Galilean Invariance: UNIZOR.COM - Relativity 4 All - Principles
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-05-22

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com
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UNIZOR.COM - Relativity 4 All - Principles - Galilean Invariance

Assume, at zero-time T=t=0 both reference frames coincide and, therefore, any point in space at zero-time has the same coordinates in both reference frames.
We will check if some familiar physical laws look the same in these two inertial systems and are invariant relative to Galilean transformation of coordinates.


Newton's First Law

The Newton's First Law of Motion (Law of Inertia) states that every object in absence of force acting on it will remain at rest or in uniform motion in a straight line.
Assume, an object in α{T,X,Y,Z} frame is moving with a constant speed
V={VX,VY,VZ}
along a straight line.
Its position then can be described as linear functions of time
X(T) = X0 + VX·T
Y(T) = Y0 + VY·T
Z(T) = Z0 + VZ·T
where {X0,Y0,Z0} are coordinates of an initial position of an object at zero-time (t=T=0) in both reference frames and {VX,VY,VZ} are constant components of an object's velocity vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes.

Let's apply the transformation to β{t,x,y,z} frame moving along the X-axis of α frame with constant speed v.
t = T
x(t) = X(T) − v·T =
= X0 + (VX − v)·T =
= X0 + (VX − v)·t
y(t) = Y(T) = Y0 + VY·T =
= Y0 + VY·t
z(t) = Z(T) = Z0 + VZ·T =
= Z0 + VZ·t

Thus, the motion in β frame described by these linear functions of time t
x(t) = X0 + (VX − v)·t
y(t) = Y0 + VY·t
z(t) = Z0 + VZ·t
is indeed a uniform motion along a straight line.
The Newton's First Law is preserved by a transformation from one inertial reference frame to another.


Newton's Second Law

The Newton's Second Law of Motion establishes the relationship between the force F, mass m and acceleration a of an object
F = m·a
where F and a are vectors, while m is a positive constant.
That means that acceleration vector is directed along the same direction as a vector of force.

This vector equation in α{T,X,Y,Z} reference frame can be rewritten in coordinate form
FX = m·aX
FY = m·aY
FZ = m·aZ
where {FX,FY,FZ} are constant components of a force vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes and
{aX,aY,aZ} are constant components of an acceleration vector, also the same in both systems for the same reason.

As we know, an acceleration is the second derivative of a position (a function of time) by time.
Therefore, in α frame the above equations can be written as
FX = m·X"(T)
FY = m·Y"(T)
FZ = m·Z"(T)

Since coordinates in β
{t,x,y,z} frame are related to coordinates in α frame as
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
the second derivative is
x"(t) = X"(T)
y"(t) = Y"(T)
z"(t) = Z"(T)

As we see, the acceleration of an object is the same in both inertial frames.

Therefore,
Fx = FX = m·X"(T) = m·x"(t)
Fy = FY = m·Y"(T) = m·y"(t)
Fz = FZ = m·Z"(T) = m·z"(t)

As we see, equations of motions in both reference frames are identical, which confirms the identical form of the Newton's Second Law if we switch from one inertial frame to another.


Velocity Addition Law

Let's examine the velocity of a moving object in two different inertial reference frames, α{T,X,Y,Z} and β{t,x,y,z}, assuming β frame is moving along X-axis of the frame α with constant speed v along X-axis.

Consider an object moving in α frame with a constant vector velocity Vα(T).
The coordinates of a velocity vector are first derivatives of coordinates of the position. So, if our object moves in the α{T,X,Y,Z} frame with velocity vector
Vα(T) = {VXα(T),VYα(T),VZα(T)}
Vβ(t) = {Vxβ(t),Vyβ(t),Vzβ(t)}
and the components of this vector are
Vxβ(t) = x'(t)
Vyβ(t) = y'(t)
Vzβ(t) = z'(t)

The equations of coordinate transformation
applied to components of velocity vector produce
Vxβ(t) = x'(t) = X'(T) − v
Vyβ(t) = y'(t) = Y'(T)
Vzβ(t) = z'(t) = Z'(T)