Conservative Force Field

Work

The work of the constant force $\mathbf{F}$ (for example, the force with which we lift up a load) on the path $\mathbf{S}=\overrightarrow{M_1{M}_2}$ is, by definition, the scalar product $A=(\mathbf{F},\mathbf{S})=|\mathbf{F}||\mathbf{S}|\cdot \cos \phi .$ Suppose we are given a vector field $\mathbf{F}$ and a curve $l$ of finite length. We approximate the curve $l$ by a polygonal line with components $\Delta {\mathbf{S}}_i$ and denote by ${\mathbf{F}}_i$ the value of the force at some particular point of $\Delta {\mathbf{S}}_i$; then the work of the field $\mathbf{F}$ on the path $l$ is by definition $A={\lim }_{|\Delta {\mathbf{S}}_i|\to 0}\sum ({\mathbf{F}}_i,\Delta {\mathbf{S}}_i).$

Theorem

A vector field $F$ is conservative if and only if its work along any path $M_1,M_2$ depends only on the endpoints of the path and not on the shape of the path.

Central Field

A vector field in the plane $E^2$ is called central with center at $0$, if it is invariant with respect to the group of motions of the plane which fix $0$.

Theorem

Every central field is conservative, and its potential energy depends only on the distance to the centre of the field.