What is Galilean transformation? Derive Galilean transformation equations for position and time.
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The Galilean transformation establishes the relationship between the coordinates x,y,z and t of an event in system S and coordinates x’,y’,z’ and t’ of the same event in system S’. Let an event occur in an inertial frame of reference S and let the event be located by the coordinates (x,y,z and t) in the system S. Let us consider another inertial frame of reference S’, which moves in a straight line with respect to the frame S at a constant speed v. Let us assume that the reference frame S’ moves so that x and x’ are in one straight line and have the same positive direction. Let the origins O and O’ coincide at the moment t=0. The coordinate x’ depends on time, since the system S’ moves with velocity v. During the time t, the moving system covers a distance equal to vt. Hence
x’=x-vt
y’=y
z’=z
t’=t
x’=x-vt
y’=y
z’=z
t’=t
Galilean transformations can be represented as a set of equations in classical physics. They are also called Newtonian transformations because they appear and are valid within Newtonian physics. The Galilean transformation equation relates the coordinates of space and time of two systems that move together relatively at a constant velocity. Galilean transformations form a Galilean group that is inhomogeneous along with spatial rotations and translations, all in space and time within the constructs of Newtonian physics. Again, without the time and space coordinates, the group is termed as a homogenous Galilean group. The law of inertia is valid in the coordinate system proposed by Galileo.
What are Galilean Transformations?
To explain Galilean transformation, we can say that the Galilean transformation equation is an equation that is applicable in classical physics. Galilean equations and Galilean transformation of wave equation usually relate the position and time in two frames of reference. These two frames of reference are seen to move uniformly concerning each other.
Thus, the Galilean transformation definition can be stated as the method which is in transforming the coordinates of two reference frames that differ by a certain relative motion that is constant.
A group of motions that belong to Galilean relativity which act on the four dimensions of space and time and form the geometry of Galilean is called a Galilean group.
All these concepts of Galilean transformations were formulated by Gailea in this description of uniform motion. The topic of Galilean transformations that was formulated by him in his description of uniform motion was motivated by one of his descriptions. The description that motivated him was the motion of a ball rolling down a ramp.
Galilean Equations
The Galilean equations can be written as the culmination of rotation, translation, and uniform motion all of which belong to spacetime. Galilean transformation derivation can be represented as such:
To derive Galilean equations we assume that ‘x' represents a point in the three-dimensional Galilean system of coordinates. ‘t ’represents a point in one-dimensional time in the Galilean system of coordinates.
A uniform Galilean transformation velocity in the Galilean transformation derivation can be represented as ‘v’.
Thus, (x,t) → (x+tv,t) ; where v belongs to R3 (vector space).
A translation is given such that (x,t) →(x+a, t+s) where a belongs to R3 and s belongs to R.
A rotation is given by (x,t)→(Gx,t), where we can see that G: R3 →R3 is a transformation that is orthogonal in nature.
Galilean and Lorentz Transformation
Both the homogenous as well as non-homogenous Galilean equations of transformations are replaced by Lorentz equations. Galilean and Lorentz transformation can be said to be related to each other. Galilean transformation of the wave equation is nothing but an approximation of Lorentz transformations for the speeds that are much lower than the speed of light.
Between Galilean and Lorentz transformation, Lorentz transformation can be defined as the transformation which is required to understand the movement of waves that are electromagnetic in nature. To explain Galilean transformation, we can say that it is concerned with the movement of most objects around us and not only the tiny particles.
For the Galilean transformations, in the space domain, the only mixture of space and time is found that is represented as,
x′=(x−vt)x′=(x−vt)
; where v is the Galilean transformation equation velocity.
In Lorentz transformation, on the other hand, both ‘x’ and ‘t’ coordinates are mixed and represented as
x′=γ(x−vt)andct′=(ct−βx)