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 :)

 

[AndyW]

Hi there,

Thank you for your question. The answer to whether Euclidean space should be thought of as a linear vector space or an affine space is not a simple one. It depends on the context in which it is being used.

In mathematics, Euclidean space is typically defined as an affine space, meaning that it allows for translations without affecting the system. However, in physics, we often use Euclidean space as a linear vector space, where translations do affect the system, as seen in the example you provided with the Lagrangian.

Noether's theorem is a fundamental principle in physics that states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity. In the case of translational invariance, this means that the system's Lagrangian is invariant under translations. However, as you pointed out, this is not the case for the Lagrangian you provided, as it does not remain the same under translations. This does not mean that the Lagrangian is a physical observable, but rather that it is not invariant under the specific type of translation being considered.

In summary, Euclidean space can be thought of as both an affine space and a linear vector space, depending on the context in which it is being used. In physics, we often use it as a linear vector space, but it is important to understand its affine properties as well. I hope this helps clarify your question. Let me know if you have any further questions.