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they say that a conservative force can be associated to a potential .. Why is that ? and what does it mean that a force has a potential ???
When we say the force field [itex] \vec F=F_x \hat x+F_y \hat y+F_z\hat z [/itex] has a potential, we mean that we can find a function like [itex] \Phi [/itex] such that [itex] F_x=-\frac{\partial \Phi}{\partial x} [/itex],[itex] F_y=-\frac{\partial \Phi}{\partial y} [/itex] and [itex] F_z=-\frac{\partial \Phi}{\partial z} [/itex]. It sometimes makes things easier because we can work with a scalar instead of a vector. Also it allows us to associate energy with forces so that we can apply conservation of energy.Sry , but what does it mean that a force has a potential ?
Sry , but what does it mean that a force has a potential ?
vanhees71 said:[itex]\vec \nabla \times \vec{g}=0 \quad , \quad \vec{\nabla} \cdot \vec{g}=-4 \pi \gamma \rho[/itex]
This implies that there's a gravitational potential [itex]\phi[/itex] such that
g⃗ =−∇⃗ ϕ
No, its not needed so don't bother. Its just mathematical details. I just wanted to get sure I'm standing on rock here!One can prove that the Helmholtz decomposition is unique up to a vectorial constant, if the vector field and its 1st derivatives vanishes at infinity. I don't know, where to find the formal proof of this. I'd have a look in textbooks on mathematical physics. If needed, I can try to find a reference.