Is Euclidean space an affine space?

[BruceW]

Hi everyone,

I have a question that I'm not sure about. I wanted to know if it is standard to think of Euclidean space as a linear vector space, or a (more general) affine space? In some places, I see Euclidean space referred to as an affine space, meaning that the mathematical definition of the space allows us to make translations without affecting our system.

But on the other hand, if we have a Lagrangian like $1/2 \ m v^2+mgy$ then a translation $y \rightarrow y+d$ changes the Lagrangian to $L \rightarrow L+mgd$. Now, I know that this does not affect the physics of the system, since it does not matter if we add a constant to the Lagrangian. But when people talk about Noether's theorem, they say that due to the form of this Lagrangian, we do not have 'translational invariance'. So are they including the Lagrangian as a physical observable of the system?? And so, is our space a linear vector space, not an affine space? Maybe they are using the term 'translational invariance' to mean something different to the 'translational invariance' that allows us to call our space affine? (and if so, then that is pretty darn confusing).

thanks in advance :)