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Homogenous Functions

  1. Jul 23, 2010 #1
    Is there such a thing as a homogenous function of degree n < 0?

    Considering functions of two variables, the expression:

    [tex]f(x,y) = \frac{y}{x}[/tex]

    is homogeneous in degree 0, since:

    [tex]f(tx,ty) = \frac{ty}{tx} = \frac{y}{x} = f(x,y) = t^0 \cdot f(x,y)[/tex]

    and the expression:

    [tex]f(x,y) = x[/tex]

    is homogenous in degree 1 since:

    [tex]f(tx,ty) = tx = t^1 \cdot f(x,y)[/tex]

    and the expression:

    [tex]f(x,y) = x^3y^2[/tex]

    is homogeneous in degree 5, since:

    [tex]f(tx,ty) = t^5 \cdot f(x,y)[/tex]

    I suppose that in the same way I could construct a function something like:

    [tex]f(x,y) = \frac{y}{x^2}[/tex]

    So that:

    [tex]f(tx,ty) = t^{-1} \cdot f(x,y)[/tex]

    or in other words, "homogenous" in degree n = -1.

    Does this ever really come up much?
     
  2. jcsd
  3. Jul 24, 2010 #2
    I think the answer is "yes"..

    For instance, the Euler theorem on homogeneous functions states, in relevant part, that if you have a function in two variables, x and y, which is homogeneous in degree n, then:

    [tex]n f(x,y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}[/tex]

    So taking the example I gave above, where you have:

    [tex]f(x,y) = \frac{x}{y^2}[/tex]

    which, ostensibly, is homogeneous in degree -1. Plugging this equation in the Euler theorem above, you would have:

    [tex]\frac{\partial f}{\partial x} = \frac{1}{y^2}[/tex]

    [tex]x\frac{\partial f}{\partial x} = \frac{x}{y^2}[/tex]

    and that:

    [tex]\frac{\partial f}{\partial y} = -2\frac{x}{y^3}[/tex]

    [tex]y\frac{\partial f}{\partial y } = -2\frac{x}{y^2}[/tex]

    so that:

    [tex]x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = -\frac{x}{y^2} = -f(x,y)[/tex]

    which is what we would expect of a function which is homogenous in degree -1..
     
  4. Jul 26, 2010 #3
    Yes.

    Consider the following:

    [tex]
    U = \sum_{1 \leq i < j \leq n}^{n} \frac{m_i m_j}{||q_i - q_j||}
    [/tex]

    It usually appears in the n-body problem in the form of...

    [tex]
    m_i \ddot{x_i} = \frac{dU}{dq_i}
    [/tex]

    The fact that U is homogeneous of degree -1 is key to prove that there are no equilibrium solutions for the n-body problem.
     
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