Chapter 5

ποταμοῖσι τοῖσιν αὐτοῖσιν ἐμβαίνουσιν ἕτερα καὶ ἕτερα ὕδατα ἐπιρρεῖ
῾Ηράκλειτος ὁ ᾿Εφέσιος

The terminology and methodologies developed for use in mechanics have left a mark on many other aspects of physical investigation, especially since mechanics have been a mainstay of physical development. While the first observations and musings concerning mechanics may be traced back as far as ancient Greece [81][82], the foundations have been laid and formalized by none other than Sir Isaac Newton [83]. His work greatly contributed to if not, along with Leibniz [84] defined, the field of calculus but also defined a new era of mechanics. The high level mathematical formalisms described in Chapter 4 are employed in the following to formulate models of physical reality up to the abstract setting of phase spaces. The first step in this endeavour is to be aware of the used model of the physical world.

The process of establishing an acquaintance with the models of the physical world can be seen as both following a historical path as well as increasing the level of abstraction. Beginning with Newton physics was given a solid enough theoretical background to separate itself from philosophy1 . Section 5.1 describes Newton’s approach and also introduces his theoretical concept of the structure of world, which shall be used throughout this whole work. Building on these foundations, the Lagrangian formalism is introduced in Section 5.2, which, while having been introduced on purely algebraic considerations2 , today already indicates geometric structures. Carrying abstractions further Hamiltonian mechanics, which are described in Section 5.3, show mechanics as geometry in phase space. Upon having described the structures of phase spaces, statistical descriptions, as shown in Section 5.4, are employed in order to deal with an exceeding number of degrees of freedom. How the statistical description can be linked to the tangible world we expect is discussed in Section 5.5. Section 5.6 revisits the evolution in phase space but moves from a local to an integral description. The strength of the concept of a phase space is shown in Section 5.7, as it is applied in the field of quantum mechanics, which also intends to illustrate that the same structures, once found, may be applicable beyond their initial context, if proper links can be found. The whole chapter should also serve to show that there are several different levels of abstraction and formalism to choose from in the field of physics. It is hoped that the field of scientific computing would take this as an example to also extend the number of choices available.

 5.1 Newton’s Mechanics and World
 5.2 Lagrangian Formalism
 5.3 Hamiltonian Formalism
 5.4 Statistical Description – Boltzmann’s Equation
 5.5 Macroscopic Quantities
 5.6 Boltzmann’s Equation in Integral Form
 5.7 The Quantum World