Newtons laws of motion are not the last word in classical mechanics. In the 250 years after Newton, physicists and mathematicians found ways to reformulate classical mechanics, providing powerful tools to solve problems but, equally as importantly, giving us a new perspective on the laws that govern our universe. This chapter takes the first step in this direction. We will introduce the wonderful principle of least action, a simple rule that underlies all known laws of physics. This will give us new insights, not least the wonderful Noethers theorem, relating symmetries to conservation laws.

Beginning with linear programming and ending with neural network training, this chapter features seven applications of the divide-and-concur approach to solving problems with RRR.

In 1925, as matrix mechanics was taking shape, Lucy Mensing (1901−1995), who earned her PhD with Lenz and Pauli in Hamburg, came to Göttingen as a postdoc. She was the first to apply matrix mechanics to diatomic molecules, using the new rules for the quantization of angular momentum. As a byproduct, she showed that orbital angular momentum can only take integer values. Impressed by this contribution, Pauli invited her to collaborate on the susceptibility of gases. She then went to Tübingen, where many of the spectroscopic data were obtained that drove the transition from the old to the new quantum theory. It is hard to imagine better places to be in those years for young quantum physicists trying to make a name for themselves. This chapter describes these promising early stages of Mensing’s career and asks why she gave it up three years in. We argue that it was not getting married and having children that forced Lucy Mensing, now Lucy Schütz, out of physics, but the other way around. Frustration about her own research in Tübingen and about the prevailing male-dominated climate in physics led her to choose family over career.

Over the past hundred years or so, physicists have developed a foolproof and powerful tool that allows us to understand everything and anything in the universe. You take the object that you’re interested in and you throw something at it. Ideally, you throw something at it really hard. This technique was developed around the turn of the 20th century and has since allowed us to understand everything from the structure of atoms, to the structure of materials, to the structure of DNA. In short, throwing stuff at other stuff is the single most important experi- mental method available to science. Because of this, it is given a respectable sounding name. We call it scattering.

The first four women to obtain a PhD in physics at Leiden University all graduated with Nobel laureate Hendrik Lorentz, among them Hendrika Johanna (Jo) van Leeuwen (1887−1974). She and her younger sister Cornelia (Nel) van Leeuwen finished their undergraduate studies in physics in Leiden in the early twentieth century. Whereas the younger sister left physics in 1917 after a relatively short period as a graduate student, Jo van Leeuwen went on to earn a PhD in 1919. Her thesis elucidates that magnetism is exclusively a quantum phenomenon – a result that was independently also obtained by Niels Bohr and that is now commonly known as the Bohr–van Leeuwen theorem. From 1920 onwards Van Leeuwen worked at the Technische Hoogeschool in Delft (now Delft University of Technology). Initially serving as an assistant, she was appointed as a reader in theoretical and applied physics in 1947, becoming the first female reader in Delft. This chapter outlines the foray into physics by the two sisters, focusing specifically on Jo van Leeuwen, detailing her work and early contributions to the quantum theory of magnetism.

In this chapter, we explore the concept of information in living organisms in its broadest sense. Biological organisms perceive the external environment, alter their own state, and take action (selection among possibilities). To capture these properties intrinsic to the organisms, we begin by discussing the “information quantity” that quantifies such situations. Starting with the definition of information quantity, we introduce Shannon entropy and provide an overview of Shannon’s information theory framework. We also discuss Kullback–Leibler divergence and mutual information. Next, moving on to information in DNA sequences, we cover various aspects such as differences in the frequency of AT and GC occurrence, the structure of genetic codes, long-range correlations in DNA sequences, and recent findings in intergenic sequences. Additionally, we explain kinetic proofreading as one candidate for achieving high accuracy in molecular recognition from a combination of unreliable elements. Furthermore, we explore the relationship between entropy in statistical mechanics and information, elucidating the connection between Maxwell’s demon and information using the Szilard engine as a mediator. Finally, we introduce intriguing points from the perspective of dynamics and information, highlighting the dynamic interplay between the two.

Chien-Shiung Wu (1912–1997) is often referred to as “the Chinese Marie Curie” even though she conducted most of her research in the US. She is best known for her discovery of the non-conservation of parity for weakly interacting particles – a finding for which she is widely regarded as having been passed over for the 1957 Nobel Prize in Physics. Seven years earlier, though, in a one-page letter to Physical Review, Wu and her graduate student also quietly reported what has come to be understood as the first conclusive evidence of entangled photons. Twenty years later, as quantum foundations research emerged from shadow, Wu revisited her 1949 experiment with a more refined approach. Wu shared the new results with Stuart Freedman, a collaborator of John Clauser. Clauser et al. would rigorously critique Wu’s experiments through at least 1978. In 2022, the Nobel Committee honored Clauser, Alain Aspect, and Anton Zeilinger, each of whom had produced increasingly convincing proof of entanglement beginning in the 1970s. Wu’s foundational work from almost seventy years earlier, however, was not mentioned. This chapter aims to help bring Wu’s entangled photons back into the light.

This chapter explores what we could do with a computer whose operating system is quantum mechanics, rather than classical mechanics. One of the answers is: factorise primes really quickly. We will explain why this is interesting.

This chapter quantitatively examines molecule numbers and reaction rates within a cell, along with thermal fluctuations and Brownian motion, from a mesoscopic perspective. Thermal fluctuations of molecules are pivotal in chemical reactions, protein folding, molecular motor systems, and so on. We introduce estimations of cell size and molecule numbers within cells, highlighting the possible significance of the minority of molecules. Describing their behaviors necessitates dealing with stochastic fluctuations, and the Gillespie algorithm, widely employed in Monte Carlo simulations for stochastic chemical reactions, is described. We elaborate on extrinsic and intrinsic noise in cells, and on why understanding how cells process fluctuations for sensing is crucial. To facilitate this comprehension, we revisit the fundamentals of statistics, including the law of large numbers and the central limit theorem. We derive the diffusion equation from random walk and confirm the dimensionality dependence of random walks, and elucidate Brownian motion as the continuous limit of random walk and explain the Einstein relation. As examples of the physiological significance of fluctuations in cell biology, we estimate the diffusion constant of proteins inside cells, diffusion-limited reactions, and introduce bacterial random walks and chemotaxis, and amoeboid movements of eukaryotic cells.