A Source of Conservative Forces do Work on a Closed Trajectory

It is proved that the power of conservative forces (including gravitational forces) performs work on closed trajectories of multiple body's motion, if these bodies are not rigidly connected and between them forces are acting, which depend on the speed of these bodies. A shortened version of this paper has been published in [1, 2] as an appendix. We shall begin with considering some examples. Example 1. There is an electrical charge Q and another charge much smaller by its size Qq 1. Coulomb forces acting of the 1q from the side of the charge Q do not perform any work on a closed path of the motion of charge 1q. Let there be another charge Qq 2, and both charges 1q and 2q are moving along near closed paths. Then between them Lorentz forces are acting. Let the medium in which the charges 1q and 2q are moving provides some resistance to their motion. Then under the influence of Lorentz forces a certain work will be performed. The energy for this work is provided from the electrical charge Q (this is similar to the fact that Lorentz forces acting as Ampere forces perform work by the energy of the power force). Thus, the source of Coulomb forces performs work on closed paths of the two charges motion. Example 2. There is a DC motor with self-excitation (in which the armature winding and the electromagnetic field are connected in series or in parallel). In such a motor the energy source is a DC voltage source, i.e. a source of Coulomb forces. This source explicitly performs work. In the general case from these examples it follows that the source of Coulomb forces performs work on closed trajectories of multiple unconnected charges motion. As the Coulomb forces are conservative, then the previous conclusion is equivalent to the following