1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Understanding Galilean structure

  1. Aug 14, 2014 #1
    I just start to read Mathematical Methods of Classical Mechanics by Arnold. And I am sort of very puzzled by all the notion.
    Firstly, if the universe is seen as a 4D affine space, why is time a mapping from [itex]R^4→R[/itex]? I mean this kind of 4D contains time, right?
    Secondly, I thought the kernel of such a mapping t should be the set of events simultaneous with a given event (affine apace), yet it saids kernel is a 3D linear subspace of a vector space [itex]R^4[/itex].
    Thirdly, I never formally took a class on group theory, and google did not exactly answer me. But what is the dimension of the galilean group (or any other group).

    Thanks in advance!
     
  2. jcsd
  3. Aug 14, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Every point in spacetime (R^4) gets exactly one real value "time". The mapping is trivial here, it will get more interesting in special relativity.

    While "t=0" is nothing special for physics, it's still the kernel by definition, and it is a 3D linear subspace of R^4.
     
  4. Aug 14, 2014 #3

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    The Galilei group is a 10 dimensional Lie group. It's the basic symmetry group of non-relativistic physics, being more important to Quantum Mechanics than to Classical Mechanics.
     
  5. Aug 14, 2014 #4
    I see. I guess I'll have to wait for the special relativity to come in.

    I still don't get it. How does "t=0" provide a fixed origin for the space (if that is indeed what it takes to change a affine space to a linear space)?

    Regards,
    hunc
     
  6. Aug 14, 2014 #5
    I dig into a Lie group textbook and find the definition. Thanks.
     
  7. Aug 16, 2014 #6

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    What do you mean with "provide"? The definition of this point is arbitrary. In every (arbitrary) fixed system, t=0 is a linear 3D subspace.
     
  8. Aug 17, 2014 #7
    Before t=0, we have an affine space. By t=0, we fixed a point in "time", not "space". So how should we end up with a linear space?
     
  9. Aug 17, 2014 #8

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    What does that mean?
    We "have" a 4D space. This 4D space has some arbitrary point defined as (0,0,0,0) with arbitrary direction definitions and so on. Now you can consider all points (0,x,y,z). They form a linear subspace.
     
  10. Aug 18, 2014 #9
    Maybe I didn't put it clear enough. A 4D space is what I had in mind before reading Arnold. But in the book Mathematical Methods of Classical Mechanics, it introduce galilean space-time structure as a affine space A4. And it kind of offers the difference between the two case in the image. What puzzles me is how are such difference generated.
     

    Attached Files:

  11. Aug 18, 2014 #10

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Sorry, I don't understand what the problem is.
     
  12. Aug 18, 2014 #11

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    I thought, mathematically Galileo spacetime is a fibrebundle, 3d affine euclidean spaces along the time axis. It's nicely explained in

    R. Penrose, Road to reality.

    Minkowski spacetime is a 4d pseudo-euclidean affine space with a fundamental form of signature (1,3).
     
  13. Aug 18, 2014 #12
    Thanks still.
     
  14. Aug 18, 2014 #13
    I'll check it out. Thanks for the input.
     
  15. Aug 18, 2014 #14
    One important note: By R^4, Arnold does not denote RxRxRxR, but any four-dimensional real vector space (he states that very early). That's not really an issue, as they are isomorphic to each other, but once you forget about the construction via cartesian products, the time mapping is not "trivial" in the sense of canonical.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Understanding Galilean structure
  1. Galilean group (Replies: 2)

Loading...