The Principia is one of the greatest books of all time. In it Newton formulates a “System of the World,” incorporating his own new mathematical ideas as well as displaying a masterful command of geometry. “Newton was the greatest genius that ever existed,” the mathematician Joseph-Louis Lagrange is alleged to have said, adding, with a grain of humor, “and the most fortunate, for we cannot find more than once a system of the world to establish.”

We include a chapter on Newton's laws of motion and the Principia even though this work is not directly related to variational principles. We do so for two reasons: the monumental importance of his work on mechanics and the fact that his ideas are crucial in the development of the principle of least action. Moreover, Newton himself was a proponent of the Aristotelian simplicity and economy (Lyssy, 2015). In his first rule for the study of Natural Philosophy we read: “As the philosophers say: Nature does nothing in vain, and more causes are in vain when fewer suffice. For nature is simple and does not indulge in the luxury of superfluous causes” (Cohen and Whitman, 1999, p. 794).

Newton's Propositions on the Laws of Motion

Book I of the Principia starts with three axioms, the proverbial Newton's laws of motion. After some definitions, Newton states (Motte, 1729, p. 83):

LAW I

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

LAW II

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed

LAW III

To every action there is always an opposed and equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

In many derivations of the Principia, Newton uses his second law following a method of impulses. The body moves on straight lines that are broken by the actions of impulses acting at regular intervals of time.

In this chapter we visit mechanics in the Age of Enlightenment. In that period, Newton's ideas, which allowed only the study of motion of bodies free in space, were extended to incorporate constraints in mechanical systems. The key figures are James Bernoulli, Jean le Rond d'Alembert and Joseph-Louis Lagrange. The central concept is the principle of virtual work, which establishes the conditions of static equilibrium and its extension to dynamics.

The Principle of Virtual Work

The idea of treating a static equilibrium problem using ideas from dynamics goes back to Aristotle's text “Mechanical Problems.” Although his authorship is disputed, it is probably the product of his contemporaries of the Peripatetic School. In the discussion of the lever – although the word equilibrium is never used – we read “the ratio of the weight moved to the weight moving it is the inverse ratio of the distances from the center” (Aristotle, 350 BC/1955, p. 353). This statement is regarded by many as a precursor to the so-called method of virtual velocities, or virtual displacements (Capecchi, 2012). In “On Mechanics,” one of his early works, (Galileo, 1600/1960) borrows Aristotle's idea and treats equilibrium on an inclined plane as an invariance under hypothetical displacements. In the “Discourses,” he uses notions of statics and dynamics in the same sentence: “when equilibrium (that is, rest) is to prevail between two moveables, their [overall] speeds or their propensities to motion – that is, the spaces they would pass in the same time – must be inverse to their weights [gravità]”(Galilei, 1638/1974, p. 173). Since Galileo talks about the “propensity” to move, and the system is at rest, the velocity refers to a hypothetical motion in a time different from the time of our universe. Galileo realized that, for weights on an inclined plane, the determining factor for equilibrium is their motion away from or “removal from the center of the earth”(Galileo, 1600/1960, p. 177). For the inclined plane with two masses of Figure 5.1, the ratio of the masses is 2. In order for the system to be at equilibrium, the ratio of the vertical velocities (if they were displaced in the same amount of time) is 1/2.

In this introductory chapter, which actually is the core of the entire book, we make a serious effort to bring together a variety of ideas from philosophy and physics. We first lay the epistemological foundations for our approach to particle physics. These philosophical foundations, combined with tools borrowed from the amazingly sophisticated, but still not entirely satisfactory, mathematical apparatus of present-day quantum field theory and augmented by some new ideas, are then employed to develop a mathematical representation of fundamental particles and their interactions. In this process, also the relation between “fundamental particle physics” and “quantum field theory” is going to be clarified.

This first chapter consists of two voluminous sections: a number of philosophical contemplations followed by a discussion of mathematical and physical elements. The reader might wonder why these massive sections are not presented as two separate chapters. The reason for keeping them together is the desire to emphasize the intimate relation between these two sections: the selection and construction of the mathematical and physical elements is directly based on our philosophical contemplations. The beauty of the entire approach is a fruit of this intimate relationship.

The presentation of the material in this chapter is based on the assumption that the reader has a basic working knowledge of linear algebra and quantum mechanics. If that is not the case, the reader might want to consult the equally entertaining and serious introduction Quantum Mechanics: The Theoretical Minimum by Susskind and Friedman [4]. The book Quantum Mechanics and Experience by Albert [5] offers a fascinating introduction with a strong focus on the measurement problem. An almost equation-free discussion of the history and foundations of quantum mechanics can be found in the book Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information by Whitaker [6]. Complex vector spaces, Hilbert space vectors and density matrices for describing states of quantum systems, bosons and fermions, canonical commutation relations, Heisenberg's uncertainty relation, the Schrodinger and Heisenberg pictures for the time evolution of quantum systems, as well as a basic idea of the measurement process are all referred or alluded to in Section 1.1; these basics of quantum mechanics are recapped only very briefly in Section 1.2. The crucial construct of Fock spaces is explained in a loose way in Section 1.1.3 and later elaborated in full detail in Section 1.2.1.