Force and Potential Energy in Three Dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function of the coordinates. The potential energy function U is also a function of all three spatial coordinates. Force in one dimension can be written as the negative ratio of potential energy change to the displacement along that coordinate. For minimal displacement, the ratios become derivatives. If a function has many variables, the derivative only takes the variable that the partial derivative specifies; the other variables are held constant. Thus, adding the force along the three spatial coordinates, the net force can be written as the vector sum of partial derivatives of potential energy with respect to each coordinate, multiplied by the corresponding unit vector. In other words, force in three dimensions can be expressed as the negative of the gradient of the potential energy function.

Let us understand this with the help of an example. The potential energy of a 3-dimensional force is given by

where k is a constant. Derive F_x, F_y, and F_z, and then describe the total force at each point in terms of its coordinates. Here, the known quantity is potential energy. The unknown quantities F_x, F_y, F_z, and the total force vector need to be calculated. Force is related to potential energy by  the following equation:

Substitution of U in the above equation gives the expression for F_x, F_y, and F_z in terms of the coordinates x, y, and z. The vector force is the sum of the force along each coordinate. In other words,

Substituting the values of  F_x, F_y, and F_z in the above equation, we get the net force vector as

This text is adapted from Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability.