Introduction
In this article Galilean transformation and inverse Galilean theory , we are going to discuss about the basic idea of Galilean transformation, various transformations , inverse Galilean transformation , also about the different laws in inverse Galilean theory.
Galilean transformation and inverse Galilean theory
Galilean transformation
If we know the position of a particle with respect to a stationary frame of reference, then a transformation relation through which we can find the position of same with respect to another Inertial frame of reference is called Galilean transformation.

Transformation of position case 1
When both frame of reference coincide initially at time =0 ,
now S’ is moving with constant velocity v then we have
P is the point of observation.
r= position vector of P with respect to S frame.
r’ = position vector of P with respect to S’ frame
From diagram we have,
OO’=Vt
From vector addition we have,
r’+ Vt= r
r’ = rVt (1)
If Vx,Vy,Vz are the components of velocity then we can write,
V= Vxiˆ+Vyjˆ+Vzkˆ (2)
Similarly, the position vector in component form,
r= xiˆ+yjˆ+zkˆ (3)
r’ = x’iˆ+y’jˆ+z’kˆ (4)
Now, put the value from (2),(3) and (4) in equation (1)
x’iˆ+y’jˆ+z’kˆ=(xiˆ+yjˆ+zkˆ) Vxiˆ+Vyjˆ+Vzkˆ
Now,compare the cofficient of i,j and k
x’= xVxt (5)
y’= yVyt (6)
z’= zVzt (7)
Equation (5),(6)&(7) are called transformation Galilean equations when S’ is moving in any direction.
If S’ is moving only in xdirection then equation will be
x’= xVt
y’= y
And, z’=z
(Where Vx,Vy and Vz are zeo)
Case 2
When both frame of reference are not coincide initially.
Now S’ is moving with constant velocity v,then we have
From diagram we have
r’= (rVt)r₀
Transformation of displacement between two points
About diagram
OP= r₁
OQ=r₂
O’P= r’₁
O’Q= r’₂
Where P,Q are point of observation
d=distance between points with respect to S frame of reference
d’= distance between two points with respect to S’ frame of reference
Now, from diagram we have
PQ = d= r₂−r₁
d = r₂−r₁ (1)(with respect to S frame of reference)
From Galilean transformation we know,
r’ = rVt (general equation)
For first point
r₁’=r₁Vt (2)
For second point
r₂’=r₂Vt (3)
d = r’₂−r’₁ (4) ( with respect to S’ frame of reference)
Now, from equation (2),(3)& (4)
d’ = r’₂r’₁
d’ = (r₂Vt )(r₁Vt)
d’=(r₂Vt )r₁+Vt
d’= r₂−r₁
d’ = d
Distance between two points remains invariant in Galilean transformation.
Transformation of velocity
Let u is the velocity of particle in S frame of reference . And u’ is velocity of particle in S’ frame of reference.
Hence,
u = dr/dt and u’ = dr’/dt (1)
Where r and r’ are position vector of S and S’ frame of reference.
We know that time coordinates can be given by
t= t’
∴ dt= dt’ (2)
From Galilean transformation we have,
r’ = rVt (3)
Differentiate equation (3) with respect to time t,
dr’/dt = dr/dt – d/dt(Vt)
dr’/dt = dr/dt V
u’= uV (4)
Where u’ = velocity of particle in S’ frame of reference
u= velocity of particle in S frame of reference
And V = velocity of S’ frame of reference
Transformation of acceleration
Let ‘a’ is the acceleration of particle in S frame of reference and ‘a” is acceleration of particle in S’ frame of reference.
From Galilean transformation of velocity we get
u’ = uV
Differentiate with respect to time t,
du’/dt = du/dtdV/dt
a’=a
(Because dV/dt =0 , V= constant)
Where a’= du’/dt
And a= du/dt
If there is no force the acceleration in both frame of reference will be zero. That means law of inertia is followed or these are Inertial frame of reference.
Newton’s relativity or Galilean invariance law / classical relativity
Results are same for an experiment in both frame of reference. One is stationary and other one is moving with constant velocity . It is impossible to determine an absolute velocity or motion in inertial frame of reference.
We can calculate only relative velocity or motion between two frame of reference, this is called Newton’s relativity or Galilean invariance law.
(It is applicable only when V<<C when V ≈ C then this theory fails)
 1) Invariance of the first the law of Newton
 We that acceleration a= a’, under Galilean transformation.if there is no force acting then acceleration becomes zero, or constant. This means acceleration in both frame of reference is constant or Zero. So that law inertia is valid.
 Law of inertia
 If a body is in rest will remain in rest or if a body is in moving, it will remain in the motion untill or unless an external force is applied on the body.
 a=a’
 2) Newton’s second law
 From Galilean transformation of force we know,
 F=F’
 Or dP/dt = dP’/dt
 This shows that rate of change of momentum is proportional to the applied force and in the direction of force ,which wtoNewton’s second law.
 3)Newton’s third law
 We have already proved that force remains invariant underiGalilean transformation that means in each frame of reference action and reaction forces are same or remains invariant. This is called third law of Newton’s.