Velocity transformation using the chain rule

In summary, the chain rule can be used to solve for velocity transformation when the coordinates in the Lorentz transformation are used to refer to the curve's parameter.
  • #1
Adel Makram
635
15

Homework Statement


How to obtain the famous formula of velocity transformation using a chain rule.
I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule.

Homework Equations


##x=\gamma(x`+vt`)##
##t=\gamma(t`+\frac{v}{c^2}x`)##

The Attempt at a Solution


I tried the following chain rule ##\frac{dx}{dt}=\frac{dx}{dx`}\frac{dx`}{dt`}\frac{dt`}{dt}## so ##u=\frac{dx}{dx`}\frac{dt`}{dt}u`##
The first term ##\frac{dx}{dx`}=\gamma(1+\frac{v}{u`})##
The second term ##\frac{dt`}{dt}## requires me to take a derivative of ##t`## with respect to ##t##. Here the equation, ##t=\gamma(t`+\frac{v}{c^2}x`)## will not help because I have to differentiate ##x`## with respect to ##t##.
 
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  • #2
Your chain rules ignores the fact that it is a differentiation wrt many variables. You need the full chain rule with all partial derivatives. An important thing to keep in mind is when t and t' are used in their capacity as coordinates in the Lorentz transformation and when they are used to refer to acurve parameter.
 
  • #3
Orodruin said:
Your chain rules ignores the fact that it is a differentiation wrt many variables. You need the full chain rule with all partial derivatives. An important thing to keep in mind is when t and t' are used in their capacity as coordinates in the Lorentz transformation and when they are used to refer to acurve parameter.
Here is another trial;
##\frac{dx}{dt}=\frac{\partial x}{\partial x`}\frac{\partial x`}{\partial t}+\frac{\partial x}{\partial t`}\frac{\partial t`}{\partial t}##
But ##\frac{\partial x`}{\partial t}=\frac{\partial x`}{\partial t`}\frac{\partial t`}{\partial t}##
So, ##\frac{dx}{dt}=\frac{\partial x}{\partial x`}\frac{\partial x`}{\partial t`}\frac{\partial t`}{\partial t}+\frac{\partial x}{\partial t`}\frac{\partial t`}{\partial t}##
Substitute ##u## for ##\frac{dx}{dt}## and ##u`## for ##\frac{\partial x`}{\partial t`}##
Also, from the two equation of LT ##\frac{\partial x}{\partial x`}=\gamma##, ##\frac{\partial t`}{\partial t}=\frac{1}{\gamma}##, ##\frac{\partial x}{\partial t`}=v\gamma## and ##\frac{\partial t`}{\partial t}=\frac{1}{\gamma}## (here, I did not use the inverse LT to substitute for ##\frac{\partial t`}{\partial t}##, instead I used the main transformation)
yields; ##u=\gamma u`\frac{1}{\gamma}+v\gamma\frac{1}{\gamma}=u`+v## This is non-relativistic transformation.
 
Last edited:
  • #4
From some help, I found a way out.
First the velocity should be represented by the total derivatives not partial derivatives.
##\frac{dx}{dt}=\frac{dx}{dx`}\frac{dx`}{dt`}\frac{d t`}{dt}##
Now ##\frac{dx}{dx`}## and ##\frac{d t`}{dt}## are expressed in term of partial derivatives;
##\frac{dx}{dx`}=\frac{\partial x}{\partial x`}\frac{\partial x`}{\partial x`}+\frac{\partial x}{\partial t`}\frac{\partial t`}{\partial x`}=\frac{\partial x}{\partial x`}+\frac{\partial x}{\partial t`}\frac{\partial t`}{\partial x`}##
##\frac{dt`}{dt}=\frac{\partial t`}{\partial t}\frac{\partial t}{\partial t}+\frac{\partial t`}{\partial x}\frac{\partial x}{\partial t}=\frac{\partial t`}{\partial t}+\frac{\partial t`}{\partial x}\frac{\partial x}{\partial t}##
Replacing; ##\frac{dx}{dt}## and ##\frac{dx`}{dt`}## by ##u## and ##u`##, and the other partial derivatives from Lorentz Transformation yields the proper result.
 

Related to Velocity transformation using the chain rule

1. What is velocity transformation using the chain rule?

Velocity transformation using the chain rule is a mathematical concept used in physics to convert the velocity of an object from one reference frame to another. It involves applying the chain rule from calculus to the equations of motion.

2. Why is velocity transformation necessary?

Velocity transformation is necessary because different observers may have different frames of reference from which they are observing the same object. In order to accurately describe the motion of the object, its velocity must be transformed to the reference frame of the observer.

3. How is the chain rule applied in velocity transformation?

The chain rule is applied by taking the derivative of the velocity with respect to time in the original reference frame and multiplying it by the derivative of the position with respect to velocity in the new reference frame. This ensures that the correct velocity is calculated in the new reference frame.

4. What are the limitations of velocity transformation using the chain rule?

Velocity transformation using the chain rule assumes that the transformation is happening in one direction only. It also assumes that the transformation is happening at a constant velocity and that the reference frames are inertial (not accelerating).

5. Can velocity transformation using the chain rule be applied to other physical quantities?

Yes, the chain rule can be applied to other physical quantities such as acceleration, force, and momentum. The same principles of transforming between reference frames apply, but the specific equations used may differ.

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