- #1
sam guns
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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?
[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?
