Physics
Galilean Transformation
The Galilean transformation is a set of equations that relate the coordinates of an event as measured in one inertial frame to the coordinates measured in another inertial frame moving at a constant velocity relative to the first. It forms the basis for classical mechanics and describes how space and time coordinates transform between different inertial reference frames.
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10 Key excerpts on "Galilean Transformation"
eBook - PDFRelativity: The Theory and Its Philosophy
Foundations & Philosophy of Science & Technology
- Roger B. Angel, Mario Bunge(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
Thus, we see that every 54 Relativity: The Theory and its Philosophy frame transformation is a coordinate transformation but not conversely. In general, the mark of a physically significant frame transformation is that the coefficients of the transformation equations involve the variable of time, e.g. as in the Galilean trans-formation. It is most important to note that such time-dependent frame transformations do have physical content. They are statements about the world. This may be illustrated in a direct and simple way. Let us suppose that a frame K has a velocity v with respect to a frame K. Furthermore, we suppose that a particle has a trajectory xt) with respect to ΤΪ. We now wish to determine the velocity of the particle with respect to K. The answer to this question is directly deducible from the Galilean Transformation in the following manner. We first write down the inverse Galilean Transformation relating the coordinates of the particle in the X-frame to those in the X-frame: X l = T + vf. Differentiating, we derive: dX l dV dv- dT dX i dt dt dt dt dt We may rewrite this result in vectorial form as W = Ü+V This is the Newtonian law for the composition of velocities. Assuming that all the vectors are parallel, we find that the composition of velocities is arithmetic. Hence, we have derived a definite physical law from the Galilean Transformation, proving that the latter is factual or, more technically, that it is a statement in the object language of Newtonian particle mechanics. All of this may strike the reader as rather dubious. In effect, I have been arguing that if a car is heading due north with a velocity of 50 m.p.h. and a rifle with a muzzle velocity of 150 m.p.h. is fired from the car in the direction of its motion, then Newtonian mechanics predicts that the bullet will travel with an initial velocity of 150 + 50 m.p.h.
eBook - PDF- T. M. Helliwell, V. V. Sahakian(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
(1.1). This can be true only if the relative frame velocity V is zero! To focus on the problem at hand, let us rephrase things in a slightly more general language. Say observer O is tracking a particle along a general trajectory x(t). The same particle is seen by O to evolve along x (t ). A Galilean Transformation tells us that x(t) = x (t )+ Vt . Taking the time derivative of both sides of this equation, we find the usual velocity addition rule (1.3) 1 There is much more on the electric potential in Chapter 8. 65 2.1 Einstein’s Postulates and the Lorentz Transformation d dt = d dt : x(t) = x (t ) + Vt ⇒ dx(t) dt = dx (t ) dt + V ⇒ v x = v x + V, (2.4) where v x = dx/dt, v x = dx /dt , and we used t = t from Eq. (1.1). So if v x = c, then v x = c + V = c for V = 0, which contradicts the postulate, and we have a problem: Galilean Transformations are incompatible with a universal speed of light. The second postulate can instead be seen as a restriction on the transformation rules relating the coordinate systems of inertial observers. The critical question is then: What are the correct transformation equations relating the coordinates of O and O that replace the Galilean Transformation? Since the Galilean Transformation arises intuitively from our basic sense of the world around us, it had better be the case that it can still be viewed as a decent approximation to the correct transformation, which we now set out to find. As shown already in Figure 2.1, frames O and O are assigned coordinate labels (t, x, y, z) and (t , x , y , z ), respectively, such that O moves with velocity V in the positive x direction as seen by observers at rest in frame O , with the x and x axes aligned, and with the y axis parallel to the y axis and the z axis parallel to the z axis.
eBook - PDFIntroduction to General Relativity
International Series of Monographs in Natural Philosophy
- H. A. Atwater(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
The coordinate t It will be recognized that here we treat time as a parameter, upon which the space coordinates depend, rather than as an independent variable of a four-dimensional continuum. Expressions in which the coordinates appear as functions of time may, in fact, be regarded as descriptions of lower-dimensional subsurfaces in the four-dimensional space. INTRODUCTION 5 transformation of eqns. (1.2) may be described by the matrix of transfor-mation coefficients: f 1 000 a cm — V 1 0 0 óx 0010 0001 (1.2a) The Galilean Transformation of eqns. (1.2) includes the expression of the Newtonian assumption that the time scale is universal, or that time flows equally for all observers. For a mass point m in coordinate frame É which is constrained about the point r 0 = (x0 , Yo, by a spring force with spring constant k the equations of motion [eqns. (1.1)] take the form mx = —k(x — x o ), my = — k(Ý — 5k), (1.3) mz = —k(z — When the Galilean Transformation, eqns. (1.2), is applied to these equa-tions of motion using x = (x — Vt), x o = (c o — N ~ , y0 = Yo, Zo = z o etc., we find mx = —k(x — x o ) , n 1 ~~ = —k(y — yo), (1.4) m2 = —k(z — z u ). The identity of the character of eqns. (1.4) with eqns. (1.3) is an expres-sion of the covariance of Newton's laws under Galilean Transformation. To the physicists of the nineteenth century the Galilean Transformation (1.2) was accepted as being manifestly correct, since it seemed to be an explicit and indisputable statement of what was actually happening, during the relative translation of two coordinate frames. A conflict with experiment appeared, however, when this transformation was applied to problems involving the propagation of light. t It may be observed that in calculating the matrix element bt M/ c 1 we associate the numerator of the differential quotient with the rows of the matrix, and the deno-minator with its columns.
eBook - PDF- Md Nazoor Khan, Simanchala Panigrahi(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
One might say that F = ma holds in any coordinate system provided the term force is re-defined to include the so-called d’Alembert forces. In a curved space–time, all frames are non-inertial. Measurements with respect to non-inertial reference frames can always be transformed to an inertial frame, directly incorporating the acceleration of the non-inertial frame as that acceleration as seen from the inertial frames. 9.3 Galilean Transformation Let us first see how we transform from one inertial frame to another in Newtonian mechanics. Suppose an inertial frame of reference S ¢ is moving with respect to another inertial frame of reference S with a constant velocity v along the x -direction and both S and S ¢ coincide with each other at t = 0. Two observers O and O ¢ are attached at the origin of S and S ¢ respectively. Let a point in space–time (called an ‘event’) has the coordinates ( x , y , z , t ) in frame S as measured by O and ( x ¢ , y ¢ , z ¢ , t ¢ ) in S ¢ as measured by S ¢ . Then from our common sense/classical ideas and with reference to Fig. 9.1, the relation between the coordinates of the event in S and S ¢ will be t ¢ = t or dt ¢ = dt x ¢ = x – vt or = − ' ' dx dx v dt dt or u ¢ = u – v (9.1) Special Theory of Relativity 703 y ¢ = y or = ' ' dy dy dt dt z ¢ = z or = ' ' dz dz dt dt Figure 9.1 Observers O and O' move with relative velocity v and each observer has its own set of coordinates ( x , y , z , t ) and ( x' , y' , z' , t'' ) These relations are called Galilean Transformations. From this, we see that the time of occurrence of an event is the same in all inertial frames. A more precise way of stating this is that the time interval between two events is invariant under Galilean Transformation. 9.4 Michelson–Morley Experiment When we say that the speed of light in a vacuum is 2.997925 × 10 8 m/s ε µ = ( 1/ ) o o , we do not mention any reference system.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Lengths remain unchanged and a single universal time scale is assumed to apply to all inertial frames. • Newton’s laws of mechanics obey the principle of having the same form in all inertial frames under a Galilean Transformation, given by x = x′ + vt, y = y′, z = z′, t = t′. The concept that times and distances are the same in all inertial frames in the Galilean Transformation, however, is inconsistent with the postulates of special relativity. • The relativistically correct Lorentz transformation equations are Lorentz transformation Inverse Lorentz transformation t = t′ + vx′/c 2 1 − v 2 /c 2 t′ = t − vx/c 2 1 − v 2 /c 2 x = x′ + vt′ 1 − v 2 /c 2 x′ = x − vt 1 − v 2 /c 2 y = y′ y′ = y z = z′ z′ = z We can obtain these equations by requiring an expanding spherical light signal to have the same shape and speed of growth, c, in both reference frames. • Relativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. • The Lorentz transformation corresponds to a space-time axis rotation, similar in some ways to a rotation of space axes, but in which the invariant spatial separation is given by Δs rather than distances Δr, and that the Lorentz transformation involving the time axis does not preserve perpendicularity of axes or the scales along the axes. • The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space and time itself, rather than from the laws of electromagnetism. 5.6 Relativistic Velocity Transformation • With classical velocity addition, velocities add like regular numbers in one-dimensional motion: u = v + u′, where v is the velocity between two observers, u is the velocity of an object relative to one observer, and u′ is the Chapter 5 | Relativity 235 velocity relative to the other observer. • Velocities cannot add to be greater than the speed of light.
eBook - PDF- Frederic V. Hartemann(Author)
- 2001(Publication Date)
- CRC Press(Publisher)
The reference frames used in special relativity are defined so that free particles, in the absence of external fields, move with constant velocities in such inertial or “Galilean,” to use Einstein’s terminology, refer-ence frames. Near the end of the nineteenth century, whereas the laws of Newtonian mechanics were thought to obey Galilean transforms from one inertial frame to another, it became clear that Maxwell’s equations could not be written in invariant form under such transformations, which Barut writes in the following form: (2.1) Here, O represents an arbitrary orthogonal transformation of the spatial coordinates; that is, a transformation conserving both lengths and angles, while v is the relative velocity between the two frames. x ′ O x ( ) v t , t ′ t . = + = 26 High-Field Electrodynamics It is very important to stress that only the spatial coordinates are mod-ified under a Galilean transform; this is directly related to the fact that, in Newtonian mechanics, time is absolute, and forces or interactions propa-gate instantaneously, in sharp contrast with Maxwell’s theory. As Pauli writes: “as long ago as 1887, in a paper still written from the point of view of the elastic-solid theory of light, Voigt mentioned that it was mathemat-ically convenient to introduce a local time t ′ into a moving reference system. The origin of t ′ was taken to be a linear function of the space coordinates, while the time scale was assumed to be unchanged. In this way the wave equation, (2.2) could be made to remain valid in the moving reference system, too. These remarks, however, remained completely unnoticed, and a similar transfor-mation was not again suggested until 1892 and 1895, when H. A. Lorentz published his fundamental papers on the subject.” At that point, Lorentz formally introduced a transformation of the space and time coordinates under which Maxwell’s equations remained unchanged.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Newton's equations of motion in the two systems are Fj = mxj = mx'j = F'j (4.3) Thus, the form of the law of motion is invariant to a Galilean transforma-tion. The individual terms are not invariant, however, but they transform according to the same scheme and are said to be covariant. 98 4 · THE SPECIAL THEORY OF RELATIVITY 4.3 The Lorentz Transformation The principle of Galilean invariance predicts that the velocity of light is different in two inertial reference frames that are in relative motion. This result is in contradiction to the second postulate of relativity. Therefore, a new transformation law must be found which will render physical laws relatiiistically covariant. Such a transformation law is the Lorentz trans-formation. Historically, the use of the Lorentz transformation preceded the development of Einsteinian relativity theory,* but it also follows from the basic postulates of relativity, and we shall derive it on this basis. If a light pulse is emitted from the common origin of the moving systems K and K' (see Fig. 4-1) when they are coincident, then according to the second postulate, the wavefronts observed in the two systems must be described by Σ x 2 j -s* 2 7 = 1 Σ x? -^' : If we define a new coordinate in each system, x 4 = ict and X4 = ict then we may write Eqs. (4.4) ast μ=1 4 Σ *; 2 μ = 1 From these equations it is clear that the two sums must be proportional, and since the motion is symmetric between the systems, the proportionality constant is unity.J Thus, Σ -^ = Σ * ; 2 (4.6) μ μ * This transformation was originally postulated by Hendrik Anton Lorentz (1853-1928) in 1904 in order to explain certain electromagnetic phenomena, but the formulae had been set up as early as 19(X) by J. J. Larmor. The complete generality of the transformation was not realized until Einstein derived the result. W. Voigt was actually the first to use the equations in a discussion of oscillatory phenomena in 1887.
eBook - PDF- L Z Fang, R Ruffini;;;(Authors)
- 1983(Publication Date)
- WSPC(Publisher)
Chapter 1 BASIC CONCEPTS OF GENERAL RELATIVITY 1 -1 The Aims and Characteristics of Astrophysics To deal generally with the s/ery large scale of spacetime is one of the aims of astrophysics. The reception of light and its emission from distant galaxies are met most generally and the relationships between these events, separated by great distances, are often discussed in astrophysics. In order to describe events, we need a frame of reference, within which events are denoted with respect to a coordinate system. Each event is specified by four numbers, usually taken as three position coordinates in r and one time coordinate t. The most convenient frame of reference for describing the recep-tion of light is a frame at rest with respect to us or to our own galaxy, or to the centre of mass of the local group of galaxies. On the other hand, the simplest frame for describing the emission of light from a distant galaxy G is a frame at rest with respect to it or to the centre of mass of the group of galaxies within which the galaxy G is located. In this frame, the emission event has coordinates f', t'. Evidently, a question which must be answered is how we can transform the coordinates of an event in one frame to get the coordinates in 1 another. It is only by solving this problem that we can find the co-ordinates r, t of this emission event in our own frame. According to Newtonian mechanics, a transformation between the so-called inertia! frames is a Galilean Transformation given by the well known formulas as follows: x' = x - v t , y' = y » Z 1 = 2 , t' = t , where (t,x,y,z) are the coordinates of an event in an inertial frame K and (t* ,x f ,y* ^z 1 ) the coordinates of the same event in another inertial frame K 1 , moving relative to K with uniform velocity v along the positive x-axis of K. Unfortunately, in astrophysics, Galilean Transformations cannot generally be employed because of two reasons.
eBook - PDF- Palash B. Pal(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
This principle was first advocated by Galileo, and was inherent in the Newtonian formulation of dynamics. Later, when Maxwell formulated his theory of electromagnetism, it was found that the equations involve a constant which has the dimension of ve-locity. Further, it was realized that the constant equalled the speed of prop-agation of electromagnetic waves in the vacuum. This created an apparent contradiction with Newtonian dynamics. To resolve the problem, Einstein 16 § 2.1. Lorentz transformation equations 17 took a second axiom, viz., that the speed of light in the vacuum is the same for all inertial observers. It is this second axiom which necessitated reformulation of the laws of dynamics. For two observers in relative motion, the notion of time and space had to be modified in order that the speed of light remains constant for both of them. The consequences of these axioms are summarized in the Lorentz transformation equations . Consider a frame of reference S , and another one called S ′ which moves with a uniform speed v along a direction which is called the common x -axis in both frames. Suppose S ′ has a co-ordinate system whose origin coincides with that of S at time t = 0. Then the relation between the location and time of any event from the two frames will be given by x ′ = x − vt √ 1 − v 2 y ′ = y z ′ = z t ′ = t − vx √ 1 − v 2 . (2.1) Box Exercise 2.1 Rewrite the Lorentz transformation equations using conventional units where the magnitudes of c and planckover2pi1 are not unity. From them, recover the Galilean Transformation equations by taking the limit c → ∞ . Note that Eq. (2.1) is a set of homogeneous linear equations. They can therefore be written in a matrix notation. Introducing a shorthand x μ ≡ parenleftBig t,x,y,z parenrightBig , (2.2) we can write the Lorentz transformation equations of Eq.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
No loss of generality occurs considering the case when the velocity of the arbitrarily chosen mobile system moves along the x direction, i.e. if v ′ O = v ′ O u x = const. The law of transformation of the velocity, evaluated in paragraph 5.2, is x y z x y z O O′ ′ ′ ′ v O ′ Relative Motion Chapter 5 100 v = v ′ O + ′ v corresponding to the scalar equations dx dt = d ′ x dt + v ′ O dy dt = d ′ y dt dz dt = d ′ z dt ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⇒ d x − ′ x ( ) dt = v ′ O d y − ′ y ( ) dt = 0 d z − ′ z ( ) dt = 0 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ , whose integration provides ′ x = x − v ′ O t ′ y = y ′ z = z ⎧ ⎨ ⎪ ⎩ ⎪ , which are the transformation laws of the coordinates between two inertial reference systems and are called Galilean Transformations. In an inertial system v ′ O = const, thus, by taking the time derivative of the transforma- tion of the velocity, we get the law of transformation of the acceleration a = ′ a . Notice that when changing inertial reference system, in general coordinates and veloci- ties change while accelerations are retained. This property is expressed by stating that the acceleration is invariant for Galilean Transformations. Chapter 5 Relative Motion 101 Example 5.1 Galilean Transformation of a Uniform Motion Consider a point-like particle moving of uniform motion with v ≡ v x , v y , v z ( ) in the absolute reference system. Its position time law is expressed by the equations x = x 0 + v x t y = y 0 + v y t z = z 0 + v z t ⎧ ⎨ ⎪ ⎩ ⎪ which, in the relative reference system moving with velocity v ′ O = v ′ O u x , become, by us- ing the Galilean Transformations, ′ x = x 0 + v x t − v ′ O t = x 0 + v x − v ′ O ( ) t ′ y = y 0 + v y t ′ z = z 0 + v z t ⎧ ⎨ ⎪ ⎩ ⎪ thus the motion in the relative system is still uniform, but with velocity ′ v ≡ v x − v ′ O , v y , v z ( ) .
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