Inertial frame of reference

As a rule, geophysical literature describes the rotation of a particle on the earth surface with the help of the attraction force and the centrifugal force. It turns out that the latter appears because we use a system of coordinates that rotates together with Earth. As we know Newton s second law, wa = F, is valid only in an inertial frame of reference, that is, the product of mass and acceleration is equal to the real force acting on the particle. However, it is not true when we study a motion in a system of coordinates that has some acceleration with respect to the inertial frame. For instance, it may happen that there is a force but the particle does not move. On the contrary, there are cases when the resultant force is zero but a particle moves. Correspondingly, replacement of the acceleration in the inertial frame by that in a non-inertial one gives a new relation between the acceleration, mass, particle, and an applied force ... 

Substitution of Equation (2.38) into the Newton s second law in the inertial frame of reference ma = F gives... 

Here p is the pressure, ga the field of attraction, 5 the density of the fluid, and r the vector directed away from the axis of rotation and it is equal in magnitude to the distance between a particle and this axis. The first two terms of Equation (2.332) characterize the real forces acting on the particle, namely the surface and attraction ones. At the same time the last term is a centrifugal force, and it is introduced because we consider a non-inertial frame of reference. It is convenient to represent Equation (2.332) as... 

As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an influence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth s surface. With this purpose in mind let us choose a non-inertial frame of reference, shown in Fig. 3.5a its origin 0 is located near the earth s surface and it rotates together with the earth with angular velocity a>. The unit vectors i, j, and k of this system are fixed relative to the earth and directed as follows i is horizontal, that is, tangential to the earth s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth s axis, drawn from south to north, I is the unit vector along OiO, and K is a unit vector parallel to SN. 

First, we derive again but in a slightly different way than in Chapter 2 the equation of a motion in a non-inertial frame of reference. As before, r is the position of the moving particle with respect to 0 and OiO = ro. The position of the particle with respect to the origin 0i of the inertial frame is... 

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. 

An absolute inertial frame of reference E must be reintroduced. In plain words, introduce a modem version of the ether. 

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 governing conservation equations in the inertial frame of reference were solved with the boundary conditions of temperature and velocity ... 

The first term can be interpreted as the vibrational kinetic enei y, and the second and third terms are the effective potential energy for the vibrational motion. The final term, which is the rotational kinetic energy, becomes an addition to the potential energy in the non-inertial frame of reference of the rotating molecule. 

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. 

James Clerk Maxwell died in 1879, the same year that Albert Einstein was born. Sixteen years later Einstein recognized that Maxwell s equations are covariant with respect to the Lorentz transformations between relatively moving inertial frames of reference, that is, reference frames that are in constant relative motion in a straight line. Thus, Einstein recognized in 1895 that the laws of electrodynamics, expressed with Maxwell s held equations, must be in one-to-one correspondence in all possible inertial frames of reference, from the view of any one of them [1]. 

The symmetry requirements of the theory of relativity have geometric and algebraic modes of expression. From the geometric view in special relativity, the continuous spacetime transformations that leave the laws of nature covariant (i.e., unchanged in form) in all possible inertial frames of reference, from the view of any one of them, are the same set of transformations that leave invariant the squared differential metric ... 

Einstein s special theory of relativity postulates that the finite velocity of light in vacuiun and the laws of nature are the same in all inertial frames of reference. The consequences unique to this theory, relative to classical quantum theory. 

Ashworth DG, Davies PA (1979) Transformations between rotating and inertial frames of reference. J Phys A Math Gen 12 1425-1440... 

The famous mathematician Leonhard Euler (1707-1783) provided the laws of motion for a rigid body. Euler s as well as Newton s laws are valid in an inertial frame of reference. A frame of reference which is at absolute rest is an inertial... 

Middle section (rows d, e, and/) Summed to yield the absolute velocity (velocity with respect to an inertial frame of reference) of point P. 

The absolute velocity and acceleration of G, are expressed in terms of the body-fixed coordinate system, b ba, b3, which is fixed to the pendulum and rotates about the b3 axis as before. Although it is equivalent to expressing within the inertial frame of reference, I, j, k, the body-fixed coordinate system, b bj, bj, uses fewer terms. The velocity and acceleration for G, are respectively as follows ... 

The e, 62, 63 coordinate system is defined to generalize the discussion of the angular velocity derivations and represents the inertial frame of reference. The 3-1-2 transformation follows an initial rotation about the third axis, Cj, by an angle of to yield the e[, 2, ej coordinate system. Then a second rotation is performed about the e, axis by an angle of yielding the e", e, e, system. Finally, a third rotation is performed about the axis by to yield the final e", e" body frame of reference. This defines the transformation from the ej, C2,63 system to the ef, e, e" system. To supplement the kinematics tables, an expression for the angular velocity vector is defined from this transformation as... 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

There are inertial systems (IS) or inertial frames of reference, in which the forceless motion of a particle is described by a constant velocity. 

Newton s first law is often denoted as the principle of inertia. However, the formulation presented above constitutes a much stronger statement. It is also an existence theorem for very special coordinate systems, the inertial systems or inertial frames of reference, in which the laws of Nature take a particularly simple form. Especially, due to Eq. (2.1) free particles move in straight lines with constant velocity in inertial systems. Here, of course, r denotes the spatial coordinate vector... 

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 ... 

Equations (2.19) and (2.20) have exactly the same form, i.e., Newton s equation of motion is indeed covariant under Galilean transformations. These two equations describe the same physical situation with respect to two different inertial frames of reference. Although the physical vectorial force is of course the same in both frames of reference, F = F, its components F, and F- are in general different functions of their arguments. This relationship is given by the second equality of Eq. (2.20). 

Quantities without any indices such as the mass m, which are not only covariant but invariant under Galilean transformations, are called Galilei scalars or zero rank tensors. They have exactly the same value in all inertial frames of reference. 

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