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This chapter reviews Newton’s definitions with emphasis on his theoretical concept of a centripetal force. It reviews his treatment of time, space, motion, Laws of Motion and their corollaries, before emphasizing the empirical support Newton cites for his Laws of Motion. The Laws of Motion, together with theorems derived from them, afford systematic dependencies that make orbital phenomena measure features of centripetal forces maintaining bodies in those orbits. This chapter gives an account of Newton’s inferences to the centripetal direction of and inverse-square variation of the forces maintaining satellites and planets in their respective orbits of Jupiter, Saturn and the sun. It also discusses a number of philosophical lessons these inferences can teach us about scientific method. Appendix 1 gives details of Newton’s appeal to pendulum experiments. Appendix 2 exhibits Newton’s proofs of propositions 1-4 of book 1. Appendix 3 extends Newton’s precession theorem to orbits of large eccentricity.

Modern science could not advance until the ideas of force and mass were understood. Newton provided this understanding by stressing the role of momentum and by creating his three laws of motion. But notions were not rigorously based and left much undefined until Mach gave a detailed operational method for defining mass and force. Newton's genius and insight was illustrated by the fact that he got everything right in spite of this. When force and mass were examined with rigorous logic, they were found to be in accord with Newton's intuition.

In the usual Newtonian-Laplacian image, Isaac Newton's three laws of motion are often taken as the basic ones. Heinrich Hertz, on the other hand, formulated one and only one law of motion: that every free system persists in its state of rest or of uniform motion in a straightest path. Hertz's fundamental law can be formulated as follows: a free system moves with constant speed along a path that is as straight as it can be without breaking the connections of the system. Hertz's formulation of the fundamental law was surprisingly stable throughout his work on mechanics. It was the geometry of systems of points that allowed Hertz to limit the laws of motion to his one simple, elegant and intuitively appealing fundamental law. Hertz mentioned one other law that could have replaced his fundamental law on free systems, namely, the law of least acceleration.

The law of gravitation could not be proved and could not be applied unless the basic laws of motion were understood. Newton realized this and in an amazing intellectual tour de force, he formulated his famous three laws of motion that govern all of mechanics. All mechanics, from falling bodies, rotating machinery and automobiles, to aircraft and planetary motion; in short everything that moves and is subject to forces, are governed by Newton's laws of motion. This was the most important and far-reaching achievement in the history of physical science. It superseded all previous attempts to understand motion, and has never been surpassed. These laws properly stressed the importance of inertia, momentum, and reaction forces for the first time.

The period until 1860 saw a steady flow of interesting new theoretical discoveries in mechanics, even in the area of mechanics of systems of finitely many mass points. Moreover, there was a continued discussion about the meaning, nature, and foundations of the principles of mechanics and the concepts entering into them. These discussions went on till well into the 20th century, and provide the background and the mechanical context for Heinrich Hertz's book (Principles of Mechanics). This chapter presents the most important elements of this mechanical context for Hertz's work. First, the development of mechanics from Isaac Newton to Hertz is summarised, focusing on the different laws and principles of mechanics discussed by Hertz. Newton's laws of motion are examined, along with the principle of virtual work, d'Alembert's principle, Lagrange multipliers, principle of live force, principle of conservation of energy, Gauss's principle of least constraint, Jacobi's version of the principle of least action, Hamilton's equations of motion, and basic notions of mechanics. A terminology of concepts such as mass, space, time, distance, and force is also presented.

Modern mechanics begins with the publication in 1687 of Isaac Newton’s Principia, in which he explains mathematically the motions of planets, moons, comets, and tides in what is now known as the three laws of motion. This chapter assumes that Newton’s three laws of motion refer fundamentally to point particles. After deriving the laws of momentum, angular momentum, and work-energy for point particles, it is shown that, given certain plausible and universally accepted additional axioms, essentially the same laws can be proved to apply to macroscopic bodies, considered as collections of the elementary point particles. This chapter also discusses Newton’s space and time, single point particle, collective variables, the law of momentum for collections, the law of angular momentum for collections, the work-energy theorem for collections, potential and total energy for collections, the center of mass, center of mass and momentum, center of mass and angular momentum, center of mass and torque, change of angular momentum, center of mass and the work-energy theorems, center of mass as a point particle, and special results for rigid bodies.

This chapter addresses Baxter’s response to Copernicanism, substantial forms, Descartes’s laws of motion, and Henry More’s variant of mechanical philosophy. Baxter was far more concerned about changing notions of substance and causality than he was about Copernicanism. His objections to mechanical philosophy stemmed from a desire to affirm secondary causes as intrinsic sources of motion, which he regarded as important to a correct understanding of God and creation. He defended a concept of substantial form and objected to the theological foundation of Descartes’s first law of motion. Baxter also argued for the plausibility of various kinds of nonliving principles of motion against Henry More’s restriction of motion to spiritual beings.

The early modern period (1543-1785) contributed many physical concepts that are simple enough to be conveyed with a diagram. Copernicus’s heliocentric universe (1543), Galileo’s discovery of the mountains on the moon (1610), Kepler’s laws of planetary motion (1620), Boyle’s law (1662), Newton’s theory of color (1666), and Bernoulli’s principle (1733) are but six of the eighteen topics covered in this period. Experiments and observations that generated measurements became the norm.

This chapter describes a planning problem that generates competitive equilibrium allocations and compares two methods for solving it. The first method uses state- and date-contingent Lagrange multipliers; the second uses dynamic programming. The first method reveals a direct connection between the Lagrange multipliers and the equilibrium prices in a competitive equilibrium to be analyzed in Chapter 7. The second method provides good algorithms for calculating both the law of motion for the optimal quantities and the Lagrange multipliers. The chapter also describes a set of MATLAB programs that solve the planning problem and represent its solution in various ways. These programs are used to solve the planning problem for six sample economies formed by choosing particular examples of the ingredients from Chapter 4.

In 1973, Gunnar Johansson launched the systematic study of the perception of the human body in motion. Merging his applied interests in traffic safety and his theoretical interests in vector analysis and decomposition, Johansson produced captivating displays of people dancing and doing other ordinary activities, but with the people represented only by lights on their joints. This chapter discusses Johansson’s research and some of my own follow-up on his ideas, serving as background for ideas in this volume.

This chapter begins by discussing Isaac Newton's biography and private life. It notes some of Newton's traits, which allowed him to face all the hurdles that came his way. It investigates how he developed his interest in the logical consistency in nature like the daily variations of the tides and the extent of the Moon's influence. It states that it was during the Bubonic plague years that he developed most of his basic mathematical ideas but remained secretive and failed to publicise them. It investigates how derived ideas such as the laws of motion and gravity. It also talks about his construction of the first reflecting telescope. This chapter also discusses how Newton gained recognition from his fellow researchers and how he tried to make more achievements, the Principia and the Opticks. It tells of some of the social acquaintances that Newton has made throughout his life as a scientist.

This chapter describes an economic environment with five components: a sequence of information sets, laws of motion for taste and technology shocks, a technology for producing consumption goods, a technology for producing services from consumer durables and consumption purchases, and a preference ordering over consumption services. A particular economy is described by a set of matrices that characterize the motion of information sets and of taste and technology shocks; matrices that determine the technology for producing consumption goods; matrices that determine the technology for producing consumption services from consumer goods; and a scalar discount factor that helps determine the preference ordering over consumption services. The chapter describes and gives examples of each component of the economic environment.

It was apparent from its beginning that special relativity developed as the invariance theory of electrodynamics would require a modification of Newton’s three laws of motion. This chapter discusses that modified theory. The relativistically modified mechanics is presented and then recast into a fourvector form that demonstrates its consistency with special relativity. Traditional Lagrangian and Hamiltonian mechanics can incorporate these modifications. This chapter also discusses the momentum fourvector, fourvector form of Newton’s second law, conservation of fourvector momentum, particles of zero mass, traditional Lagrangian and traditional Hamiltonian, invariant Lagrangian, manifestly covariant Lagrange equations, momentum fourvectors and canonical momenta, extended and invariant Hamiltonian, manifestly covariant Hamilton equations, the Klein-Gordon equation, the Dirac equation, the manifestly covariant N-body problem, covariant Serret-Frenet theory, Fermi-Walker transport, and example of Fermi-Walker transport.

This chapter presents an account of the origins and development of one's ability to classify moveable entities as either animate or inanimate. The account builds on the known abilities of young infants to find three dimensional objects and to reason about some of their fundamental physical characteristics, for example, that they occupy space, move as a whole, or cannot pass through each other. This chapter also shows that motion paths are ambiguous for adults, not just infants. A moving object is perceived as inanimate when its motion path is consistent with Newtonian laws of motion. If the motion path violates Newtonian principles, then animacy is perceived.

Deborah Brown looks at how Hobbes and Descartes used the language of ‘tendency’ in their natural philosophies. She contrasts the problems that arise for Hobbes as he tries to reduce tendencies away with a Descartes, for whom ‘tendency talk is not a mere façon de parler’ but rather is ‘real and causally explanatory’, and who strives to incorporate inherent tendencies into his broader mechanistic ontological framework. The resulting interpretation of Descartes sees him as much closer in his conception of natural laws to Nancy Cartwright than to David Lewis. One of the real benefits of this interpretation, claims Brown, is that it ‘might just help to demystify Descartes’s references to active forces’. Having thus establishing that Descartes was a realist about tendencies who nevertheless remained committed to a non-teleological, mechanistic account of nature, Brown contrasts this Cartesian picture with Hobbes’s reductive mechanics that eliminated forces and tendencies by equating them with actual motions, including even the ‘force of a body at rest’.

In his Principia (1687), Sir Isaac Newton laid out his discovery of the laws of motions and the law of universal gravitation. His historic journey involved a critical moment when, aided by discussions with Robert Hooke, he conquered the challenge of circular motion, e.g. one body circling another, by introducing the concept of force. The Principia was a tour-de-force demonstration of the intelligibility of the universe and ultimately broke physics away from philosophy. This work led directly to the concept of energy.