My name is Alexander FufaeV and here I write about:

What are Conservative Forces?

Table of contents

Why is force the negative gradient of potential energy? Why can the force be written as a gradient of potential energy? Or electric field as a gradient of potential? And why is there a minus sign in front of the gradient?
How do you prove that a force is NOT conservative? Here you can find out how to check whether a force is conservative or not.

A conservative force $ \boldsymbol{F} $ always has a potential $ \varphi $, meaning it can be written as the gradient of a potential $ \varphi $:

$$ \begin{align} \boldsymbol{E} ~=~ -\nabla \, \varphi \end{align} $$

Here, $ \nabla $ is the Nabla operator. An electric field is "force per charge", a gravitational field is "force per mass". We can also express the above equation with the force $ \boldsymbol{F} $ and the potential energy $ W_{\text{pot}} $:

$$ \begin{align} \boldsymbol{F} ~=~ - \nabla \, W_{\text{pot}} \end{align} $$

In a system in which conservative forces act, the sum of potential and kinetic energy is conserved (i.e. constant over time). The work performed in such a conservative system is independent of the chosen path - only the path difference is important!

Examples of conservative forces

Gravitational force, spring force, Coulomb force. As soon as frictional forces occur, the system is no longer conservative.

Why is force the negative gradient of potential energy?

The nabla operator $\nabla$ applied to a scalar function $W_{\text{pot}}(x,y,z)$: $\nabla \, W_{\text{pot}}$ is intended for a multidimensional force: $\boldsymbol{F} = (F_1, ~F_2, ~F_3)$. If we consider the force $F_1 := F$ in ONE dimension only, the Nabla operator becomes the partial derivative:

$$ \begin{align} F ~=~ - \frac{\partial W_{\text{pot}}}{\partial x} \end{align} $$

Equation 2 is the definition of the work that is differentiated on both sides to obtain equation 2 for the force:

$$ \begin{align} W_{\text{pot}} ~&=~ - \int F \, \text{d}x \leftrightarrow \\\\
\frac{\partial}{\partial x} \, W_{\text{pot}} ~&=~ - \int \frac{\partial F}{\partial x} \, \, \text{d}x \leftrightarrow \\\\
- \frac{\partial W_{\text{pot}}}{\partial x} ~&=~ F \end{align} $$

However, in the definition 3 of potential energy, a minus sign is included, and thus also in the gradient 1 of potential energy. The minus sign is a convention. However, this definition is motivated by the natural principle that nature always seeks to minimize the (potential) energy of a body. For example, an apple falls to the ground to minimize its potential energy. Without the minus sign, 1 would imply that as the apple reduces its height above the ground, its potential energy increases. To minimize its energy, the minus sign is introduced in 1.

How do you prove that a force is NOT conservative?

To check whether a force $\boldsymbol{F}$ acting on an object is conservative or non-conservative, the work $W$ along the considered closed path $L$ must be calculated.

$$ \begin{align} W ~=~ \oint_L \, \boldsymbol{F} \cdot \text{d}\boldsymbol{l} \end{align} $$

If the integral is NOT zero, then the force is NOT conservative, that is, the sum of kinetic and potential energy is not conserved. However, the procedure with the path integral only applies to the closed path $L$.

To check whether the path integral for all closed paths $L$ does result in zero, the rotation of $\boldsymbol{F}$ must be calculated. The rotation of the force must result in zero if it is a conservative force:

$$ \begin{align} \nabla \times \boldsymbol{F} ~=~ 0 \end{align} $$

Condition for a conservative force

If the curl $\nabla \times \boldsymbol{F}$ of the force field $\boldsymbol{F}$ is zero, then the force field is conservative.