PC1672 Advanced dynamics

3.2 Conservative forces

The work done by a force on a particle moving from A to B is given by the line integral

$\begin{displaymath}W=\int_A^B{\bf F}\cdot{\rm d}{\bf x}\end{displaymath}$

This is equal to the change in the particle's kinetic energy. A force field ${\bf F}({\bf x})$ is a force that depends on the particle's position but not its velocity. Such a field is conservative if the work done depends only on the starting and finishing points and not on the path taken between them,

$\begin{displaymath}\int_A^B{\bf F}\cdot{\rm d}{\bf x}=U({\bf x}_A)-U({\bf x}_B)\end{displaymath}$

where $U({\bf x})$ is the potential energy corresponding to the force.

For a conservative force, the total energy

$\begin{displaymath}E={1\over 2}mv^2+U({\bf x})\end{displaymath}$

is conserved, or a constant of motion.

An equivalent local (differential) definition of a conservative field is that it can be written in terms of the gradient of the potential energy as

$\begin{displaymath}{\bf F}({\bf x})=-\hbox{\boldmath $\nabla$}U\end{displaymath}$

(provided the potential energy $U({\bf x})$ is a single-valued function of position).

For a conservative force we do not need to go back to Newton's second law and integrate with respect to time in order to find the speed of an object. We can use the fact that the total energy is constant to get an equation relating the speed of an object to its position.

Textbook references

K&K 4.5-4.8, 5.5, 5.6
F&C 4.2
B&O 2.5
M 2.5
M&T 2.5
B 6.8, 6.11 for a mathematical treatment of conservative fields

Mike Birse
17th May 2000