Galilean transformation

In summary, the electromagnetic wave equation is not invariant under Galilean transformation, as shown by the fact that it does not retain the same form when expressed in terms of the new variables x', y', z', and t'. This can be seen by applying the chain rule and using the Galilean transformation equations for dx'/dx, dx'/dt, dt'/dt, dx'/dy, dx'/dz, dy'/dx, and dt'/dx. The derivative of d(phi)/dt is incorrect and can be corrected by using the fact that the derivative of v is zero in a Galilean transformation. By applying this correction, it is possible to derive the third term -2v[d^2(phi)/dx'dt'] from
  • #1
427
2

Homework Statement


1) Show that the electromagnetic wave equation,
d^2(phi)/dx^2 + d^2(phi)/dy^2 + d^2(phi)/dz^2 –(1/c^2)( d^2(phi)/dt^2) = 0
is not invariant under Galilean transformation.
Note: here d is a partial differential operator.



Homework Equations





The Attempt at a Solution



I have the solution but I couldn’t understand one particular step. The solution is as follows:
The equation will be invariant if it retains the same form when expressed in terms of the new variables x’,y’,z’,t’. From Galilean transformation we have,
dx’/dx=1, dx’/dt=-v, dt’/dt=dy’/dy=dz’/dz=1, dx’/dy= dx’/dz= dy’/dx= dt’/dx=0
From chain rule and using the above results we have,
d(phi)/dx= [d(phi)/dx’][dx’/dx] + [d(phi)/dy’][dy’/dx] + [d(phi)/dz’][dz’/dx] + d(phi)/dt’][dt’/dx] = d(phi)/dx’
And,
d^2(phi)/dx^2= d^2(phi)/dx’^2
Similarly,
d^2(phi)/dy^2= d^2(phi)/dy’^2 &
d^2(phi)/dz^2= d^2(phi)/dz’^2

Moreover,
d(phi)/dt= -v[d(phi)/dx’] + d(phi)/dt’
Differentiating the above equation with respect to t ,
d^2(phi)/dt^2 = d^2(phi)/dt’^2 -2v[d^2(phi)/dx’dt’] + v^2[d^2(phi)/dx’^2]
This is where I have a doubt. I differentiated in the following way:

d^2(phi)/dt^2= -v[d^2(phi)/dx’dt] - [d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]
= -v[(d^2(phi)/dx’^2)(dx’/dt)] –[d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]
= (v^2)[ d^2(phi)/dx’^2] –[d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]
I am able derive 2 of the terms but how to derive the third term -2v[d^2(phi)/dx’dt’] from –[d(phi)/dx’][dv/dt]. Could somebody please help me with this derivation?
 
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  • #2
Amith2006 said:

Homework Statement


1) Show that the electromagnetic wave equation,
d^2(phi)/dx^2 + d^2(phi)/dy^2 + d^2(phi)/dz^2 –(1/c^2)( d^2(phi)/dt^2) = 0
is not invariant under Galilean transformation.
Note: here d is a partial differential operator.



Homework Equations





The Attempt at a Solution



I have the solution but I couldn’t understand one particular step. The solution is as follows:
The equation will be invariant if it retains the same form when expressed in terms of the new variables x’,y’,z’,t’. From Galilean transformation we have,
dx’/dx=1, dx’/dt=-v, dt’/dt=dy’/dy=dz’/dz=1, dx’/dy= dx’/dz= dy’/dx= dt’/dx=0
From chain rule and using the above results we have,
d(phi)/dx= [d(phi)/dx’][dx’/dx] + [d(phi)/dy’][dy’/dx] + [d(phi)/dz’][dz’/dx] + d(phi)/dt’][dt’/dx] = d(phi)/dx’
And,
d^2(phi)/dx^2= d^2(phi)/dx’^2
Similarly,
d^2(phi)/dy^2= d^2(phi)/dy’^2 &
d^2(phi)/dz^2= d^2(phi)/dz’^2

Moreover,
d(phi)/dt= -v[d(phi)/dx’] + d(phi)/dt’
Differentiating the above equation with respect to t ,
d^2(phi)/dt^2 = d^2(phi)/dt’^2 -2v[d^2(phi)/dx’dt’] + v^2[d^2(phi)/dx’^2]
This is where I have a doubt. I differentiated in the following way:

d^2(phi)/dt^2= -v[d^2(phi)/dx’dt] - [d(phi)/dx’][dv/dt] + [d^2(phi)/dt’^2]
That's incorrect. Recall that, as you already used for the first derivative,
[tex] \frac{d}{dt} (anything) = \frac{d}{dt'} (anything) -v \frac{d}{dx'}(anything) [/tex]
In addition, you may use that the derivative of "v" is zero (v is a constant in a Galilean transformation).

Apply what I just wrote above to the time deriavtive of your two terms appearing in d(phi)/dt= -v[d(phi)/dx’] + d(phi)/dt’ and you will get the answer.

Patrick
 
  • #3
nrqed said:
That's incorrect. Recall that, as you already used for the first derivative,
[tex] \frac{d}{dt} (anything) = \frac{d}{dt'} (anything) -v \frac{d}{dx'}(anything) [/tex]
I didn't get your point. Could u please explain it in detail?
 
  • #4
bump... have same question
 

1. What is the Galilean transformation?

The Galilean transformation is a mathematical concept that describes how the position, velocity, and time measurements of an object in motion change when observed from different frames of reference. It was first proposed by Galileo Galilei in the 17th century and was later refined by Isaac Newton.

2. How does the Galilean transformation differ from the Lorentz transformation?

The Galilean transformation applies to situations where objects are moving at speeds much slower than the speed of light, while the Lorentz transformation is used for objects moving at speeds close to the speed of light. The Galilean transformation assumes that time and space are absolute, while the Lorentz transformation takes into account the effects of time dilation and length contraction.

3. What is the formula for the Galilean transformation?

The formula for the Galilean transformation is x' = x - vt, where x' is the position of the object in the moving frame of reference, x is the position in the stationary frame of reference, v is the relative velocity between the two frames, and t is the time measured in the stationary frame.

4. Can the Galilean transformation be applied to all types of motion?

No, the Galilean transformation is only valid for objects moving at constant velocities. It cannot be used for objects undergoing acceleration or moving at relativistic speeds.

5. How is the Galilean transformation used in real-world applications?

The Galilean transformation is commonly used in classical mechanics and everyday situations where objects are moving at low speeds. It is also used in navigation systems, such as GPS, to calculate the position of objects relative to a stationary frame of reference.

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