Dot product of Force and Position as a constant of motion - physical significance?

  1. Reason I posted this in the maths help forum is that an equation of this form randomly popped up in a homework I was doing on differential geometry. I started with a one-form ω=dβ (β is a scalar function) and found that if for a random vector v, ω(v) = 0, then

    [itex]\frac{d}{dt} \left( \gamma^{i}\frac{\partial\beta}{\partial x^{i}} \right) = 0[/itex]

    where γ is the integral curve of v (aka the position if you interpret v as a velocity)

    If you interpret the scalar field β as a potential field, then this says that the dot product of position and force is a constant of motion. Understanding it is not really significant to what I am expected to turn in, but regardless, does it have any physical significance?
    1. The problem statement, all variables and given/known data
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,043
    Science Advisor
    Homework Helper

    welcome to pf!

    hi sam! welcome to pf! :smile:

    it looks like the formula for a bead sliding along a frictionless rod forced to rotate (irregularly) about a pivot

    but, so far as i know, it has no practical significance​
     
  4. Re: Dot product of Force and Position as a constant of motion - physical significance

    Thanks for your reply! It's kind of what I suspected, for a second I thought it could be some important constant of motion related to the virial theorem or something like that, but I couldn't find anything in my old mechanics textbooks. I guess it's just a curiosity then :)
     
    Last edited: Mar 8, 2012
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