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

 P: 3 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