Here we discuss the growing dominance of teams in science. Importantly, this shift toward collaborative work is not unique to fields where experimental challenges are becoming more complex and expensive. Rather, we see a universal rise in team science even in “pencil and paper” disciplines like mathematics and sociology. We find that teams tend to produce more impactful science, garnering more citations than solo-authored work at all points in time and across all disciplines. What has driven the shift toward collaboration in science? The increasing complexity and expense of scientific experimentation forces communities to share resources and knowledge effectively. Additionally, the ever-broadening body of knowledge has made specialization necessary, which means that each person has command of a small piece of a larger puzzle. We also discuss what we call “the death of distance” created by advancing technologies, which has made collaboration easier both among institutions and across international borders. While the advantages of these types of collaboration are clear, there are some potential drawbacks which we detail here.

Here we introduce Part IV, where we will discuss the work at the frontiers of the science of science, the future of the discipline, and how knowledge of the doings of science may change how science is done.

In this chapter we define and detail the Matthew effect, exploring the role that status plays in success. We use the absence and presence of Lord Rayleigh’s authorship on a paper to introduce the idea of reputation signaling, and look at how reputation signaling plays out in randomized control experiments. We then discuss the implications of reputation signaling for both single and double-blind review processes. We find that the Matthew effect applies not just to scientists themselves, but also to their papers through a process known as preferential attachment. To see how an author’s reputation affects the impact of her publications, we look at how her citation patterns deviate from what preferential attachment would predict. We also explore the drivers behind the Matthew effect, asking whether status alone dictates outcomes or whether it reflects inherent talent.

Here we outline our aims for the book and provide a definition for the science of science. We also identify our audience – scientists and students, science administrators, and policymakers, and those already working on science of science research. We explain that the book is structured into four parts: The Science of Career, The Science of Impact, The Science of Collaboration, and an Outlook on the future of the science of science.

We introduce Part II by sharing the story of the LIGO experiment which validated Einstein’s theory of general relativity and which many consider to be the “discovery of the twenty-first century.” While Einstein’s discovery was made by a single scientist, the LIGO experiment involved the contributions of over 1,000 authors. These two discoveries, made 100 years apart, speak to the changing nature of science, where 90% of papers now are written by teams. In Part II we will explore the implications of collaborative work, the benefits and challenges of working in teams, and the factors that help and hinder team effectiveness.

Here we explore peer effects in science, outlining the ways in which scientists affect each other’s outcomes and behavior. In particular, we look at the influence of star scientists on their colleagues, showing that both the productivity of a department and the quality of future faculty increases after a luminary is hired. We detail the negative affect that a star scientist’s death can have on her colleagues, effects which speak to the power of the “invisible college” that binds scientists together in shared interests and ideas. Lastly we outline examples of how changes to the invisible college can have far-reaching impact, demonstrating the highly connected nature of science.

The exponential growth of science has continued, virtually uninterrupted, for decades. What does this mean for contemporary scientists? With the scientific literature doubling every 12 years, the practice of science is characterized by immediacy: 80 to 90 percent of all scientists who have ever lived are alive now. That means that science is becoming more competitive. If one invididual doesn’t make a discovery, someone else likely will. We explore the implications of this growth and competition for scientists from a training and employment standpoint, finding that it is increasingly difficult to earn a PhD and to locate a job in academia after a doctorate is earned. That doesn’t mean that making it as a scientist is impossible – the dearth of jobs in academia has led to a shift toward industry, where many scientists thrive. We end the chapter by asking if new discoveries require more effort than they did in the past. We can answer this question by comparing the growth rate of the workforce compared with the growth rate of producitivity, finding that there is relative stability in individual productivity over a wide range of disciplines.

Deterministic evolution is a hallmark of classical mechanics. Given a set of exact initial conditions, differential equations evolve the trajectories of particles into the future and can exactly predict the location of every particle at any instant in time. So what happens if our uncertainties in the initial position or velocity of a particle are tiny? Does that mean that our uncertainties about the subsequent motion of the particle are necessarily tiny as well? Or are there situations in which a very slight change in initial conditions leads to huge changes in the later motion? For example, can you really balance a pencil on its point? What has been learned in relatively recent years is that, in contrast to Laplace’s vision of a clock-like universe, deterministic systems are not necessarily predictable. What are the attributes of chaos and how can we quantify it? We begin our discussion with the notion of integrability, which ensures the absence of chaos.

In this chapter we describe motion caused by central forces, especially the orbits of planets, moons, and artificial satellites due to central gravitational forces. Historically, this is the most important testing ground of Newtonian mechanics. In fact, it is not clear how the science of mechanics would have developed if the earth had been covered with permanent clouds, obscuring the moon and planets from view. And Newton’s laws of motion with central gravitational forces are still very much in use today, such as in designing spacecraft trajectories to other planets. Our treatment here of motion in central gravitational forces is followed in the next chapter with a look at motion due to electromagnetic forces, which can also be central in special cases, but are commonly much more varied, partly because they involve both electric and magnetic forces. Throughout this chapter we focus on nonrelativistic regimes. The setting where large speeds are involved and gravitational forces are particularly large is the realm of general relativity, where Newtonian gravity fails to capture the correct physics. We explore such extreme scenarios in the capstone Chapter 10.

Some general features of special polynomials.

In this final chapter we introduce Hamilton--Jacobi theory along with its special insights into classical mechanics, and then go on to show how Erwin Schrödinger used the Hamilton--Jacobi equation to learn how to write his famous quantum-mechanical wave equation. In doing so, we will have introduced the reader to two of the ways classical mechanics served as a stepping stone to the world of quantum mechanics. Back in Chapter 5 we showed how Feynman’s sum-over-paths method is related to the principle of least action and the Lagrangian, and here we will show how Schrödinger used the Hamilton--Jacobi equations to invent wave mechanics. These two approaches, along with a third approach developed by Werner Heisenberg called “matrix mechanics,” turn out to be quantum-mechanical analogues of the classical mechanical theories of Newton, Lagrange, Hamilton, and Hamilton and Jacobi, in that they are describing the same thing in different ways, each with its own advantages and disadvantages.

As we saw in Chapter 1, Newton’s laws are valid only for observers at rest in an inertial frame of reference. But to an observer in a non-inertial frame, like an accelerating car or a rotating carnival ride, the same object will generally move in accelerated curved paths even when no forces act upon it. How then can we do mechanics from the vantage point of actual, non-inertial frames? In many tabletop situations, the effects of the non-inertial perspective are small and can be neglected. Yet even in these situations we often still need to quantify how small these effects are. Furthermore, learning how to study dynamics from the non-inertial vantage point turns out to be critical in understanding many other interesting phenomena, including the directions of large-scale ocean currents, the formation of weather patterns -- including hurricanes and tornados, life inside rotating space colonies or accelerating spacecraft, and rendezvousing with orbiting space stations. There is an infinity of ways a frame might accelerate relative to an inertial frame. Two stand out as particularly interesting and useful: linearly uniformly accelerating frames, and rotating frames.

Basics of fractal dimension.

Watching a shoe tumble erratically as it flies through mid-air may be entertaining, but -- to anyone without a background in rigid-body dynamics -- it can look quite troubling. There is no net torque acting on the shoe, yet the rotational motion looks and is rather complicated. However, with the powerful tools provided by the Lagrangian formalism, we are well equipped to tackle this subject, and go beyond it to more complicated examples of rotational motion. We start with a definition of a rigid body, and then proceed to introduce the Euler angles that can be used to describe the orientation of an object in three-dimensional space. With this scaffolding established, we can go on to describe torque-free dynamics, and then full rotational evolution with nonzero torque. For simplicity, throughout this chapter we restrict our discussion to nonrelativistic dynamics.