Golstein

The motion of material bodies formed the subject of some of the earliest research
pursued by the pioneers of physics. From their efforts there has evolved a vast
field known as analytical mechanics or dynamics, or simply, mechanics. In the
present century the term "classical mechanics" has come into wide use to denote
this branch of physics in contradistinction to the newer physical theories, espe-
especially quantum mechanics. We shall follow this usage, interpreting the name to
include the type of mechanics arising out of the special theory of relativity. It is
the purpose of this book to develop the structure of classical mechanics and to
outline some of its applications of present-day interest in pure physics. Basic to
any presentation of mechanics are a number of fundamental physical concepts,
such as space, time, simultaneity, mass, and force. For the most part, however,
these concepts will not be analyzed critically here; rather, they will be assumed as
undefined terms whose meanings are familiar to the reader.

MECHANICS OF A PARTICLE
Let r be the radius vector of a particle from some given origin and v its vector
velocity:
v=?. A-1)
dt
The linear momentum p of the particle is defined as the product of the particle
mass and its velocity:
p = mv. A.2)
In consequence of interactions with external objects and fields, the particle may
experience forces of various types, e.g., gravitational or electrodynamic; the vec-
vector sum of these forces exerted on the particle is the total force F. The mechanics
of the particle is contained in Newton's second law of motion, which states that
there exist frames of reference in which the motion of the particle is described by
the differential equation

The equation of motion is thus a differential equation of second order, assuming
F does not depend on higher-order derivatives.
A reference frame in which Eq. A.3) is valid is called an inertial or Galilean
system. Even within classical mechanics the notion of an inertial system is some-
something of an idealization. In practice, however, it is usually feasible to set up a co-
coordinate system that comes as close to the desired properties as may be required.
For many purposes, a reference frame fixed in Earth (the "laboratory system") is
a sufficient approximation to an inertial system, while for some astronomical pur-
purposes it may be necessary to construct an inertial system by reference to distant
galaxies.
Many of the important conclusions of mechanics can be expressed in the form
of conservation theorems, which indicate under what conditions various mechan-
mechanical quantities are constant in time. Equation A.3) directly furnishes the first of
these, the

Conservation Theorem for The Linear Momentum of a Particle: If the total force,
F, is zero, then p = 0 and the linear momentum, p. is conserved.
The angular momentum of the particle about point O, denoted by L, is defined
as
L=rxp, A.7)
where r is the radius vector from O to the particle. Notice that the order of the
factors is important. We now define the moment of force or torque about O as
N=rxF. A.8)
The equation analogous to A.3) for N is obtained by forming the cross product of
r with Eq. A.4):