism and friction, and start to understand some of their features. For each, we will solve F=ma in some simple settings.

The harmonic oscillator is, by some margin, the most important system in physics. This is partly because its easy and we can solve it. And partly because, under the right circumstances, pretty much anything else can be made to look like a bunch of coupled harmonic oscillators. In this chapter, we look at what happens when a bunch of harmonic oscillators – or springs – are connected to each other.