- #36
AmphibiousBat
- 1
- 0
I've been meaning to think of this for some time. I have a deep feeling that Affine methods
are the natural ones for Newtonian physics and for Minkowsky physics. Affine methods are a nice marriage of the manifold picture and the vector space picture, that works in flat spaces.
When you use affine space, you don't forget about the vector space "its always there floating in the back ground" What you do, is imagine you have geometric flat space
no preferred origin, and when you take two points in that space, you get a difference vector (the same as you's do in the vector space method) you can't however "add two points"
but you can add a vector to a point to get another point. To emphasize: an affine space is a "pair" of a geometric flat space (whaterver flatspace may mean) and a vector space.
Naturally, Minkowsky space has no origin, we just imagine an origin to be there for simplicity.
by the way if you have a parametrized curve in an affine space you may define its derivative,
which "feels" just like the abstract tangent vector to a curve on a manifold.--(in fact that is what it is if you look at the affine space as a manifold) this derivative may be defined directly without the use of coordinates, because you can take differences between points.
I know this much, but I hav'n t explored it any further.
are the natural ones for Newtonian physics and for Minkowsky physics. Affine methods are a nice marriage of the manifold picture and the vector space picture, that works in flat spaces.
When you use affine space, you don't forget about the vector space "its always there floating in the back ground" What you do, is imagine you have geometric flat space
no preferred origin, and when you take two points in that space, you get a difference vector (the same as you's do in the vector space method) you can't however "add two points"
but you can add a vector to a point to get another point. To emphasize: an affine space is a "pair" of a geometric flat space (whaterver flatspace may mean) and a vector space.
Naturally, Minkowsky space has no origin, we just imagine an origin to be there for simplicity.
by the way if you have a parametrized curve in an affine space you may define its derivative,
which "feels" just like the abstract tangent vector to a curve on a manifold.--(in fact that is what it is if you look at the affine space as a manifold) this derivative may be defined directly without the use of coordinates, because you can take differences between points.
I know this much, but I hav'n t explored it any further.
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