The assumption is that the expression multiplying the vector
(multivector) is a scalar and the
result is a vector. The geometric algebra is instantiated by defining a
set of abstract vectors
say e1, e2, e3, e4 and the dot product of all pairs of the vectors.
These dot products can be
symbols or numbers. The default option is that the dot product of ei
and ej is the symbol eidotej
and printed as ei.ej. The operator for the dot product is "|" so that
"print e1|e2" returns "(e1.e2)".
Thus if you define vectors - (a1,a2,b1,and b2 are sympy symbols)
a = a1*e1+a2*e2
b = b1*e1+b2*e2
Then there are three products
print a|b -> a1*b1*(e1sq)+a2*b2*(e2sq)+(a1*b2+a2*b1)*(e1.e2)
where in the dot product e1.e2 = e2.e1
print a*b -> a1*b1*(e1sq)+a2*b2*(e2sqr)+a1*b2*(e1e2)+a2*b1*(e2e1)
where in the geometric product e1*e1=e1sq and e2*e2=e2sq
print a^b -> (a1*b2-a2*b1)*(e1^e2)
where in the wedge product e1^e1=e2^e2=0 and e1^e2=-e2^e1
The geometric product of two vectors can be reduced via
a*b = a|b+a^b
using e1*e1 = e1sq, e2*e2 = e2sq, e1*e2 = (e1.e2)+e1^e2, and e2*e1 =
(e1.e2)+e2^e1 = (e1.e2)-e1^e2.
Also
a|b = (a*b+b*a)/2 and a^b = (a*b-b*a)/2
On 05/09/2011 08:51 PM, Luke wrote:
Can the vectors and multivectors in the GA module work with arbitrary
sympy expressions? i.e, if v is a GA vector, and s is a sympy
expression, does it make sense to do: s*v? Is the result of type Mul
or of something else?
~Luke
On Mon, May 9, 2011 at 5:42 PM, Alan Bromborsky<abro...@verizon.net> wrote:
On 05/09/2011 08:30 PM, Luke wrote:
Here is a more explicit example of what Gilbert is talking about:
In [1]: from sympy import *
In [2]: from pydy import *
In [3]: x = symbols('x')
In [4]: N = ReferenceFrame('N')
In [5]: type(N[1])
Out[5]: pydy.pydy.UnitVector
In [9]: test = x*N[1] + x*x*N[2]
In [10]: type(test)
Out[10]: sympy.core.add.Add
In [11]: test2 = Vector(test)
In [12]: test2
Out[12]: x*n1> + x**2*n2>
In [13]: type(test2)
Out[13]: pydy.pydy.Vector
In [14]: test3 = x*test2
In [15]: test3
Out[15]: x*x*n1> + x**2*n2>
In [16]: type(test3)
Out[16]: sympy.core.mul.Mul
As you can see, in Out[15], the multiplication of x and test2 is being
printed in a way that doesn't make sense, and in Out[16] we can see
that this multiplication isn't resulting in another Vector object,
instead it is of type Mul.
Currently, to create a Vector object, you need to explicitly call
Vector on a sympy expression that is composed from terms that have
UnitVectors in them, or pass a dictionary with UnitVectors as the keys
and sympy expressions as the values.
Once you have that Vector object, you might want to multiply it by a
scalar (sympy expression) and have it return another Vector object.
This could be done using a somewhat user unfriendly approach:
In [22]: Vector(dict([(k, x*v) for k, v in test2.dict.items()]))
Out[22]: x**2*n1> + x**3*n2>
This gets the job done, but it isn't very convenient.
So basically, the question is whether
1) Is it easy enough in Sympy to make something like x*aVectorObject
evaluate to another Vector object? Where the x is a "scalar part"
(probably a sympy Expression) and aVectorObject is of type Vector
(which currently subclasses from Basic)?
2) Or is it ok to have to be more explicit about creating Vector
objects by using the notation Vector(x**2*N[1] + x**3*N[2]) or
Vector({N[1]: x**2, N[2]:x**3})?
Additionally, there are other types of products that make sense for
Vector and UnitVector objects, namely dot, cross, and outer products.
So the stuff above would only be for multiplying a Vector by a scalar.
I think all the other types of products have to make use of explicit
method calls since there would be no way to know which type of product
would be implied.
~Luke
On Mon, May 9, 2011 at 4:23 PM, Gilbert Gede<gilbertg...@gmail.com>
wrote:
In PyDy (which we plan to merge into SymPy.physics.classical this summer)
Vector is one of the classes already implemented (along with UnitVector).
Vectors extend Basic, and UnitVector extend Expr.
The way it works now, you can't use Vector as part of a SymPy expression:
In [57]: test
Out[57]: x*n1> + x*a1>
In [58]: x*test*x
Out[58]: x**2*x*n1> + x*a1>
Do people want to be able to use Vector (which comes with dot, cross,
outer
products, and derivative (between reference frames) functions
implemented)
in SymPy expressions? Or is it the type of thing no one really wants?
I'm looking forward to getting to work on PyDy& SymPy this summer.
-Gilbert
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GA module has vectors and multivectors. See documentation
http://docs.sympy.org/dev/modules/galgebra/GA/GAsympy.html
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