3.4 Conservative Forces

We have seen that in some cases the total energy is conserved and in others it is not. As we have seen above, work done is defined by a line integral, so in general it depends on the path chosen for integration, which corresponds to the path of the object through space. However, we have seen for the case of gravity and the spring force that the work done depends only on the initial and final positions; the integral is path independent. We also saw that in this case we can define a potential energy function for which

W=\int_{r_{1}}^{r_{2}}\boldsymbol{F}\cdot\operatorname{\mathrm{d}}\!% \boldsymbol{r}=-(U(r_{2})-U(r_{1})),

(3.51)

where $\boldsymbol{F}=-\nabla U$ (mathematicians will note that this is just a special case of the gradient theorem, a multidimensional generalisation of the fundamental theorem of calculus). We call forces which have this property conservative forces. They are called this because by the work-energy theorem:

W=-(U(r_{2})-U(r_{1}))=-\Delta U=\Delta K,

(3.52)

and hence the total energy $E=K+U$ is conserved.

To fully define a potential energy function, we must explicitly say where potential energy is zero. Suppose we choose the point $\boldsymbol{r}_{0}$ , so $U(\boldsymbol{r}_{0})=0$ . Then we can define

U(\boldsymbol{r})=-\int_{\boldsymbol{r}_{0}}^{\boldsymbol{r}}\boldsymbol{F}% \cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}.

(3.53)

We can choose whatever point is most convenient because it won’t affect the physics, since force is minus the derivative of the potential energy any constant value we add to it will disappear. Only differences in potential (a.k.a. potential difference) have physical significance.

3.4.1 Force in Different Coordinate Systems

The gradient formula $\boldsymbol{F}=-\nabla U$ is the most general way to define potential energy, but it can be written more explicitly by choosing a coordinate system. For example, in 1D it reduces to

F=-\frac{\operatorname{\mathrm{d}}\!U}{\operatorname{\mathrm{d}}\!x},

(3.54)

in 3D cartesian coordinates it is

\boldsymbol{F}=-\frac{\partial U}{\partial x}\hat{\boldsymbol{\imath}}-\frac{% \partial U}{\partial y}\hat{\boldsymbol{\jmath}}-\frac{\partial U}{\partial z}% \hat{\boldsymbol{k}},

(3.55)

in 2D polar coordinates it is

\boldsymbol{F}=-\frac{\partial U}{\partial r}\hat{\boldsymbol{r}}-\frac{1}{r}% \frac{\partial U}{\partial\theta}\hat{\boldsymbol{\theta}},

(3.56)

and so on.

3.4.2 Properties of Conservative Forces

These discoveries unlock a very powerful and intuitive way to look at dynamics. Rather than looking at the force and trying to figure out at what points it is pushing in which direction, we can simply look at the graph of the potential energy, specifically the slope. For example, suppose a force has the following potential energy function as a function of radial distance $r$ . At the points B and D, the gradient is flat, so $\frac{\operatorname{\mathrm{d}}\!U}{\operatorname{\mathrm{d}}\!r}=0$ and $F=0$ , there is no force at these points. At A and E we have a positive gradient, which means force is negative, and at C the gradient is negative which implies a positive force. We can immediately see that an object under the influence of this force will be pushed towards B or to $r=0$ , depending on where it starts and how much kinetic energy it has.

We will now list a few equivalent properties of conservative forces.

Definition 3.3.

A force $\boldsymbol{F}$ is conservative if it satisfies any of the following conditions, which are all equivalent.

•

The work done by the force is path independent.
•

The work done by the force on a closed loop is zero.

$W=\oint\boldsymbol{F}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r}=0.$ (3.57)
•

A potential energy function $U$ can be defined such that $\boldsymbol{F}=-\nabla U$ .
•

The work done by the force is equal to minus the difference in $U$ between the start and end points.
•

$\boldsymbol{F}$ is irrotational, i.e. the curl of $\boldsymbol{F}$ is zero.

$\nabla\times\boldsymbol{F}=0.$ (3.58)

3.4.3 Examples of Conservative Forces

For a central force, looking at equation 3.35 we can see that the second integral will be zero since there is no force in the tangential direction. Thus the work done by a central force is always given by

W=\int_{r_{1}}^{r_{2}}F(r)\operatorname{\mathrm{d}}\!r.

(3.59)

So the work depends only on the initial and final radial distance and is thus path independent, which implies that central forces are always conservative.

Another category of conservative forces are those that point along a single axis and where the strength of the force depends only on the distance along that axis, if it depends on anything at all (constant forces are conservative!). These are forces of the form

\boldsymbol{F}=F(x)\hat{\boldsymbol{\imath}}.

(3.60)

Hooke’s law and the constant gravitational field fall into this category of force. The proof is basically the same as for central forces, but we will show that the work done over a closed path is zero.

$\displaystyle W$	$\displaystyle=\oint\boldsymbol{F}\cdot\operatorname{\mathrm{d}}\!\boldsymbol{r% }=\oint F(x)\hat{\boldsymbol{\imath}}\cdot(\operatorname{\mathrm{d}}\!x\hat{% \boldsymbol{\imath}}+\operatorname{\mathrm{d}}\!y\hat{\boldsymbol{\jmath}}+% \operatorname{\mathrm{d}}\!z\hat{\boldsymbol{k}})$	(3.61)
	$\displaystyle=\int_{x_{1}}^{x_{2}}F(x)\operatorname{\mathrm{d}}\!x+\int_{x_{2}% }^{x_{1}}F(x)\operatorname{\mathrm{d}}\!x$	(3.62)
	$\displaystyle=\int_{x_{1}}^{x_{2}}F(x)\operatorname{\mathrm{d}}\!x-\int_{x_{1}% }^{x_{2}}F(x)\operatorname{\mathrm{d}}\!x=0.$	(3.63)

Another two familiar examples of conservative forces are the universal law of gravitation and Coulomb’s law. Being central forces, they are therefore conservative. Let’s calculate their potential energy functions. For gravity, we get

$\displaystyle U(\boldsymbol{r})$	$\displaystyle=-\int_{\boldsymbol{r}_{0}}^{\boldsymbol{r}}\boldsymbol{F}\cdot% \operatorname{\mathrm{d}}\!\boldsymbol{r}=-\int_{\boldsymbol{r}_{0}}^{% \boldsymbol{r}}-\frac{GMm}{r^{2}}\hat{\boldsymbol{r}}\cdot(\operatorname{% \mathrm{d}}\!r\hat{\boldsymbol{r}}+r\operatorname{\mathrm{d}}\!\theta\hat{% \boldsymbol{\theta}})$	(3.64)
	$\displaystyle=GMm\int_{r_{0}}^{r}\frac{1}{r^{2}}\operatorname{\mathrm{d}}\!r$	(3.65)
	$\displaystyle=\left.-\frac{GMm}{r}\right\|_{r_{0}}^{r}.$	(3.66)

We can choose $r_{0}$ for our convenience, in this case if we choose $r_{0}=\infty$ then the lower limit vanishes and the potential energy has the simple form

U(r)=-\frac{GMm}{r}.

(3.67)

Similarly, for Coulomb’s law $\boldsymbol{F}=\frac{q_{1}q_{2}}{4\pi\varepsilon_{0}r^{2}}\hat{\boldsymbol{r}}$ we get (taking the same zero point $r_{0}=\infty$ )

U(r)=\frac{q_{1}q_{2}}{4\pi\varepsilon_{0}r}.

(3.68)

In the case where $q_{1}$ and $q_{2}$ have opposite signs, the numerator $q_{1}q_{2}$ is negative and we get a graph that looks just like the one for gravity. The potential energy is negative for all values of $r$ , the gradient is positive and therefore the force is always negative. This means that the Coulomb force is attractive in the case of opposite charges. Where $q_{1}$ and $q_{2}$ have the same sign, the numerator is positive and we have $U(r)>0$ for all $r$ . This implies that the gradient is always negative so the force is positive, indicating a repulsive force.