What is meant by equivalent directions of space?

Question

My Mechanics professor assigned me this question but I needed a bit of clarification on a given condition, the question is as follows:

Compute the Lagrangian for a free particle in an $(n+m)$-dimensional universe with the following property:

• $n$ directions of space are equivalent to each other and the remaining $m$ directions are equivalent to each other. However, no direction of the first set is equivalent to any direction of the other set.

I was confused about what is meant by $n$ equivalent directions of space. Are they all collinear? And with the remaining $m$ equivalent directions, if this implies that they are collinear wouldn't this just be a 2-dimensional axis question?

Answer 1

As a simple example, imagine rubbing your hand on some wood. Running your hand along the grain of the wood feels different from running your hand against the grain of the wood. Another example would be in your every day life: motion in the plane of the earth is quite different than perpendicular to it due to gravity! Thinking more mathematically, you could imagine that, for instance, there is a different potential in the 𝑥 direction than the 𝑦 direction. Or in 3 dimensions, a potential that depends only on 𝑥2+𝑦2 and 𝑧; indeed this is how you could implement gravity on Earth