Newtonian inertial frame

The problem relates directly to the constancy of c, which implies that the velocity of light is independent of both the motion of its source and the direction of propagation, a condition that cannot hold in more than one Newtonian inertial frame if the Galilean principle of relativity applies. Since there is no evidence that the laws of physics are not identical in all inertial frames of reference the only conclusion is that the prescription for Galilean transformations needs modification to be consistent, not only with simple mechanics, but also with electromagnetic effects. 

We consider two inertial frames of reference, S and S, with origins O and O and axes Ox, Ov, Oz in S and Ox. OV, O in S. (An inertial frame of reference is defined as a coordinate frame in which the laws of Newtonian mechanics hold one of the consequences of the special theory of relativity is that any pair of such inertial frames can only move with a uniform velocity relative to each other.) Now an observer at the origin O will describe an event in his frame by values of x, y, z, t where t is the time measured by a clock at rest in S. Similarly an observer at O will describe the same event in terms of the corresponding values x, y, z, t measured in S. ... 

The fictitious forces are conventionally derived with the help of the framework of classical mechanics of a point particle. Newtonian mechanics recognizes a special class of coordinate systems called inertial frames. The Newton s laws of motion are defined in such a frame. A Newtonian frame (sometimes also referred to as a fixed, absolute or absolute frame) is undergoing no accelerations and conventionally constitute a coordinate system at rest with respect to the fixed stars or any coordinate system moving with constant velocity and without rotation relative to the inertial frame. The latter concept is known as the principle of Galilean relativity. Speaking about a rotating frame of reference we refer to a coordinate system that is rotating relative to an inertial frame. 

To examine the elementary mathematical operations involved in Newtonian mechanics, for example, we describe the motion of a material particle by the Newton s second law of motion. The Newtonian frame of reference adopted is henceforth named O. The moving relative reference frame is designated O. The basic task is thus to transform the Newton s second law of motion as formulated in an inertial frame of reference into a relative rotating frame of reference. 

The Newtonian laws of motion have been formulated for inertial frames of reference only, but no special IS has been singled out so far, since classical nonrelativistic mechanics relies on the Galilean principle of relativity ... 

We now exploit the relativity (in the sense of Galilei) of Newton s laws for the determination of the most general Galilean transformations. Newtonian physics crucially relies on the concept of absolute time the time difference df between two events is the same in all inertial frames. The time shown by a clock is in particular independent of the state of motion of the clock. As a consequence the most general relation between the time f in IS and the time f in IS is given by... 

Another formal deficiency of Newtonian mechanics and therefore of the Galilei transformations is the existence of Maxwell s equations (2.94)-(2.97), which contain the speed of light c as a parameter. Maxwell s equations directly yield the wave equation (2.121), which states that the propagation of electromagnetic fields in vacuum occurs with speed c. Either Maxwell s equations are not valid in all inertial frames of reference or the Galilei transformations cannot be the correct coordinate transformations between inertial frames. Experimentally one finds that the Maxwell equations are valid for any inertial system. 

In order to derive the relativistic analog of the Newtonian equation of motion, we again consider the motion of a single particle with velocity v with reference to the inertial frame IS. The instantaneous rest frame of the particle at time t is denoted as IS Since the particle is at rest in IS at time t (and will therefore move only with small velocities classical equation of motion as given by Eq. (2.2) to hold exactly in IS, ... 

Here, F denotes the familiar nonrelativistic Newtonian force, which is independent of the chosen frame of reference (cf. section 2.1.2) and does therefore not carry a prime. The classical nonrelativistic Newtonian momentum is denoted as = mv, and m is again the rest mass of the particle. Newton s equation of motion is well tested and established for small velocities and we therefore have no reason to question its validity in IS. It cannot be overemphasized, however, that Eq. (3.123) is only valid in IS but not in other inertial frames of reference. In order to find the correct equation of motion for general inertial frames of reference we will establish a covariant 4-vector equation which reduces to Eq. (3.123) if the particle is at rest. 

We now understand that the first law assures the existence of inertial frames, and the second and third laws are valid in the inertial frames. This is the essence of Newtonian mechanics. 

In classical Newtonian mechanics, relations between the spatial parameters and time in two inertial frames S and S are expressed in terms of the Galilean transformations. Assume that S is moving with constant speed v in the direction of the positive x axis of S. If the coordinate axes of S are parallel to those of S, the Galilean coordinate transformations are... 

In this case, the rotational inertia is a scalar, as the two vectors L and Q are colinear. As each body is featured in translational mechanics by an inertial mass (mechanical inductance) M (which is also a scalar in the Newtonian frame), relating the velocity v to the momentum P, the relationship between the two inductances can be written as ... 

Restriction to Newtonian frame. It is only in assuming constant the inertial mass M that Newton s second law as written in Equations Fl.l and FI.2 is retrieved from the expression F1.3. In the Formal Graph theory, one is therefore invited to indicate this restriction of validity range in writing a //n under the equal sign of Newton s second law (when classically composed with Equations Fl.l and FI.2), to recall the linearity condition of the operator that is an inertial mass. 

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