Chapter 2
Vector Analysis and Classical and
Relativistic Mechanics
2.1 Kinematics and Dynamics of a Point Mass
Kinematics is the study of motion without reference to forces; the Greek
word kinema means "motion." Dynamics is the study of motion under the
action of forces; the Greek word dynamis means "force." Also, the Greek
word mechanikos means "machine."
A curve C, defined by a vector-valued function of time r = r(t) provides
the mathematical description of the trajectory in M^3 of a particle (point
mass) so that the location of the particle at time t is at the end point
of the vector r(t). The initial point O of the vector is the origin of the
corresponding frame in which the motion is studied. It is clear that the same
motion can be studied in different frames and in different coordinates. The
Prenet trihedron (page 32) is an example of a coordinate system in which
the particle is at rest, but the coordinate system is moving. The objective
of this section is to derive the rules for describing the motion of a point
mass in different coordinate systems.
Unless explicitly mentioned otherwise, we assume that the curve C is
smooth, that is, the unit tangent vector u exists at every point of the
curve; see page 27.
2.1.1 Newton's Laws of Motion and Gravitation
The motion of a point mass m is related to the net force F acting on the
mass. In an inertial frame, this relation is made precise by Newton's three
laws of motion, and conversely, every frame in which these laws hold is
called inertial. These laws were first formulated by Newton around 1666,
less than a year after he received his bachelor's degree from Cambridge
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