Relation between Force and Potential Energy

Conservative Force

A force for which the work done in moving an object between two points is independent of the path taken. The work done in a closed path is zero.

Potential Energy

Energy possessed by an object due to its position or configuration within a force field.

Gradient

In vector calculus, the gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is that rate of increase.

Force from Potential Energy (1D)

Applies to conservative forces. U is potential energy, x is position.

F = -\frac{dU}{dx}

Potential Energy from Force (1D)

C is the constant of integration, determining the zero potential energy reference.

U(x) = -\int F(x) dx + C

Force from Potential Energy (3D)

\nabla is the gradient operator. \frac{\partial}{\partial x} is partial derivative.

\vec{F} = -\nabla U = -\left(\frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k}\right)