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

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

$\frac{d}{dt} \left( \gamma^{i}\frac{\partial\beta}{\partial x^{i}} \right) = 0$

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
 PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor hi sam! welcome to pf! 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
 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 :)