Motion equations Galilean transformation

The Maxwell equations are valid only in the unique inertial coordinate system, but they are not invariant for the Galilean transformation (1). This means that the Maxwell equations do not satisfy the requirements of classical equation of motion. This problem was apparently solved by the introduction of the concept of ether, the bearing substance of light. The challenge was to determine ether as the unique inertial system, or earth s motion in this ether. 

In a strict sense, the classical Newtonian mechanics and the Maxwell s theory of electromagnetism are not compatible. The M-M-type experiments refuted the geometric optics completed by classical mechanics. In classical mechanics the inertial system was a basic concept, and the equation of motion must be invariant to the Galilean transformation Eq. (1). After the M-M experiments, Eq. (1) and so any equations of motion became invalid. Einstein realized that only the Maxwell equations are invariant for the Lorentz transformation. Therefore he believed that they are the authentic equations of motion, and so he created new concepts for the space, time, inertia, and so on. Within... 

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-... 

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. 

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). 

If the relative velocity V of the Galilean ttansformation (2.4) approaches the speed of light c, the uniformity of time is not applicable, and the Newtonian framework is no longer valid. That is, the Galilean transformation is changed into the Lorentz transformation under invariance of Maxwell s electtomagnetic equations, and the equation of motion is now described in relation to Einstein s theory of relativity. ... 

The linear stability of inviscid jets has been presented in the previous Section, and it is straightforward to extend the results to the viscous equations of motion and boundary conditions. Of relevance in this Section are the temporal results since we want to describe pinching in the absence of a convective fiow - this can arise either from a Galilean transformation to remove the convective uniform component of the fiow, or we can think of examples such as the breakup of liquid bridges. The viscous dispersion relation has been given by Rayleigh and further discussion can be found in the texts of Chandrasekhar [8], Lamb [39] and Middleman [48], for example. 

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-... 

Greenspan [31] outline the transformation of the Eulerian equations governing the motion of an incompressible viscous fluid from an inertial to a rotational frame. The transformation of the Navier-Stokes equations simply results in adding the artificial forces in the momentum balance. The additional equations are apparently not changed as the substantial derivative of scalar functions are Galilean invariant so the form of the terms do not change. 

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