📂Classical Mechanics

Conservative Force

Definition

When a force $\mathbf{F}$ moves an object from a point $\mathbf{a}$ to $\mathbf{b}$, if the work done by that force has a value independent of the path, then that force is called a conservative force.

$$ \underset{\text{path I}}{\int_{\mathbf{a}}^{\mathbf{b}}} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = \text{work} = \underset{\text{path II}}{\int_{\mathbf{a}}^{\mathbf{b}}} \mathbf{F} \cdot \mathrm{d}\mathbf{r} $$

Explanation

Being independent of the path is the same as saying that it depends only on the starting point $\mathbf{a}$ and the ending point $\mathbf{b}$. Mathematically it is the same concept as a conservative field, and one may understand a conservative force as a conservative field whose unit is force.

It can be shown that a force $\mathbf{F}$ being a conservative force, its curl being $\mathbf{0}$, its closed-path integral being $0$, and the existence of a scalar potential $V$ are all equivalent.

$$ \begin{array}{ccc} \mathbf{F} \text{ is conservative} & \iff & \nabla \times \mathbf{F} = \mathbf{0} \\[1em] \Updownarrow & & \Updownarrow \\[1em] \displaystyle \oint_{C} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = 0 & \iff & \text{There exists $V$ such that } \mathbf{F} = -\nabla V \end{array} $$