The periodic table is one of the most iconic images in science. All elements are classified in groups, ranging from metals on the left that go bang when you drop them in water through to gases on the right that don’t do very much at all. The purpose of this chapter is to start to look at the periodic table from first principles, to understand the structure and patterns that lie there.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

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.

The purpose of this chapter is to understand how quantum particles react to magnetic fields. There are a number of reasons to do be interested in this. First, quantum particles do extraordinary things when subjected to magnetic fields, including forming exotic states of matter known as quantum Hall fluids. But, in addition, magnetic fields bring a number of new conceptual ideas to the table. Among other things, this is where we first start to see the richness that comes from combining quantum mechanics with the gauge fields of electromagnetism.

The difference between quantum and classical mechanics does not involve just a small tweak. Instead it is a root and branch overhaul of the entire framework. In this chapter we introduce the key concept that underlies this new framework: the quantum state, as manifested in the wavefunction.

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.

Physicists have a dirty secret: we’re not very good at solving equations. More precisely, humans aren’t very good at solving equations. We know this because we have computers and they’re much better at solving things than we are. This means that we must develop a toolbox of methods so that, when confronted by a problem, we have some options on how to go about understanding whats going on. The purpose of this chapter is to develop this toolbox in the guise of various approximation schemes.

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.

The essence of dimensional analysis is very simple: if you are asked how hot it is outside, the answer is never “2 o’clock”. You’ve got to make sure that the units, or “dimensions”, agree. In this chapter, we understand what it means for quantities to have dimensions and how getting to grips with this can help solve problems without doing any serious work.