Chapter 1 provides an exposition of the general breadth and scope of the contemporary prohibition of the threat or use of force. After setting the prohibition in its historical context, it then examines the sources of the prohibition and its status as a ‘peremptory’ or ‘jus cogens’ norm of international law and what this means in terms of the prospects for its adaptation and modification in the future. While much focus has been traditionally placed upon the use of force, by comparison relatively little attention has been given to the threat of force, despite both coming under the contemporary rule against force. As such, the chapter sets out and provides an examination of the concept of a threat of force. While the prohibition is of undoubted importance, the decentralised nature of the international legal system means that accountability and responsibility for its violation are uncertain. The chapter seeks to examine this issue in detail, while also providing attention to the ‘human element’ to the prohibition, as well as assessing the continued relevance and effectiveness of the norm.

Chapter 2’s purpose is to provide some focused discussion on the meaning of the ‘force’ prohibited in Article 2(4) and customary international law. In seeking to expose the uncertainties regarding this particular term, various factors or common elements of ‘force’ are distilled. After looking at the prohibition of force in the context of the principle of non-intervention, the chapter moves on to look at the type of force that the prohibition is concerned with and, concluding that it is ‘armed force’, then moves on to attempt to distil the key elements of such force, including whether it is the means used or the effects created which is of importance, and whether force can be used indirectly. The chapter then addresses the ‘gravity’ or severity aspects of a use of force, in particular by distinguishing it from an armed attack or act of aggression, but also by examining whether there is a level of force – or de minimis threshold – below which an action falls out of the remit of the prohibition. Finally, and having distilled the key practical components of a prohibited use of force, the chapter focuses upon the mens rea component.

Just as force is a ubiquitous concept in linear mechanics, torque is ubiquitous in rotational mechanics. We, therefore, begin this chapter with the definition and detailed description of torque, which we then use to study static equilibrium. Our discussion includes descriptions of common forces and their points of application, as well as subtleties associated with studying systems of objects in static equilibrium. The chapter ends with some useful theorems commonly found in the literature.

In this chapter, we begin by examining the work due to a torque. We then define the concept of the rotational kinetic energy for a point mass, systems of discrete masses, and continuous rigid bodies. We develop the angular work-kinetic energy theorem and use it to study the conservation of energy and the conservation of mechanical energy in systems involving rotational motion. To develop these theorems, we draw from our understanding of the analogous theorems in linear motion.

Gravity has an irresistible grip on our curiosity and is able to drive our imagination to completely different theoretical spaces. This very fact alone sets gravity aside from all other types of physical interactions we know. Indeed, gravity is the only physical interaction of which we have a conscious experience and this awareness is with us every second of our life. In this book we set out to try to address the question: '…what is gravity and how does gravity actually work?'. This book is meant as a guide in a journey that will take us from our basic understanding of gravity, the one that is deeply coded in our brains even at an instinctive level, to the more physically detailed and yet incorrect description provided by Newton’s theory of gravity. The journey will then lead us to the mathematically beautiful and physically profound description that Einstein has proposed with his 'general theory of relativity', and that is elegantly embodied in his field equations.

The rhetoric of royal legitimacy emphasised divine selection and personal merit, but the proof of these qualities lay in the ruler’s measured but effective exercise of his power. For mirror-writers, evidence of fitness to rule depended on not only military success but also, and more particularly, the stability and prosperity of the realm. Foremost among the principles of governance, for many mirror-writers, was justice, understood partly in legal terms – the ruler was responsible for upholding the law – and partly in terms of the king’s judicious management of the multiple constituencies who made up his realm. The four texts in this chapter address the underlying supports of royal authority and the principles that rulers should adopt in their governance. Justice, central to the theory of virtue, appears in this context as central to the theory of governance. In enumerating the pillars necessary for the support of sovereignty, the authors represented in this chapter emphasise, in addition to justice, religion, force, wealth and kindness. The texts are drawn from Pseudo-Aristotle, Sirr al-asrār; al-Māwardī, Tashīl al-naẓar wa-taʿjīl al-ẓafar; Niẓām al-Mulk, Siyar al-mulūk; and al-Ṭurṭūshī, Sirāj al-mulūk.

How can the use of force by states be constrained under international law?  Under what circumstances has the use of force by states been deemed "legitimate"?  How are rules about the legitimate use of force changing?  These questions are examined in depth, along with the relevant instruments of international law.  The chapter details the various ways that "force" and "aggression" have been defined and used.  It treats all the major forms of force or coercion, including full-scale military operations, economic sanctions and reprisals, proxy and clandestine forces, small-scale conflicts.  It then traces growing prohibitions on the use of force from the League, the UN, the ICJ, and other sources.  Finally, the Laws of Armed Conflict are highlighted as are the Geneva Conventions and instruments limiting munitions and conflict on the land, air, and sea.  The use of drones and autonomous weapons systems, guided by AI, are highlighted as a growing area of concern for international law.

Contemporary thought on human rights is rooted in significant part in the soil of natural law theory. Some natural law theorists embrace the idea of enforceable rights to a range of desirable socioeconomic outcomes. But natural law theory is best understood as grounding rights that foster these outcomes indirectly. Such rights, as I envision them, qualify as socioeconomic rights because they directly concern the socioeconomic sphere and because they further those aspects of flourishing which alternative schemes of socioeconomic rights are often intended to protect. There is, I suggest, a plausible natural law case to be made for indirectly promoting these dimensions of well-being by enforcing legal rights to bodily integrity, property, and labor, and so accepting robust limits on the use of force. In this chapter, I lay the groundwork for an exploration of that case by elaborating the variety of natural law theory in which I seek to ground my understanding of socioeconomic rights. I go on to discuss norms germane to the institutional context of socioeconomic life and to propose a set of fundamental socioeconomic rights and briefly consider the significance of these rights.

The relation between neutrality and the use of force is better to be kept within the law of armed conflict rather that the law on the use of force between States. This means that the right of self-defence cannot be the indispensable legal basis for the use of force between belligerents and neutrals. On the contrary it appears that neutral due diligence has been relied on as a basis to expand the scope of the right of self-defence. The latter is admissible as the basis for resort to force only in the case of resistance of a neutral State to repel belligerent violation of its territory or by a belligerent that fully complies with its duties under the law of neutrality.

Just as a debate about the fundamental nature of physical entities arose after Descartes, a similar issue arose after Newton. Like Descartes, but of course with very different epistemological and methodological considerations, Newton held that the most fundamental conserved quantity was “quantity of motion” or momentum. Leibniz opposed this, arguing instead for vis viva or “living force.” This controversy introduced two kinds of problems: 1) whether and how empirical proofs could be generated for metaphysical conceptions, and consequently 2) how to understand the relationship between metaphysics and experimental philosophy. These concerns were handled quite differently by two important philosophers: Gravesande and Du Châtelet. Their moves partly resolved older debates, but also partly reconfigured them into new questions we are still attempting to answer.

In this chapter, I analyze the connections that Hume draws between utility as a source of merit, the principle of humanity as that which elicits our approval of useful mental qualities and blame for pernicious ones, and the role of reason, both in determining usefulness and in reaching just evaluations of merit or demerit. I contrast this set of connections with another set of moral sentiments that have as their object the immediately agreeable or disagreeable qualities. I reconstruct the argument meant to establish the principle of humanity as the foundation of morality, in part with the aim of showing that the sentiment of humanity gains force through our collective participation in the common point of view from which we form a general standard of virtue.

Chapter 5 is mainly devoted to the interaction between waves and immersed bodies. In general, an immersed body may oscillate in six different modes, three translating modes (surge, sway, heave) and three rotating modes (roll, pitch, yaw). An oscillating body radiates waves, and an incident wave may induce a corresponding excitation force for each one of the six modes. When a body oscillates, it radiates waves. Such radiated waves and excitation forces are related by so-called reciprocity relationships. Such relations are derived not only for a single oscillating body but even for a group (or 'array') of immersed bodies. Axisymmeric bodies and two-dimensional bodies are discussed in separate sections of the chapter. Although most of this chapter discusses wave-body dynamics in the frequency domain, a final section treats an immersed body in the time domain.

Referring to a simple illustration, a verbal explanation is given by the essential, but perhaps paradoxical, statement that to absorb wave energy from a wave by means of an oscillating system, it is required that the system radiates a wave which interferes destructively with the incident wave. Then various mathematical relations are derived concerning the conditions for an oscillating body to remove energy from an incident wave. The mathematical conditions for wave-power absorption may be illustrated as a paraboloid-shaped 'island' on an infinite complex-plane 'ocean' surface. The top of this 'island' corresponds to maximum absorbed power. An additional matter is the optimum control of a wave-energy converter (WEC) body. Thus far, the WEC body's shape and oscillation mode have been taken into account, but not its physical size. The latter is an important parameter related to the cost of the WEC, when the Budal upper bound is explained and discussed. Another important phenomenon, related to the Keulegan–Carpenter number, is discussed, in relation to an example of a WEC body. In a final section of the chapter, a WEC body, oscillating in several modes of motion, is discussed.