A Modern Approach to Classical Mechanics

The following sections are included:

• From the Preface of the first edition
• Preface to the second edition
• Contents

The following sections are included:

• Why classical mechanics is still challenging
• The birth of classical mechanics
• Observations and the resulting pictures
• Time, space and motion
  • Newton's concepts
  • The mathematical pictures of space and time
  • Kinematics

Before establishing the laws, Newton introduces the definitions of mass, force, and quantity of motion. We shall take these as our starting point. Again we also mention Mach's criticisms of Newton's formulation, from his book, “The Science of Mechanics” (see Bibliography). Then we present the Newton's laws in the form used usually in classical mechanics and derive rather general consequences.

The simplest system consists of a single particle whose motion only has one degree of freedom. In the following, we investigate the constants of the motion and the conserved quantities for Newton's equation in one dimension. Since the representation in phase space here is two-dimensional, the particle's motion in phase space can be visualized graphically.

To solve the equations of motion for a particle restricted to move in two-dimensional configuration space, even if the system is conservative, one requires at least one further integral of the motion in addition to the energy, .

But if no more isolating (i.e. 'usable') integrals exist, and the system is nonlinear, chaotic behavior may appear.

The central forces hold a preeminent position in mechanics. These forces produce a second conserved quantity, the angular momentum, which is a vector. This number of conserved quantities is sufficient to render the equation of motion for a particle in a central force field integrable. However, if an additional force breaks the spherical symmetry — for instance, if a homogeneous magnetic field is switched on — then integrability vanishes…

So far we have considered one particle moving under the influence of an ‘external’ force. Now we want to apply the results of the last chapter to the planetary system, considering for the moment only the motion of a single planet around the sun. Therefore we have to show first that the motion of two interacting bodies can be related to the one of a single body in a field of force. In fact, under suitable assumptions about the force, the interaction of two point masses can be reduced to the motion of one point mass in a central field of force.

The following sections are included:

• Unbounded motion in a central force
• Kinematics of two-particle-collisions
  • Elastic collisions of two particles
  • Kinematics of elastic collisions
• Potential scattering
  • The scattering cross section
  • Scattering in the 1/r potential

In our treatment of physical processes so far, we have always tacitly chosen some particular point of reference. The positions of physicals events — for instance, the instantaneous location of a point particle — have been referred to a point of reference that coincides with the origin of the coordinate system. Until now, we have not given much thought to how the choice may influence the result of an observation or the mathematical form of physical laws. Only when considering collisions and scattering, we distinguished between kinematics in the laboratory frame and in the center of mass frame — and found that the scattering angles and also the differential cross sections differ in these two frames. We give two further examples…

At the end of the 18th century, an important stage in the development of classical mechanics was achieved. In 1788 Lagrange presented mechanics in a uniform, clear and very elegant manner in his “Mécanique analytique”. Lagrange's book is the first comprehensive representation of analytical mechanics. It is an edifice of ideas: the building blocks are logical related to each other. The representation of physical processes by mathematical theories and models is unified and elegant. This “Analytical Mechanics” is a ‘phenomenal achievement in the economy of thought’ (E. Mach, “The Science of Mechanics”).

The treatment in this chapter differs from the historical development: Lagrange's work is based on D'Alembert's principle and on the concept of virtual displacements; we consider neither in this book. Instead, after a brief account of the calculus of variations, Lagrangian mechanics is introduced by identifying Newton's equation of motion with the Euler-Lagrange equations for the Lagrangian.

Until now, we have investigated the dynamics of at most two particles. Henceforth, we focus on the dynamics of more than two particles. First, in this chapter, we will study general properties of systems of N interacting point masses.

In this chapter, we consider the mechanics of a rigid body. We imagine a rigid body as an ensemble of point masses that are fixed relative to each other by their mutual forces. This is the limiting case of the more realistic situation, in which the N point masses at positions r_α, α = 1, … , N, interact only via two-particle interactions, so that the total potential is given by (cf. Eq. (10.48)) , where r_αβ = |r_α − r_β|. The values r_γ = a_γ, γ = 1, … , N, for which all internal forces vanish,

In the last chapter we have used a system of N particles with fixed distances as a model for a rigid body, having only six degrees of freedom. However, considering the actual motions of the particles about their mean (equilibrium) positions, still using classical concepts, we are immediately confronted with a system possessing 3N coupled degrees of freedom.

In this chapter motion in a system of N interacting particles is studied. We assume that the interaction is such that there exist equilibrium positions for the particles; for small energies the particles move only in the vicinity of their equilibrium positions. However, even in the simple approach of the harmonic approximation, one has a coupled system of linear equations. It turns out, though, that one regains independent equations of motion if one transforms to so-called normal coordinates. The normal coordinates describe collective (i.e. coherent) displacements – the 'phonons' – of the particles from their equilibrium positions.

In Lagrangian mechanics, the equations of motions are Lagrange's equations for the action integral S (cf. Chapter 9). For the actual orbit, the action integral is extremal

Starting from this principle (see Section 9.3.4), W.R. Hamilton developed the most unified and elegant account of classical dynamics. Hamilton's approach to classical mechanics was an essential prerequisite for the development of quantum mechanics – the theory of 'atomic mechanics' – that started to grow at the beginning of the 20th century.

In the previous chapter, we saw that canonical transformations permit a clear presentation of the connection between symmetries of a system and constants of the motion. At least equally importantly, canonical transformations may serve also as a tool for systematically calculating the constants of the motion. In the following, we show how this can be done…

We will now go beyond the two-body problem. Though two-body systems with central field interactions are integrable, in the case of the long-ranged gravitational interaction, they are unrealistic. The gravitational force of further bodies can change the dynamics of the two bodies considered in an essential way. If one takes for instance the Earth and its moon one encounters very quickly irregularities of the moon's motion. Effects which are due to the presence of the sun (at least) appear. Thus the tides cannot be explained by the action of the sun only — as Galileo supposed —, but one has to take into account also the moon. Already Newton had a closer look at three-body systems in Proposition. LXVI, Book I of the “Principia”. In Book III he explained thus the tides (Prop. XXII) and the motion of the moon's nodes…

Only a few dynamical systems in classical mechanics are solvable exactly, while very many more can be solved only approximately or numerically. Some of these are extensions of an exactly solvable system. In such cases, the exactly solvable system can be taken as the starting point in a perturbative calculation. Even if the perturbative expansion does not converge, it may shed light on the behavior of the system.

In Newton's lex secunda, the interrelation between motion and force in different mechanical systems is ingeniously formulated as a universal law, yielding the basic equations of motion in mechanics. With the computing power we have at hand today, one could, without further considerations, solve the equations of motions even for very complex dynamical systems in which analytical methods are hopelessly ineffective for producing the solution. Over the years, the dominance of analytical methods in establishing general properties of the equations of motion and solving them has waned, in favor of computer calculations backing up the argumentation. Nevertheless, the insights obtained from analytical methods provide a more comprehensive view and deeper understanding of dynamical systems than numerical calculations can hope to allow…

The following sections are included:

• Appendix A Coordinates; vector analysis
• Appendix B Rotations and tensors
• Appendix C Green's functions

The following sections are included:

• Bibliography
• Index