For work done by conservative forces ($W = F.S$), we consider $S$ as the displacement and not the actual path travelled. However for non conservative forces we consider the total path length and not just the initial and final point. So can we say that all Conservative forces have fixed direction whereas all non-conservative forces are always parallel or antiparallel to path length being travelled?


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The definition you gave is a bit confused. For any force, conservative or not, its work along a trajectory $\Gamma$ is defined as: $$W=\int_\Gamma\vec{F}.d\vec{l}$$ where $d\vec{l}$ is a small fragment of the trajectory.

After that, if $\vec{F}$ is conservative, there exists a potential energy $V$ that depends only on position such that: $$\vec{F}=-\vec{\nabla}V$$ Therefore: $$W =\int_\Gamma-\vec{\nabla}V.d\vec{l} =-\int_\Gamma dV =-\Delta V$$ So work is just the variation of potential energy between the starting and ending points.

Conservative forces don't have to maintain a constant direction. Counter-examples are easy to find:

  • elastic force: $\vec{F}=-k\vec{r}$
  • newtonian force: $\vec{F}=-\dfrac{k}{r^2}\,\hat{r}$

Both forces don't have a constant direction, and they're famous examples of conservative forces.


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