# Why is a conservative force defined as the negative gradient of a potential?

I'm learning about work in my dynamics class right now. We have defined the work on a particle due to the force field from point A to point B as the curve Integral over the force field from point A to B. From math I know that if a vector field has a potential, we only need to evaluate the potential at point B minus the potential at point A to get the result of the curve Integral. In the text that I'm reading, it's explained that if the integral over a force-field is path-independent, then the force field $F = -{\rm grad}(V)$, where $V$ is the potential. Why is it defined as the negative gradient? Doesn't one determine the potential from $F$ mathematically. Why do we impose the sign on the potential?

We introduce a minus sign to equate the mathematical concept of a potential with the physical concept of potential energy.

Take the gravitational field, for example, which we approximate as being constant near the surface of Earth. The force field can then be described by $\vec{F}(x,y,z)=-mg\hat{e_z}$, taking the up/down direction to be the $z$ direction. The mathematical potential $V$ would be $V(x,y,z) = -mgz+\text{Constant}$ and would satisfy $\nabla V=\vec{F}$. This would correspond with decreasing in height increasing in potential energy which would make us have to redefine mechanical energy as $T-V$ in order to maintain conservation.

Instead of redefining mechanical energy, we introduce the minus sign $\vec{F} = -\nabla V$ which equates the physical notion of potential energy with the mathematical notion of the scalar potential.

If the resultant force acting in a body is give by minus the gradient of potencial you can show that $\frac{dE}{dt} = 0$. Where E is the total energy of particle. So total energy, kinect + potencial is conserved.

In 1-dimensional case:

$\frac{dE}{dt}=\frac{d(\frac{1}{2}mv^2+V(x))}{dt}=mv\dot{v}+\frac{dV}{dx}\frac{dx}{dt}$

$=v(ma + \frac{dV}{dx})$

Answer you people have given is right, but it is from mathematical background. Physically, conservative force is a dissipative force. Due to dissipation properties we right as gradient of potential and negative sign comes from its opposite direction of action.

• A conservative force is not a dissipative force. – pfnuesel Jan 24 '16 at 5:17