2.3. Conservative Actions

In general, the work of a force field acting on a point \(P\) moving in space from point \(A\) to point \(B\) along a path \(\ell_{AB}\) represented by integral (2.2) depends on the path, and this dependence on the path is usually highlighted with the use of the symbol \(\delta\) in the elementary work (2.1).

If the work of a force field does not depend on the path \(\ell_{AB}\) but only on the endpoints \(A\), \(B\), for all pairs of points within a region of space \(\Omega\), the force field is said to be conservative in the region \(\Omega\) of space. In this case, the work integral can be written as the difference of a scalar field, \(U(P)\) or its opposite \(V(P) := - U(P)\),

\[\begin{split}\begin{aligned} W_{AB} & = \int_{\ell_{AB}} \vec{F} \cdot d \vec{r} = \\ & = \int_{\ell_{AB}} \delta W = \\ & = U(B) - U(A) = \Delta_{AB} U \\ & = V(A) - V(B) = -\Delta_{AB} V \\ \end{aligned}\end{split}\]

The functions \(U\), \(V\) are respectively defined as the potential and potential energy of the force field. From the definition of a conservative force field it readily follows that

\[\oint_{\ell} \vec{F} \cdot d \vec{r} = 0 \ .\]

The elementary work can thus be expressed in terms of the differential of these functions,

\[\begin{split}\begin{aligned} \delta W & = \ \ \ d U =\ \ \ d \vec{r} \cdot \nabla U = \\ & = - d V = - d \vec{r} \cdot \nabla V \\ \end{aligned}\end{split}\]

Comparing this relation with the definition of work \(\delta W = d \vec{r} \cdot \vec{F}\), it is possible to identify the force field with the gradient of the potential function, and the opposite of the gradient of the potential energy,

\[\vec{F} = \nabla U = - \nabla V \ .\]

Since the force field can be written as the gradient of a scalar field, and the curl of a gradient is identically zero, the curl of a potential force field is identically zero,

\[\nabla \times \vec{F} = \vec{0} \ .\]

Note

The reverse logical process - \(\nabla \times \vec{F} = \vec{0}\) implies \(\vec{F} = \nabla U\) implies \(\vec{F}\) conservative, i.e. independence of the work from the path - requires the domain continaing the integration path \(\ell\) to be simply connected.

Example 2.1 (Force fields in non-simply connected domains)

In a region of \(E^2\), described with Cartesian coordinates, containing the origin \(O \equiv (x_0, y_0) \equiv (0,0)\) the vector field

\[\vec{F}_1(\vec{r}) = \frac{x}{x^2 + y^2} \hat{x} + \frac{y}{x^2+y^2} \hat{y}\]

is conservative, while the vector field

\[\vec{F}_2(\vec{r}) = -\frac{y}{x^2 + y^2} \hat{x} + \frac{x}{x^2+y^2} \hat{y}\]

is not conservative, even though their curl is zero in all the points of the domain where the field is defined - they’re not defined in the origin.