Classical Dynamics of Particles and Systems · Problem 2-53

A force is conservative if and only if its curl vanishes: $\nabla \times \mathbf{F} = 0$. This is the key test—if it passes, the force derives from a potential energy $U$ via $\mathbf{F} = -\nabla U$, which you can recover by integrating each component. For radial forces like $\mathbf{F} = f(r)\mathbf{e}_r$, the curl is automatically zero, so they're always conservative.

A force is conservative if and only if its curl vanishes: $\nabla \times \mathbf{F} = 0$. This is the key test—if it passes, the force derives from a potential energy $U$ via $\mathbf{F} = -\nabla U$, which you can recover by integrating each component. For radial forces like $\mathbf{F} = f(r)\mathbf{e}_r$, the curl is automatically zero, so they're always conservative.