Question: With(Phyiscs[Vectors]) can I create vectors in multiple different cartesian coordinate systems? Example: Unprimed System (i,j,k) and Primed System (i',j',k').

Unit_vectors_from_different_coordinate_systems.mw
 

restart

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with(Physics)``

with(Vectors)

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It's common in mathematical physics to use cartesian unit vectors to describe the position of a point in space.

 

r_(t) = x(t)*_i+y(t)*_j

r_(t) = x(t)*_i+y(t)*_j

(1)

Sometimes it neccessary to convert a position vector like `#mover(mi("r"),mo("→"))`(t) to another cartensian coordinate system with different unit vectors, I call the primed system. In the primed system the position vector looks like:

"(r')(t)=x'(t) (i')+y'(t) (j')"

When using Physics[Vectors] and the unit vector hat notations to define vectors in cartesian space, can I define more than one cartesian space such as:

`#mover(mi("r"),mo("→"))`(t) = x(t)*`#mover(mi("i"),mo("∧"))`+y(t)*`#mover(mi("j"),mo("∧"))`

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and

  "(r')(t)=x'(t) (i')+y'(t) (j')"?

Another way to ask the same thing: Can I define the position vector in different coordinates, each system having a distinct pair of orthogonal unit vectors?

 

The short answer I think is no. Given the current implementation it's not clear how one would go about defining the relationships between unit vectors from different coordinate systems. See below.

 

In 2D the transformation corresponds to a rotation of a vector the plane. The tranformation is characterized by the rotation angle α.

 

 

 

The unit vectors from different systems are related through scalar products.

 

"(i)*i' =(|i|)*|i'|*cos(alpha)=cos(alpha)"``NULL

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"(j)*(j)' =(|j|)*|(j)'|*cos(alpha)=cos(alpha)"NULLNULL

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"(j)*(i)' =(|j|)*|(i)'|*cos(3 alpha)=cos(3 alpha)"``NULL

 

Is there a way to implement scalar products of vectors from different coordinate systems using the Physics Tensors package? Here I create three different coordinate systems. I don't know whether the unit vectors systems X and Y have the same (i, j, k) unit vectors or does each system have its own triplet?

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Setup(coordinates = cartesian, metric = Euclidean, dimension = 3, spacetimeindices = lowercaselatin, geometricdifferentiation = true)

[coordinatesystems = {X}, dimension = 3, geometricdifferentiation = true, metric = {(1, 1) = 1, (2, 2) = 1, (3, 3) = 1}, spacetimeindices = lowercaselatin]
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(2)

Coordinates(Y, Z, Z = cylindrical)

{X, Y, Z}

(3)

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Download Unit_vectors_from_different_coordinate_systems.mw

 

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