Build an affine/vector space for physics

[Aleoa]

We have a simple table , for example a kitchen table, with some objects on it. If we consider the table having two dimension (1) does the table with objects represent an affine space ? Why ?
I want to do some calculation, so i choose a point as origin and i place a vector space in the affine plane.
(2)Placing a vector space in the affine plane is equal to put a cartesian coordinate system in the affine plane?
(3) If there is an isomorphism beetween points and vectors why affine spaces and vectors spaces are different concepts ?

 

[fresh_42]

We have a simple table , for example a kitchen table, with some objects on it. If we consider the table having two dimension (1) does the table with objects represent an affine space ? Why ?

What do you know about affine spaces? Also it's not quite clear, how you want to handle the different dimensions: the two dimensional table surface with imagined two dimensional objects on it, or the entire three dimensional room, where the table surface is just one plane among many?

I want to do some calculation, so i choose a point as origin and i place a vector space in the affine plane.

How are affine spaces and vector spaces related? If you place the origin on the table, don't you simply have a vector space?

(2)Placing a vector space in the affine plane is equal to put a cartesian coordinate system in the affine plane?

You don't place a vector space "in" the affine plane.

(3) If there is an isomorphism beetween points and vectors

This cannot be, since points are zero dimensional and vectors one dimensional.

why affine spaces and vectors spaces are different concepts ?

They are not really different concepts. An affine space is a vector space displaced by a constant, fixed vector.
See: https://en.wikipedia.org/wiki/Affine_space

 

[mathwonk]

as fresh says affine and vector spaces are not so different, a vector space has only slightly more data, i.e. an origin: thus a line is an affine space, and a line plus a chosen point on it (to serve as "origin") is a vector space.

 

[Aleoa]

Ok, in physics calculations, which is the principal "geometric container" ( that contain vectors spaces and affine spaces) ?
Is it a vector space, is it an affine plane or is it the Cartesian Coordinate System (that can be a different structure respect of affine and vector spaces ) ?
I've always seen Cartesian coordinate system in books, what i want to know is how the geometry space in physics is build ? What geometric structures it contains and who contains who ?

 

[fresh_42]

If you place your origin at the bottom of one of the table's leg, then the table surface is an affine space. If you place it on the table, then it's a vector space. For both you can choose a Cartesian coordinate system or not.

 

[mathwonk]

I would suggest that in physics, the affine space is the more fundamental object, since there is no god given location for the origin of the coordinate system. I.e. we on Earth might think (the center of) our sun should be the origin, but a being on another solar system might prefer their sun, or something else. So since the choice of origin is a bit capricious, I would say that space as given is more close to an affine space, but that to do any calculations, we choose an origin and introduce cartesian coordinates to suit the given problem. E.g. when computing trajectories of missiles fired from Earth we often take the Earth to be the x,y plane (assuming for convenience that the Earth is locally flat), and place the origin at the source of the fired missile. Or we often even assume the phenomenon takes place in a plane, with the (line in the direction of the motion on the) Earth as x-axis and the y-axis as vertical. This results in the conclusion that a projectile moves in a parabola, as first computed by Galileo, who himself actually observed that this statement depends on the simplifying assumption that the Earth is flat.

Of course having chosen an origin for a computation, one should argue that the result is independent of the choice of origin, in order for the calculation to have physical meaning. But I am not a physicist.

 

[Aleoa]

So the difference between points and vectors consist in the fact that the difference of 2 points is a vector independent from the reference system ?

ps: why, mathematically, in a vector space the vectors start from the zero vector ?

 

[fresh_42]

You still confuse points and vectors.

So the difference between points and vectors consist in the fact that the difference of 2 points is a vector independent from the reference system ?

No. The difference is that a point is zero dimensional and a vector one dimensional. They are two completely different objects. You can, however, take two points and the oriented line between them, which results in a vector. This result is a new object: length and direction. It is as such now independent from the fact that you got it from attaching it to a certain given starting point.

ps: why, mathematically, in a vector space the vectors start from the zero vector ?

They do this only, if you insist of interpreting them as a difference between points: ##\vec{v}=\overline{PQ}##. Then we can choose ##P=0## to make them all start there and compare them by direction and length. You can attach them all at the origin, but there is no need to.

Now a affine space is a set of the form ##p_0 + V##, i.e. a fixed point ##p_0## plus a vector space ##V##. This means our coordinate system has the point ##p_0## as origin and elements of the space all have the form ##\overline{0p_o} + \vec{v}## or short ##p_0+\vec{v}##. In my example above, ##p_0## could be the end of a table's leg and the table surface in ##V##. If we choose ##p_0=0##, then we choose a coordinate system with an origin on the table. Physically, the table hasn't changed at all, not even moved. It is the same thing, and this is meant by independent of the coordinate system. It's a deliberate choice, whether we measure points on the table starting at the bottom of a leg or at the tabletop. If we choose the affine variant, i.e. an origin at a table's leg, then all points on the table look like ##\text{ leg }+\text{ oriented distance on the table}=\text{ leg }+\vec{v}##. Now the difference between two of those cancels the leg out of the calculation and we have a distance on the table alone, just as if we had chosen an origin on the table.

 

[mathwonk]

As a mathematician, I look at these things a little differently, although some of these differences are quibbles. I think of an affine space as a more primitive object, just a flat space S given perhaps axiomatically, (as in say Geometric Algebra, E. Artin, chapter II), consisting of points and lines, but without any specific point being chosen. That said, it does determine a unique vector space V, which can be defined in several equivalent ways. One way is to take all ordered pairs of points (P,Q) of the space S and define an equivalence relation on them, where two pairs (P,Q) and (M,N) are equivalent if and only if either P=Q and M=N, or if the oriented segments PQ and MN are parallel and point in the same direction. Another approach is to consider the set V of translations of the space S. This is an abstract vector space in which addition is composition. The two definitions are related by considering for a given pair (P,Q) the unique translation of S taking P to Q. Then two pairs define the same translation if and only if they are equivalent in the sense defined earlier. So consider what it means to say that the point pair is zero dimensional and yet the translation they determine is one dimensional. I.e. two things with different properties can nonetheless be equivalent in terms of their algebraic structure. Perhaps the two viewpoints can be reconciled by considering each point as zero dimensional, and yet the translation from one to the other as one dimensional.

Thus an affine space S does determine a vector space V, such that for any point P in S, the other points of S do have the form P +v for some vector v in V, but there is no distinguished such point P. Hence although every affine space S does have the form P+V for a vector space V, it is not presented uniquely in this way, since the point P is not given. Now in the case where we are already given a vector space W, if we choose as our point P, the head of any vector in W, and if we also choose a subspace V of W, then it is true that the set of all sums P+vectors in V, does give an affine space embedded in W.

But I think it may be misleading to present an affine space in the form P+V, since in that case, the special point P can be considered as an origin, and then the affine space is essentially equivalent to the vector space V, i.e. one has an origin and one can define addition. I.e. the choice of a point in an affine space gives that affine space the structure of a vector space. So although perhaps to a physicist a vector should be distinguisehed from a point or pair of points by dimension considerations, to a mathematician, it is the abstract structure of the operations available that determine the nature of the object. I.e. a vector space is a space with certain operations, regardless of the nature of the elements. E.g. the space of polynomials is a vector space, and it is somewhat meaningless to me at least, to ask what is the dimension, or the direction, of a polynomial.

So there are several points of view available, and each is useful in different settings and for different purposes.

So I guess one must know how one is looking at the objects. Perhaps originally vectors were arrows, one dimensional objects with length and direction, and which could then be added and multiplied by scalars, but later mathematicians broadened the concept to be any set of things that have the algebraic structure of the earlier geometric vectors.

But even in the geometric sense, an affine space is a set S with all the properties of a vector space except an origin. There is always however a unique associated vector space V, such that an ordered pair of points of S determines a vector in V, and any choice P of point in S sets up a one one correspondence between point of S and vectors in V, i.e. then S = P+V. Once such a choice of point is made however, then we can consider S as the same as V, i.e. the choice of P gives S an origin and hence the structure of a vector space isomorphic to V.

 

[mathwonk]

Perhaps I should apologize for the overly picky mathematical stuff above. Getting my physicist's hat on, as far as possible, I agree completely with Fresh that it is key in physics to distinguish the nature of the objects one wants to describe. Some concepts are intrinsically (represented by) points and some are intrinsically (represented by) vectors, and probably he is right that this corresponds to their dimension. E.g. in physical space, which I think of as an affine 3 dimensional realm where the planets live, position is to me intrinsically represented by a point, whereas velocity is intrinsically represented by a vector. Now in my calculus book, an origin is chosen for problems in space and then position is represented by a vector, a "position vector". To me the term "position vector" is something of an oxymoron, but perhaps it is chosen for that purpose, i.e. the phrase "position vector" tips you off that it is not a real vector. Note if we move the origin, then the vector reprsenting the position changes, hence position is not intrinsically a vector concept. In practice of course when asking where something is, we can only give meaning to that by comparing its location to the location of something familiar, hence locating an object is usually done by giving the arrow from where we are ourselves, to where the object is.

But when we move the choice of origin, a velocity vector will still be the same vector. I.e. the velocity vector represented by 2 points P,Q, the velocity needed to move P to Q in unit time, equals the difference of the two position vectors OQ and OP, and this stays the same even when we change O. That's why we can write it just as Q-P or PQ. In this sense, to me, a vector can be given as the difference of two points. But as Fresh says, physically we need to know the nature of what we are describing. I.e. sometimes 2 points are just 2 separate points, and sometimes we use them to represent the velocity vector needed to go from one to the other, i.e. sometimes 2 points represent a displacement and sometimes they don't. So although to a mathematician a vector can be represented by the difference of two points, perhaps to a physicist they should not be considered as quite the same thing. I will bow out now, since I am not really a physicist. But these duelling concepts of points versus vectors have plagued and interested me ever since I started trying to teach vectors in vector calculus. Even full professors of my acquaintance have made mistakes on qualifying exams by confusing a point with a vector in a calculation. I finally arrived at a mantra for teaching it, namely (Point - Point) = vector, and (Point + vector) = Point, but maybe this is inadequate for some purposes.

 

[mathwonk]

having read the OPs questions a bit more, he seems to be asking very basic questions. In answer I would say that there are three increasingly structured spaces involved, 1) affine space, which is simply the physical space in which bodies have place. It makes sense here to draw straight lines and determine parallelism, but there are no coordinates and no algebraic operations.
2) a vector space which looks exactly like affine space except for the extra structure of an origin, a fixed point. Now one can make algebraic operations like adding vectors using the origin and the parallelogram law. I.e. now each point represents a vector by drawing an arrow from the origin to that point. And given two arrows starting from the origin, we can complete that angle to a parallelogram and then the diagonal starting from the origin is the sum of the first two vectors. There are still no coordinates.

These two spaces are related, since we may use the points of the original affine space as the points of our vector space, just with the extra data of having chosen one of them. This is often done, and then one acts as if the points now have either of two possible identities, points or vectors. I.e. even after being given an origin O, if we look at another point P. we can use that point to represent the location of an object, in which case P is being thought of only as a point of the original affine space, or we can think of P as representing the displacement vector OP, representing perhaps the constant velocity required to go from O tp P in unit time.

Now in this setting some confusion still can arise, e.g. if we want to represent displacement vector PQ between two points other than the origin, as a vector. It is natural to use the arrow PQ from P to Q, but this has not been defined as a vector in our space, since it does not begin at the origin. So technically, we must represent PQ by the arrow OR starting from the origin and parallel and similarly oriented to PQ. Then PQ and OR will be opposite sides of a parallelogram. Another approach is common if a little abstract, and that is to just agree that all arrows in our vector space represent vectors, but the two arrows are considered equal, or equivalent, if they are parallel and of same length. Now things are a little confusing, namely what is a vector in our vector space, a single point P, representing the vector OP, or is it an arrow PQ? This confusion stems from the dual nature of our space, namely it was originally an affine space of points, but now is also thought of, with its origin, as representing vectors.

This situation, although confusing, actually has some advantages, since we can make algebraic computations in the space using vector algebra, and this is useful. The confusion is what to consider a given object, a point or a vector? For this it helps to know the physical object beimng represented, i.e. is it a position, or a velocity/ displacement? For this reason, some people like to define the vector space associated to the affine space as determined by ordered pairs of points, to keep them separate from the actual points of the affine space, or to consider them even more abstractly, but naturally, as translations of that affine space. The problem still arises to calculate with those translations, and for that representing them by pairs of points or arrows is helpful.

3) Coordinate space. This is affine space together with an origin and also a set of axes with units marked on them, i.e. it is a vector space together with an ordered basis of that space. Now one can assign numbers to points and to vectors and can actually carry out calculations. Because this is so useful, it is common to use this coordinate space to reprsent both the affine space of points and the vector space of velocities. Now the user has to struggle to remember in each situation whether he is dealing with points or vector concepts, and remember as well that the actual numbers associated to them are somewhat artificial as well, depnding on choices of units. I.e. if two points with coordinates are both representing position vectors, it probably makes no sense to add their coordinates.

so there are three spaces, the affine space where bodies live, the associated vector space of translations on that space, and the more refined numerical coordinate grid which we carry around and impose on either of these to make calculations. This is all Newtonian physics, i.e. based on (coordinatized) Euclidean geometry. If we want to move into a consideration of curved space, we need to introduce a new space, maybe curved like a sphere, and now at each point we have a different vector space approximating it, where the velocity vectors acting on that point live. In a sense we just "warp" the original affine space, losing its affine qualities, but we keep the vector spaces, but consider there to be a separate vector space at each point. I.e. now it is not possible to equate PQ with any OR, since there is no translation from P to O. In this setting the distinction Fresh emphasized is that the points live in the curved space, and the vectors live in the vector spaces attached tangentially at each point of the curved space. Of course one can use differential geometry to introduce "parallel translates" and make further metric assumptions and conclusions, but I understood your question to be about pre-relativistic phenomena.

Anyway it is a wonderful question. Apologies for the long and possibly unhelpful answer. Actually I have tried now to answer all 3 of your original questions in some form. In particular, 1: what is an affine space, 3: why are vector and affine spaces different, and 2: is a vector space already a coordinate space (no).

 

[Aleoa]

You still confuse points and vectors.

No. The difference is that a point is zero dimensional and a vector one dimensional. They are two completely different objects. You can, however, take two points and the oriented line between them, which results in a vector. This result is a new object: length and direction. It is as such now independent from the fact that you got it from attaching it to a certain given starting point.

If points are 0 dimensional objects , why is it possible to identify a point with a list of number ? i.e. point P = [4,5].

For example, consider an hyperspace Ax = b , if x has dimension 2, it is a affine plane of R^(2); however, since there isn't a point of reference in an affine plane, how it possible to define the point P = [4,5] ?

 

[Aleoa]

@mathwonk , your answers are fantastic. Thanks so much.
I'll take 2-3 days to understand and metabolize them. And then if i need, i'll make you some questions :)

 

[fresh_42]

If points are 0 dimensional objects , why is it possible to identify a point with a list of number ? i.e. point P = [4,5].

Because it is an object in space, resp. on a plane in your example. You simply measure distances from the origin in a coordinate system, as a street map does. That this is equally the representation of the vector from the origin to the point doesn't change the fact, that they are different. So you shouldn't say point if you actually mean the location vector, sometimes called point vector. A vector has only length and direction. If we fix the starting point of such a directed length to be the origin of our coordinate system, then the endpoint of the vector will be the point. But you see, there have been made choices: an origin, and a vector placed there, before we could say the vector ends at the given point. By themselves, they are zero dimensional - eventually described by coordinates, which is again a choice, e.g. you've chosen a plane whereas the world outside is a space - and one dimensional as a length with a direction.

For example, consider an hyperspace Ax = b , if x has dimension 2, it is a affine plane of R^(2); however, since there isn't a point of reference in an affine plane, how it possible to define the point P = [4,5] ?

In this case it is an affine straight in ##\mathbb{R}^2##, a point ##p=(0,b)## and a line ##V## through the origin, which is a one dimensional subspace of ##\mathbb{R}^2##, and together they build ##p+V## the solution line, which is an affine space in case ##b\neq 0##.