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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.