Energy Conservation: Relationship between Potential and Force

      The potential energy between two particles is the function of distance between them, and is dependent on the characteristics of the particles. From the principle of conservation of energy, the sum of kinetic energy KE (= mv2/2) and potential energy U is constant, as shown in Eq. 1.
..........(1)

The velocity term in Eq. 1 is combined with Eq. 2 and Eq. 3 is obtained.
..........(2)

..........(3)

If Eq. 3 is differentiated with respect to time t, Eq. 4 is derived.
or ..........(4)

Because the potential energy U is a function of position, Eq. 4 can be rearranged as follows
..........(5)

This relationship is reduced by Eq. 2 and Eq. 6 to produce Eq. 7.
..........(6)

or..........(7)

where k denotes a coordinate direction, Fij is the force between particles i and j, and rij is the distance between particles i and j. Therefore, force is calculated from the potential energy U using the Eq. 7.