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Holonomic constraints integrating factor question

  1. Sep 23, 2009 #1
    Ok, heres the question, have patience with my terrible latex skills...

    1. The problem statement, all variables and given/known data
    The equations of constraint of the rolling disk:
    dx - asin(theta)d(phi) = 0 -> 1.
    dy + acos(theta)d(phi) = 0 -> 2.
    are special cases of general linear diff-eqs of constraint of the form:
    [tex]\sum[/tex]g_i(x1,...,xn)dxi = 0 -> 3.

    A constraint condition of this tupe is holonomic only if an integrating function f(x1,...,xn) can be found that turns it into an exact differential. Clearly the fn. must be such that
    [tex]\delta[/tex](fg_i)/[tex]\delta[/tex]x_j = [tex]\delta[/tex](fg_j)/[tex]\delta[/tex]x_i -> 4.

    for all i[tex]\neq[/tex]j. Show that no such integrating factor can be found for either of equations 1 or 2.

    2. Relevant equations
    above


    3. The attempt at a solution
    I have found that when I put either equation 1 or 2 into 3, i can cancel the df, and im left with the original equation, but with the sign backwards (ie 1 becomes dx + ..., and 2 becomes dy - ...). I dont know if this actually means anything...

    I am wondering if I can possibly arrange eq 4. so that i have something to integrate, and show that f diverges...? I dont know how to show that there is no such f......

    Anybody have any tips or ideas?
    Thanks for your time.
     
  2. jcsd
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