Classical Mechanics

The following sections are included:

• Preface
• Contents

Classical Mechanics describes the rules and laws governing the nonrelativistic motion of material bodies, where motion is defined as the change of location as a function of time. This motion is subject to forces that are presumed to be known. The mere description of the motion of a body is called kinematics. The study of the underlying laws that govern a mechanical motion is called dynamics. A large part of mechanics deals with point masses and systems of point masses, where the bodies are idealized as point-like objects without spatial extension, with only one intrinsic property, called the mass m.

The elementary formulation of mechanics given in the previous chapter presumes that the forces are known. In practice, however, one very often encounters situations where some of the forces are not known a priori. This typically involves constraints that limit the allowed motion such that, in many instances, one may not be able to formulate the corresponding constraining forces independent of the dynamical situation. In such cases, one needs to adapt Newton’s second law so that the determination of the constraining forces becomes an integral part of the solution of the equations of motion. The formal framework to achieve this is furnished by the Lagrangian approach to mechanics, named after Joseph Louis Lagrange.

The formulation of mechanics given in the previous sections is based on Newton’s axioms. In an equivalent axiomatic approach, one may postulate that the physical trajectories of a system are given by stationary solutions of a variational principle called Hamilton’s Principle. This approach and its implications will be the subject of the following sections. The great importance of Hamilton’s Principle lies in fact that it provides the mathematical basis for quantum-mechanical and field-theoretical formulations of modern physics that go beyond classical mechanics and the applicability of Newton’s axioms.

Yet another alternative approach to classical mechanics, Hamiltonian Mechanics, is formulated in terms of a function — the Hamilton function H — that is closely related to the total mechanical energy of the system. Because of the paramount importance of the concept of energy in physics in general, Hamiltonian Mechanics is particularly well suited for going beyond classical mechanics. Indeed, the formulation of quantum mechanics as conceived in the 1920s by Heisenberg and Schrödinger, in particular, was strongly influenced by Hamiltonian mechanics.

A rigid body is a system of n point masses m_i at positions r_i such that all pairwise distances

Many problems involve small displacements from a stable equilibrium position, with a system-dependent restoring force that seeks to drive the system back to equilibrium. The problems may also include additional external forces or internal friction, etc. In many instances, the system’s response produces small oscillations or vibrations about the equilibrium point. Simple examples are a small ball rolling around the bottom of a bowl or an atom or molecule distorted by a weak external electric or magnetic field. In this part, we consider situations for systems where the unperturbed system can be described in terms of a potential. To illustrate the basic approach, we first look at the one-dimensional case.

The central objective of the description of mechanical phenomena given in the preceding chapters was to obtain equations of motions for point masses whose solutions — the trajectories — allow one to follow the time evolution of individual particles (or, as in the case of rigid bodies, a whole collection of particles whose relative positions are fixed). There is, however, a large class of problems where this is not a very useful answer. Consider, for example, an elastic string stretched taut between two fixed points. If the string is plucked, it vibrates. In order to describe this vibration, what we need is a description of the (continuous) mass distribution of the string as a function of position and time. This is an example for the type of problems treated in continuum mechanics. In general, this area of mechanics deals with the dynamics of elastic bodies, of liquids, and of gases. The description aims to provide equations for fields, rather than trajectories…

In this part, we will explore two issues that go beyond the traditional range of validity of classical mechanics, namely the chaotic behavior of some classical systems and the resulting unpredictability of trajectories, and the non-Galilean transformations between inertial frames for objects moving at high speeds that follow from the theory of special relativity.

The following sections are included:

• Appendices
  • Coordinates, Vector Operations, etc.
    • Cartesian coordinates x, y, z
    • Cylindrical coordinates ρ, ϕ, z
    • Spherical coordinates r, φ, θ
    • Arbitrary curvilinear coordinates q₁, q₂, q₃
    • Vector-operator identities
    • Integral theorems
  • Dirac δ Distribution
  • Green’s Function Method: An Example
• Index