Newtonian Dynamics |

In law variety a pair of, the acceleration lasts solely
whereas the applied force lasts. The applied force needn't, however, be
constant in time — the law is true in the slightest degree times throughout the
motion. Law variety three applies to “contact” interactions. If the bodies area
unit separated, and also the interaction takes a finite time to propagate
between the bodies, the law should be changed to incorporate the properties of
the “field “ between the bodies.

Although our discussion of the geometry of motion has led to
major advances in our understanding of measurements of space and time in different
inertial systems, we have yet to come to the crux of the matter, namely — a
discussion of the effects of forces on the motion of two or more interacting
particles. This key branch of Physics is called Dynamics. It was founded by
Galileo and Newton and perfected by their followers, most notably Lagrange and
Hamilton. We shall see that the Newtonian concepts of mass, momentum and
kinetic energy require fundamental revisions in the light of the Einstein’s Special
Theory of Relativity. The revised concepts come about as a result of Einstein's
recognition of the crucial rôle of the
Principle of Relativity in unifying the dynamics of all mechanical and optical
phenomena. In spite of the conceptual difficulties inherent in the classical
concepts, (difficulties that will be discussed later), the subject of Newtonian
dynamics represents one of the great triumphs of Natural Philosophy. The
successes of the classical theory range from accurate descriptions of the
dynamics of everyday objects to a detailed understanding of the motions of
galaxies.

**The law of inertia**

Galileo (1544-1642) was the first to develop a quantitative
approach to the study of motion. He addressed the question — what property of
motion is related to force? Is it the position of the moving object? Is it the
velocity of the moving object? Is it the rate of change of its velocity? ...The
answer to the question can be obtained only from observations; this is a basic
feature of Physics that sets it apart from Philosophy proper.

Galileo observed that force influences the changes in
velocity (accelerations) of an object and that, in the absence of external forces (e.g: friction),
no force is needed to keep an object in motion that is travelling in a straight line with
constant speed. This observationally based law is called the Law of Inertia. It
is, perhaps, difficult for us to appreciate the impact of Galileo's new ideas
concerning motion. The fact that an object resting on a horizontal surface
remains at rest unless something we call force is applied to change its state
of rest was, of course, well-known before Galileo's time. However, the fact
that the object continues to move after the force ceases to be applied caused considerable
conceptual difficulties for the early Philosophers (see Feynman The Character
of Physical Law). The observation that, in practice, an object comes to rest due
to frictional forces and air resistance was recognized by Galileo to be a side
effect, and not germane to the fundamental question of motion. Aristotle, for
example, believed that the true or natural state of motion is one of rest. It
is instructive to consider Aristotle's conjecture from the viewpoint of the
Principle of Relativity —- is a natural state of rest consistent with this
general Principle? According to the general Principle of Relativity, the laws
of motion have the same form in all frames of reference that move with constant
speed in straight lines with respect to each other. An observer in a reference
frame moving with constant speed in a straight line with respect to the
reference frame in which the object is at rest would conclude that the natural
state or motion of the object is one of constant speed in a straight line, and
not one of rest. All inertial observers, in an infinite observations; this is a
basic feature of Physics that sets it apart from Philosophy proper.

Galileo observed that force influences the changes in
velocity (accelerations) of an object and that, in the absence of external forces (e.g: friction),
no force is needed to keep an object in motion that is travelling in a straight
line with constant speed. This observationally based law is called the Law of
Inertia. It is, perhaps, difficult for us to appreciate the impact of Galileo's
new ideas concerning motion. The fact that an object resting on a horizontal
surface remains at rest unless something we call force is applied to change its
state of rest was, of course, well-known before Galileo's time. However, the
fact that the object continues to move after the force ceases to be applied
caused considerable conceptual difficulties for the early Philosophers (see
Feynman The Character of Physical Law). The observation that, in practice, an
object comes to rest due to frictional forces and air resistance was recognized
by Galileo to be a side effect, and not germane to the fundamental question of
motion. Aristotle, for example, believed that the true or natural state of motion
is one of rest. It is instructive to consider Aristotle's conjecture from the
viewpoint of the Principle of Relativity —- is a natural state of rest consistent
with this general Principle? According to the general Principle of Relativity,
the laws of motion have the same form in all frames of reference that move with
constant speed in straight lines with respect to each other. An observer in a
reference frame moving with constant speed in a straight line with respect to
the reference frame in which the object is at rest would conclude that the
natural state or motion of the object is one of constant speed in a straight
line, and not one of rest. All inertial observers, in an infinite number of
frames of reference, would come to the same conclusion. We see, therefore, that
Aristotle's conjecture is not consistent with this fundamental Principle.

**Newton’s laws of motion**

During his early twenties, Newton postulated three Laws of
Motion that form the basis of Classical Dynamics. He used them to solve a wide
variety of problems including the dynamics of the planets. The Laws of Motion,
first published in the Principia in 1687, play a fundamental rôle in Newton’s Theory of Gravitation;
they are:

- In the absence of an applied force, an object will remain at rest or in its present state of constant speed in a straight line (Galileo's Law of Inertia)
- In the presence of an applied force, an object will be accelerated in the direction of the applied force and the product of its mass multiplied by its acceleration is equal to the force. And,
- If a body A exerts a force of magnitude$\left | F_{AB} \right |$ on a body B, then B exerts a force of equal magnitude $\left | F_{BA} \right |$ on A.. The forces act in opposite directions so that $\mathrm{F}_{AB}=\mathrm{F}_{BA}$ .

In law number 2, the acceleration lasts only while the
applied force lasts. The applied force need not, however, be constant in time —
the law is true at all times during the motion. Law number 3 applies to
“contact” interactions. If the bodies are separated, and the interaction takes
a finite time to propagate between the bodies, the law must be modified to include the properties of the “field “ between the bodies.

Reference:

FRANK W. K. FIRK

Professor Emeritus of Physics

Yale University