Suppose that we have found an inertial frame of reference. Let us
set up a Cartesian coordinate system in this frame. The motion
of a point object can now be specified by giving its position vector,
,
with respect to the origin of the coordinate system, as a function of time, .
Consider a second frame of reference moving with some
*constant* velocity
with respect to the first frame. Without loss of generality,
we can suppose that the Cartesian axes in the second frame are parallel
to the corresponding axes in the first frame, that
,
and, finally, that the origins of the two frames instantaneously coincide at --see Figure 1. Suppose that the position vector
of our point object is
in the second frame of reference.
It is evident, from Figure 1, that at any given time, , the coordinates of the
object in the two reference frames satisfy

This coordinate transformation was first discovered by Galileo Galilei, and is known as a

By definition, the instantaneous velocity of the object in our first reference frame is given by
, with an analogous
expression for the velocity, , in the second frame.
It follows from differentiation of Equations (1)-(3) with respect to time that the velocity components in the two frames satisfy

These equations can be written more succinctly as

Finally, by definition, the instantaneous acceleration of the object in our first reference frame is given by
, with an analogous
expression for the acceleration, , in the second frame.
It follows from differentiation of Equations (4)-(6) with respect to time that the acceleration
components in the two frames satisfy

(8) | |||

(9) | |||

(10) |

These equations can be written more succinctly as

According to Equations (7) and (11), if an
object is moving in a straight-line with constant speed in our original
inertial frame (*i.e.*, if
) then it also
moves in a (different) straight-line with (a different) constant speed
in the second frame of reference (*i.e.*,
). Hence,
we conclude that the second frame of reference is *also* an inertial frame.

A simple extension of the above argument allows us to conclude that there
are an *infinite* number of different inertial frames moving with *constant
velocities* with respect to one another. Newton through that one of these inertial frames was special, and
defined an absolute standard of rest: *i.e.*, a static object in this frame was in a state of absolute rest.
However, Einstein showed that this is not the case. In fact, there is no absolute standard of rest: *i.e.*, all
motion is relative--hence, the name ``relativity'' for Einstein's theory. Consequently, one inertial frame is
just as good as another as far as Newtonian dynamics is concerned.

But, what happens if the second frame of reference *accelerates* with
respect to the first? In this case, it is not hard to see that Equation (11)
generalizes to

(12) |

A simple extension of the above argument allows us to conclude that any frame of reference which accelerates with respect to a given inertial frame is not itself an inertial frame.

For most practical purposes, when studying the motions of objects close to the
Earth's surface, a reference frame which is fixed with
respect to this surface is approximately inertial. However,
if the trajectory of a projectile within such a frame is measured to high
precision then it will be found to deviate slightly from the predictions
of Newtonian dynamics--see Chapter 7. This deviation
is due to the fact that the Earth is rotating, and its surface is therefore
accelerating towards its axis of rotation. When studying the motions of
objects in orbit around the Earth, a reference frame whose origin
is the center of the Earth, and whose coordinate axes are fixed with respect
to distant stars, is approximately inertial. However, if such
orbits are measured to extremely high precision then they will
again be found to deviate very slightly from the predictions of Newtonian
dynamics. In this case, the deviation is due to the Earth's orbital
motion about the Sun. When studying the orbits of the planets
in the Solar System, a reference frame whose origin is the center of the Sun, and whose coordinate axes are fixed with respect
to distant stars, is approximately inertial. In this case, any deviations
of the orbits from the predictions of Newtonian dynamics
due to the orbital motion of the Sun about the galactic center are
far too small to be measurable. It should be noted that it is impossible
to identify an *absolute* inertial frame--the best approximation to such
a frame would be one in which the cosmic microwave background appears
to be (approximately) isotropic. However, for a given dynamical problem, it is always
possible to identify an *approximate* inertial frame. Furthermore, any
deviations of such a frame from a true inertial frame can be incorporated
into the framework of Newtonian dynamics via the introduction of so-called fictitious forces--see Chapter 7.