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Symmetries in DE

  1. Aug 6, 2008 #1
    Hello everybody!

    I have a general question concerning DEs :0

    Can one use the symmetry of the equation to somehow get the solution faster?
    What does such symmetry tell us?
    e.g.:
    [tex]\dot x=y[/tex]
    [tex]\dot y=x[/tex]

    is the symmetrical system to the second order DE

    [tex]\ddot x-x=0[/tex]

    Now we can easily see the solutions (whether e^t or e^(-t)) actually have the same properties as functions. They are even one and the same function, rotated over the y-axis!

    So, is the symmetry really providing help or this is just a coincidence?
     
  2. jcsd
  3. Aug 6, 2008 #2

    tiny-tim

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    Hello Marin! :smile:

    Well … if x' = y2

    y' = x2

    then x'' = 2y y' = 2x2 √x'

    so that's a symmetry which is no help at all! :cry:

    (I suspect there's a condition that makes it helpful … perhaps something like the Jacobian being unitary … but I'll let someone else answer that! :wink:)
     
  4. Aug 6, 2008 #3
    Sorry,tiny-tim, couldn't quite get it :(

    What's the purpose of "then x'' = 2yy' = 2x2 √x'"

    When I look at the system itself, what I see is that what's true for y should be true for x which to my understanding implies that the two functions are somehow similar to one another..

    And the big question is, if so, then HOW?

    **maybe my question above should be: Does the symmetry of a system of simultaneous DEs provide us somehow to find the solution faster?
     
  5. Aug 6, 2008 #4

    tiny-tim

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    oh I see!

    Then, yes, both x and y are solutions to the same equation, so they will be different combinations of the same basic solutions. :smile:

    (But I don't see how that would generally help.)
     
  6. Aug 7, 2008 #5
    well, if we could find one solution, e.g.:

    dy/dx=x^2 => y=1/3 x^3 +c

    it is true then that x=1/3y^3 +c

    but if x and y are basically the same functions, do we have?:

    1/3 x^3=1/3y^3 +k /.3
    x^3=y^3 +c

    which I think is the solution to the DE, from which the system has been derived, cuz:

    the system was:

    dx/dt=y^2
    dy/dt=x^2

    now dividing the second equation by the first one (to eliminate dt):

    dy/dx=x^2/y^2 - which is same with the result above.

    Was it just a coincidence or is there some symmetry in it?

    EDIT: Sorry, I didn't pay attention I used different variables ( first x and then t)
     
  7. Aug 7, 2008 #6

    tiny-tim

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    Hi Marin! :smile:
    Yes, I didn't think of that. :redface:

    So long as the right-hand side is a function of only one variable,

    we can always divide one equation by the other (as you did):

    if dx/dt = f'(y), dy/dt = f'(x), say

    then f'(y)dy = f'(x)dx,

    so f(y) = f(x) + constant. :smile:

    You're right … the symmetry does help! :smile:
     
  8. Aug 7, 2008 #7
    And what about the other cases?

    consider the system:

    [tex]\dot x=x+y^2-2t[/tex]
    [tex]\dot y=x^2+y-2t[/tex]

    to be honest, I don't have an idea how to solve it analytically :(

    But it's symmetrical... You were talking about the Jacobian hmmm could it be an ansatz here ?
     
  9. Aug 7, 2008 #8

    tiny-tim

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    Sorry, I've no idea.

    Just guessing about the Jacobian … someone else wil have to answer that. :redface:
     
  10. Aug 9, 2008 #9
    Does anybody know something about it?
     
  11. Aug 16, 2008 #10
    Hm.... look like a challenging problem. Never seen before. Is there any application for this system?

    Look like you all been thinking of reflection symmetry [tex] x \leftrightarrow y[/tex] before. May be we should be looking at other transformation such that system remain invariant. Is Lie symmetry is of any used here ? I don't know.

    I will monitor this thread. Hopefully somebody could answered it.
     
  12. Aug 16, 2008 #11
    Well, these systems have no physical meaning (at least are not meant to have here). I am interested in the problem from a pure mathematical point of view.

    - absolutely true - I consider it the most obvious one - if we could find something interesting about it, maybe we could then ask for partial symmetries or negative symmetry, etc.

    I know many DEs are not analytically solvable, and many others take a lot of time to find a solution. That's why I'm asking about these symmetrical systems. I think there must be something 'invisible' to us, but hidden in the system.


    I would be glad to see every comment or idea - more or less probable :)

    best regards, Marin
     
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