In $\textbf{f} = -\boldsymbol{\nabla} u$, what is $u$?

Question

I know that force is the negative gradient of the potential:

$$\textbf{f} = -\boldsymbol{\nabla} u$$

where force $\textbf{f}$ is a vector and $u$ is a scalar.

This is a relatively soft question, but what is $u$? I frequently hear it referred to as "the potential." But is it actually the potential energy?

For example, suppose I have a system consisting of several classical particles that interact. Suppose that I can calculate the potential energy at the position of each particle, because I know how they interact (e.g., by gravity, by Coulomb's Law, by the Lennard-Jones potential, and so on). Can I then determine the force $\textbf{f}_\boldsymbol{1}$ on particle 1 by simply calculating the negative gradient of the potential energy $u_1$ at the position of particle 1?

Answer 1

Yes, u is indeed the potential energy. And yes, you can calculate the force acting on a particle by calculating the gradient of the potential energy field at the position the particle is in.

Computationally you will want to calculate the force on particle 1, by taking the gradient at the position particle 1 is in, of the potential energy field created by all particles except 1 itself, in order to avoid nasty singularities.