Question

Explain invariant of length under Galelian transformation?

Answer 1

Galilean invariance or Galilean relativity states that the laws of motion are the same in allinertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systemsusing the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary.

Answer 2

ANSWER


Consider two frames S and S' of reference one at rest and other is moving with uniform velocity v.

Let O and O' be the observers situated at the origins of S and S' respectively.

They are observing the same event at any point P.

Let two frames be parallel to each other i.e. X'-axis is parallel to X-axis . Y'-axis is parallel to Y-axis, Z'-axis is parallel to Z-axis.

Let the coordinates of P(x,y,z,t) and (x',y',z',t') relative to origins O and O' respectively.

The choice of the origins of two frames is such that their origins concide at time

t=0

and

t'=0

Case 1
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Let the frame S' have the velocity v only in X' direction.

Then O' has velocity v only along X'-axis. 

The two systems can be combined to each other by the following equations

(x'=x-vt 
y'=y
z'=z
t'=t)...........(1)

Case 2
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Let the frame S' have velocity v along any straight line in any direction such that 

v= ivx+jvy+kvz

After time t, the frame S' separated from S by distance tvx,tvy,tvz along x,y,z axes respectively.

then two systems can be related by the following equations.

(x'=x-tvx

y'=y-tvy

z'=z-tvz

t'=t)............(2)

Transformations (1) and (2) are called galilean transformations.