m (ẋẍ + ẏÿ + żz̈) = Xẋ + Yẏ + Zż,
(25)
whence, on integration with respect to t,
| 1⁄2m (ẋ2 + ẏ2 + ż2) | = ∫ (Xẋ + Yẏ + Zż) dt + const. |
| = ∫ (X dx + Y dy + Z dz) + const. |
(26)
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force; i.e. the force acting on the particle is always the same at the same place. The work which must be done by forces extraneous to the field in order to bring the particle from rest in some standard position A to rest in any other position P will not necessarily be the same for all paths between A and P. If it is different for different paths, then by bringing the particle from A to P by one path, and back again from P to A by another, we might secure a gain of work, and the process could be repeated indefinitely. If the work required is the same for all paths between A and P, and therefore zero for a closed circuit, the field is said to be conservative. In this case the work required to bring the particle from rest at A to rest at P is called the potential energy of the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs or
| S = − | ∂V | . |
| ∂s |
(27)
In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we find