How Does Translational Invariance Influence Variable Definitions in Physics?

In summary, the exercise asks us to find a solution for the constraints ##x_{IJ}=-x_{JI}## and ##x_{IJ}+x_{JK}+x_{KI}=0##, given that ##x_{1}=0## and ##x_{I1}=-x_{1I}##. One possible solution is ##x_{IJ}=x_{I}-x_{J}##, and we can prove this by substituting the solution into the constraints and showing that they are satisfied for all indices ##I, J, K##.
  • #1
spaghetti3451
1,344
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Homework Statement



Consider a system of objects labeled by the index ##I##, each object located at position ##x_{I}##. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different directions.) Because of translational invariance

##x'_{I}=x_{I}+\delta x##

where ##\delta x## is a constant independent of ##I##, we are led to define new variables

##x_{IJ} \equiv x_{I}-x_{J}##

invariant under the above symmetry. But these are not independent, satisfying

##x_{IJ}=-x_{JI}, \qquad x_{IJ} + x_{JK} + x_{KI} = 0##

for all ##I,J,K##. Start with ##x_{IJ}## as fundamental instead, and show that the solution of these constraints is always in terms of some derived variables ##x_{I}## as in our original definition. (Hint: What happens if we define ##x_{1}=0##?) The appearance of a new invariance upon solving constraints in terms of new variables is common in physics: e.g., the gauge invariance of the potential upon solving the source-free half of Maxwell’s equations.

Homework Equations



The Attempt at a Solution



If ##x_{1}=0##, then ##x'_{1}=\delta x##.

Not sure where to go from here.
 
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  • #2
failexam said:

Homework Statement



Consider a system of objects labeled by the index ##I##, each object located at position ##x_{I}##. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different directions.) Because of translational invariance

##x'_{I}=x_{I}+\delta x##

where ##\delta x## is a constant independent of ##I##, we are led to define new variables

##x_{IJ} \equiv x_{I}-x_{J}##

invariant under the above symmetry. But these are not independent, satisfying

##x_{IJ}=-x_{JI}, \qquad x_{IJ} + x_{JK} + x_{KI} = 0##

for all ##I,J,K##. Start with ##x_{IJ}## as fundamental instead, and show that the solution of these constraints is always in terms of some derived variables ##x_{I}## as in our original definition. (Hint: What happens if we define ##x_{1}=0##?) The appearance of a new invariance upon solving constraints in terms of new variables is common in physics: e.g., the gauge invariance of the potential upon solving the source-free half of Maxwell’s equations.

Homework Equations



The Attempt at a Solution



If ##x_{1}=0##, then ##x'_{1}=\delta x##.

Not sure where to go from here.
If you set ##x_1=0##, how could you define ##x_I## such that the constraints ##x_{I1}=-x_{1I}## are verified?
 
  • #3
Aren't the constraints ##x_{I1} = - x_{1I}## always satisfied regardless of whether ##x_{1} = 0## or ##x_{1} \neq 0##, simply because ##x_{IJ} = x_{I} - x_{J}##?
 
  • #4
failexam said:
Aren't the constraints ##x_{I1} = - x_{1I}## always satisfied regardless of whether ##x_{1} = 0## or ##x_{1} \neq 0##, simply because ##x_{IJ} = x_{I} - x_{J}##?
What the exercise asks you to do is the following:
Assume that you have the numbers ##x_{IJ}##, and that they satisfy the constraints ##x_{IJ}=-x_{JI}, \qquad x_{IJ} + x_{JK} + x_{KI} = 0##.
Now show that there exists numbers ##x_I##, such that ##x_{IJ}=x_I-x_J##.

In other words, the ##x_I## are not given, you have to define them to suit the given ##x_{IJ}##.

The hint is to set one value, ##x_1=0##.
Now for an arbitrary index ##I##, how could you define ##x_I## so that they suit the given ##x_{IJ}## and satisfy the constraint ##x_{I1}=-x_{1I}##.
(I use the index 1 here because you start by setting ##x_1=0##. Of course you will have to show that the ##x_I## you construct satisfy the constraints ##x_{IJ}=-x_{JI}, \ x_{IJ} + x_{JK} + x_{KI} = 0##).
 
  • #5
I need to find the solutions ##x_{IJ}## (in terms of some variables ##x_{I}## and ##x_{J}##) which satisfy the constraints ##x_{IJ}=-x_{JI}## and ##x_{IJ}+x_{JK}+x_{KI}=0##.

The funny part is that we already know that the solution is ##x_{IJ} = x_{I} - x_{J}##, but we have to assume that we don't know the solution ##x_{IJ}## and set about finding it.
For ##x_{1}=0## and ##x_{I1} = - x_{1I}##, one possible choice of ##x_{I1}## and ##x_{1I}## are ##x_{I}## and ##-x_{I}## respectively.

##x_{IJ}## is antisymmetric, and ##x_{I} - x_{J}## is also antisymmetric, therefore ##x_{IJ} = x_{I} - x_{J}## is a possible solution of the constraints.

To prove that the solution ##x_{IJ} = x_{I} - x_{J}## satisfies the constraints, we need to plug in the solution into the constraints.
Am I on the right track?

My idea is to conjecture a possible form of ##x_{IJ}## and then to check if the conjectured solution satisfies the constraints.
 
  • #6
failexam said:
I need to find the solutions ##x_{IJ}## (in terms of some variables ##x_{I}## and ##x_{J}##) which satisfy the constraints ##x_{IJ}=-x_{JI}## and ##x_{IJ}+x_{JK}+x_{KI}=0##.

The funny part is that we already know that the solution is ##x_{IJ} = x_{I} - x_{J}##, but we have to assume that we don't know the solution ##x_{IJ}## and set about finding it.

For ##x_{1}=0## and ##x_{I1} = - x_{1I}##, one possible choice of ##x_{I1}## and ##x_{1I}## are ##x_{I}## and ##-x_{I}## respectively.
You misunderstand the exercise.
First they show you how the numbers ##(x_I)_I## allow to define the numbers ##(x_{IJ})_{IJ}## by setting ##x_{IJ}=x_I-x_J##.
It then appears that these numbers ##(x_{IJ})_{IJ}## satisfy the constraints ##x_{IJ}=-x_{JI}, \ x_{IJ} + x_{JK} + x_{KI} = 0##.

Now they want you to reverse this process.
Assume you have numbers ##(x_{IJ})_{IJ}## that satisfy the constraints ##x_{IJ}=-x_{JI}, \ x_{IJ} + x_{JK} + x_{KI} = 0##. No ##x_I## are given or assumed.
What you are asked to do is: show that you can define numbers ##(x_I)_I## such that ##x_{IJ}=x_I-x_J##.
 
  • #7
I still don't see how my understanding of the exercise differs from yours.

Perhaps if you provide a few initial lines of the solution, that will clarify things.
 
  • #8
failexam said:
I still don't see how my understanding of the exercise differs from yours.

Perhaps if you provide a few initial lines of the solution, that will clarify things.
Set ##x_1=0##. Now, you need ##x_I## to satisfy ##x_{1I}=x_1 - x_I##. That should give you a clue about how to define ##x_I##.
Once that is done, you will have to show that ##x_{IJ}=x_I-x_J## holds for all indices ##I,J##.
 
  • #9
Let's define an origin such that ##x_{1} = 0## and let's define a coordinate ##x_{I}## for particle I such that ##x_{I} = x_{I1}##.

Now, the first constraint ##x_{IJ} = -x_{JI} \implies x_{I1} = -x_{1I} \implies x_{1I} = - x_{I}##.

Next, the second constraint ##x_{1I} + x_{IJ} + x_{J1} = 0 \implies - x_{I} + x_{IJ} + x_{J} = 0 \implies x_{IJ} = x_{I} - x_{J}##.

Am I correct?
 
  • #10
failexam said:
With ##x_{1} = 0## and ##x_{1I} = x_{1} - x_{I}##, we have ##x_{1I} = - x_{I}##.
As the ##x_{IJ}## are given, and you have to define the ##x_I##, it would make more sense to write the last equation as ##x_I=-x_{1J}##. But given that, that is a good definition for ##x_I##.
failexam said:
Now, the first constraint ##x_{IJ} = -x_{JI} \implies x_{I1} = -x_{1I} \implies x_{I1} = x_{I}##.

Next, the second constraint ##x_{1I} + x_{IJ} + x_{J1} = 0 \implies - x_{I} + x_{IJ} + x_{J} = 0 \implies x_{IJ} = x_{I} - x_{J}##.

Am I correct?
Yes.
 
  • #11
I modified my post a little (before I was alerted about your new post). Would you check if my modified post is a better answer?
 
  • #12
failexam said:
Let's define an origin such that ##x_{1} = 0## and let's define a coordinate ##x_{I}## for particle I such that ##x_{I} = x_{I1}##.

Now, the first constraint ##x_{IJ} = -x_{JI} \implies x_{I1} = -x_{1I} \implies x_{1I} = - x_{I}##.

Next, the second constraint ##x_{1I} + x_{IJ} + x_{J1} = 0 \implies - x_{I} + x_{IJ} + x_{J} = 0 \implies x_{IJ} = x_{I} - x_{J}##.

Am I correct?
failexam said:
I modified my post a little (before I was alerted about your new post). Would you check if my modified post is a better answer?
Yes, totally correct.
 

1. What is translational invariance?

Translational invariance refers to the property of a system or equation to remain unchanged under translations in space or time. This means that the system or equation will produce the same results regardless of where or when it is observed.

2. How is translational invariance different from rotational invariance?

Translational invariance deals with changes in position or time, while rotational invariance deals with changes in orientation or direction. Both properties are important in physics and can be observed in various systems and equations.

3. Why is translational invariance important in science?

Translational invariance is important because it allows us to make predictions and understand the behavior of physical systems in different locations or times. It simplifies the analysis of complex systems and helps us identify the underlying patterns and relationships.

4. Can translational invariance be violated?

Yes, translational invariance can be violated in certain systems or equations. This usually happens when there are external forces or factors that affect the system, making it dependent on its position or time. Violations of translational invariance can lead to unexpected or inaccurate results.

5. How is translational invariance applied in real-world applications?

Translational invariance is applied in various fields such as physics, engineering, and economics. In physics, it is used to describe the laws of motion and conservation of energy, while in engineering it is utilized in designing systems that can function in different locations or times. In economics, it is used to model the behavior of markets and predict outcomes under different conditions.

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