Galilean frame of reference

In this section an alternative derivation of the governing equations for granular flow is examined. In this alternative method the peculiar velocity C, instead of the microscopic particle velocity c, is used as the independent variable in the particle property and distribution functions. The transformation of these functions and the governing equation follows standard mathematical procedures for changing the reference frame. The translational motion of an individual particle may be specified either by its microscopic velocity c relative to a fixed or Galilean frame of reference, or by its velocity relative to a frame of reference moving with the local velocity of the granular material Yd-... 

Transformations that preserve the relativity principle are called Lorentz transformations. The form of these looks complicated at first (see diagram). However, they arise from the simple requirement that there can be no experiment in dynamics or electromagnetism that will distinguish between two different Galilean frames of reference. 

Principle of relativity.- All of the laws of nature are identical in all Galilean frames of reference it follows that the equation of a law retains its form in time and space when we change the inertial frame of reference. The rate of propagation of the interactions is the same in all inertial frames of reference. 

Now consider the Galilean frame of reference 91 moving at a constant velocity V such that, at time t, we have V = v. The time r measured in relation to such a framework (one exists at all times) linked to the material point is called the proper time of that material point. 

Galilean frame of reference which coincides with M and has the same velocity as M at time t. 

Figure 1.2. Change of the Galilean frame of reference. The new Galilean frame of reference SRj has the velocity of translation Vj in the frame of reference...

It must be noted here that, if the Maxwell relations are independent of the chosen Galilean frame of reference, the above relations are valid for a given fluid, and are written as follows, tracking the motion of that fluid ... 

In reality, the Maxwell pressure tensor in relation to the Galilean frame of reference which, at time t and at the point in question, has the velocity v of the fluid, is expressed not by equation [3.60], but rather ... 

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. 

Superluminal dynamics is the dynamics of particles in motion with v > c where c is the speed of light in vacuum. To establish the basic concepts of SLRT, it is necessary to introduce superluminal transformation. Superluminal transformation gives the relation between the magnitudes of the frame of reference where v < c only is possible, with the same magnitudes in the frame of reference where v > c is possible. Superluminal transformation correlates with Galilean and Lorentz transformations. 

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. 

Recently, an Eulerian derivation of the Coriolis force has been reported by Kageyama and Hyodo [45]. They present a general procedure to derive the transformed equations in the rotating frame of reference based on the local Galilean transformation and rotational coordinate transformation of field quantities. 

Galilean invariance (Rothman Zalesky, 1997) is a fundamental tenet of Newtonian mechanics. It is invariance under the transformation x = x - wt, where w is the constant velocity of a moving frame of reference, and embodies the concept that only the relative velocities and positions of two bodies determine their interaction. Galilean invariance is lost in lattice gas simulations because every particle has only one possible speed. This loss is an artifact that can be eliminated for incompressible fluids by re-scaling the velocity. According to Boghosian (1993), more sophisticated lattice gas models overcome this problem. Appropriate application of lattice gas models also requires certain restrictions on the mean free path of a particle (Rothman, 1988). 

The familiar Galilean law of relative motion dictates that a stationary observer measures the position of an object in relative motion, at constant speed V, to change by an amount vt during time t. In the moving frame of reference, where the position P remains constant, the relative motion is described correctly by ... 

Lorentz transformations A set of equations for transforming the position and motion parameters from a frame of reference with orij at O and coordinates (x,y,z) to a frame moving relative to it with origin at O and coordinates (x. /.zO. They replace the Galilean transformations us in Newton-... 

The special theory. For Galileo and Newton, all uniformly moving frames of reference (Galilean frames) are equivalent for describing the dynamics of moving bodies. There is no experiment in dynamics that can distinguish between a stationary laboratory and a laboratory that is moving at uniform velocity. Einstein s special theory of relativity takes this notion of equivalent frames one step further he required all physical phenomena, not only those of dynamics, to be independent of the uniform motion of the laboratory. 

The connection between velocities is represented as an addition of pole velocities, weighted by positive and negative unit coefficients, notwithstanding the nullity of the velocity of the frame of reference. Normally, this is an assumption corresponding to the Galilean principle of additivity of velocities, which has no effect here owing to the nullity of one of the velocities. 

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. 

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. 

For the reasons discussed above we have to abandon the Galilean principle of relativity and accept that Newton s laws cannot be fundamental laws of Nature. We thus consider Maxwell s equations for electromagnetic fields to be valid in all inertial frames of reference and consequently obtain the relativity... 

Therefore, if there are two frames of reference moving with respect to each other with constant velocity, then the basic property and value of the physical parameters do not change as for the velocity of light. Now, this Lorentz transformation is simplified in to Galilean transformation when n c as... 

Suppose we offset this motion by applying a Galilean transformation x = x +Pt ). In the new reference frame, the system will move just as it did in the old reference frame but, because a — /pqt = / i P )t/A, its diffusion is slowed down by a Lorentz-Fitzgerald-like time factor 1-/3. Intuitively, as some of the resources of the random walk computer are shifted toward producing coherent macroscopic motion (uniform motion of the center of mass), fewer resources will remain available for the task of producing incoherent motion (diffusion). [tofI89]... 

The electronic subsystem in the wire 1 is in equilibrium in the reference frame moving with the drift velocity Vd = Ii/eni in the direction of the current. Therefore the structure factor Si isjrbtained from the equilibrium value Si using the Galilean transformation Si(k,u) = Si(k,u — qvd). Equations (1) and (5) then yield... 

On the other hand, according to the law of conservation of matter, the vector sum of components fluxes in the laboratory reference frame (connected with one of the ends of an infinite, on diffusion scale, diffusion couple) must be equal to zero. According to Darken, the alloy provides fluxes balancing at the expense of lattice movement as a single whole at some certain velocity, u, which is measured by inert markers ( frozen in lattice) displacement. Correspondingly in the laboratory reference frame, components fluxes acquire a drift component (similar to Galilean velocity transformation equations) as... 

In the nineteenth century, Maxwell was able to compress the physics of electromagnetism into a set of four equations, now referred to as the Maxwell s equations. However, in striking contrast to Newtonian physics, Maxwell s expressions were not invariant to a Galilean transformation. One possible explanation is that the notion of translational invariance was incorrect, and there was indeed an absolute reference frame. However, all attempts to establish such an absolute frame failed, and the logical consequence had to be that Newton s or Maxwell s equations (or both) needed reformulation. 

A very important condition comes from Galilean invariance. Let us look at a system from the point of view of a moving reference frame whose origin is given by x t). The density seen from this moving frame is simply the density of the reference frame, but shifted by x t)... 

---