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Geometric algebra cross product

  1. Feb 27, 2008 #1
    geometric algebra cross product

    --------------------------------------------------------------------------------

    The problem statement, all variables and given/known data[/b]

    my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D)

    d/dt (fsubk) = omega X fsubk omega being angular velocity

    then

    omega X fsubk = (-I omega) dot fsubk = fsubk dot (I omega), where I is pseudoscalar and (I omega) is I think the geometric product of I and omega.

    I don't understand these algebraic steps. Can someone explain?

    I did follow earlier explanations of why with vectors a,b that a X b = -I(a wedge b)

    If I have posted this to the wrong forum, would the moderators kindly forward it to a more appropriate one?


    Thank you all very much!

    Ken Cohen
     
  2. jcsd
  3. Feb 28, 2008 #2
    [tex]\omega \times f = -I (\omega \wedge f) = -(I\omega)\cdot f [/tex]

    The first equality is just the definition of the cross product as the dual of
    a wedge product (outer product). The next
    step invokes a duality relationship between the dot product and the outer
    product: the dual of the outer product of a blade A with blade B is
    just the dot product of the dual of A with B (Do you know about this?).
    On the left of the equals
    sign we see the dual of an outer product and on the right we have the
    dual of one of the factors dotted with the second factor. For the final
    step, use the general relationship

    [tex]A_r \cdot B_s = (-)^{r(s-1)} B_s \cdot A_r[/tex] for [tex]r \leq s[/tex]

    Identify the vector f with Ar and the other factor with
    Bs, so that s=2 (bivector!) and r=1 The sign is then -1:

    [tex] -(I\omega)\cdot f = f \cdot (I\omega)[/tex].
     
  4. Feb 28, 2008 #3
    Thank you very, very much.

    I am struggling to teach my self math to understand physics, and it's a lonely stumbling effort.

    what program do you use to enter notation?

    the equality of the dual of outer product equalling the dot of the dual was not presented as far as I can tell up to that point in the text I'm using.

    do you think I'm barking up the wrong tree with geometric algebra entirely?

    thanks, again.
     
  5. Feb 28, 2008 #4
    This forum allows an implementation of TeX. You just sandwich the TeX expression
    between two markup symbols: (tex) ab = a \cdot b + a \wedge b (/tex) to get

    [tex] ab = a \cdot b + a \wedge b [/tex].
    But use square brackets [] instead of round brackets (). You can also click on an equation and a window pops up
    showing you how it is done. For more info: https://www.physicsforums.com/misc/howtolatex.pdf

    Look for section 4.1.4 on pages 96 to 97 of your book. Indeed, skip to chapter 4 and
    read it right away. I think it will answer a lot of your questions. You can always come
    back to where you are. Chapter 4 is pretty well self-contained.

    Not at all! For all of physics that has to do with vectors (mechanics, spacetime,
    special relativity, electrodynamics) and for other topics (point groups, lie algebra,
    conformal geometry, computer graphics, robotics), geometric algebra offers
    a better understanding and provides better tools for attacking problems. Anything to do with rotations is best done with GA (quaternions are just the even-grade sub-algebra
    of 3D space). Once you have the GA perspective on something like electrodynamics,
    you will never look back. There is also a lot of baggage that you can forget about: e.g.
    the vector cross product, axial as opposed to polar vectors. On the other hand, some
    topics, such as quantum mechanics, although they can be approached by GA, would be
    very difficult to learn entirely from the GA point of view, in my opinion. Also, although
    GA is very valuable, you will not readily find others willing to talk physics in this language.
    Which brings me to your remark:

    Essentially that is what I am doing, and I find it lonely as well. There are three other
    very good books that I can recommend but they are very expensive; however, I don't know what I would do without them:

    "Clifford Algebra to GA" by David Hestenes and Garret Sobczyk
    (Kluwer, 1984, reprinted in 2002)
    This has it all but is not an easy read.

    "New Foundations for Classical Mechanics" (second edition)
    David Hestenes, (Kluwer, 1999, reprinted in 2003)
    A great deal on GA in addition to a wonderful treatment of mechanics.

    "GA for Computer Science" by
    Leo Dorst, Daniel Fontijne, and Stephen Mann
    (Morgan Kaufmann, Elsevier, 2007)
    A very good introduction with many things spelled out. It
    includes many exercises and an internet web-site with software
    for viewing many of the figures dynamically and interactively:
    http://www.geometricalgebra.net/

    Good luck in your endeavours. If you have specific questions to me, why not
    send me a private communication within physics forums---I think this is possible.
     
    Last edited: Feb 29, 2008
  6. Feb 28, 2008 #5
    You are very generous. I will!
     
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