Principle of Energy Conservation:

It is possible to define an energy for every system such that the energy of the system is conserved if the system is isolated and that the energy of the system plus its environment is conserved if there is energy transfer between the system and its environment. In more detail, this means the following.

For different kinds of systems, energy has to be defined differently, e.g., as mechanical, electric, or thermal energy. One kind of energy can be transformed into another kind, and energy can be transferred from one system to another. The principle of energy conservation says that energy can be defined in such a way that the energy of an isolated system is conserved while processes may be occurring inside the system in which energy of one kind is transformed into energy of another kind and that the total energy of a system and its environment is conserved in processes involving energy transfer between the system and its environment.

Example. Imagine a system consisting of two stars revolving around each other, a so-called binary star system. Let us assume this to be an isolated system without any external forces acting on it. In response to the gravitational forces that the stars exert on each other (internal forces), the stars in general will be following elliptical paths and at certain times will be getting closer to each other and at other times will be getting farther away from each other. When they are getting closer to each other, they are getting faster and their kinetic energies increase. When they are moving farther away from each other, the kinetic energies of the stars decrease.

If we defined the total energy of the binary star system to be the sum of the kinetic energies of the two stars, this energy would not be conserved. However, the principle of energy conservation tells us that it must be possible to define an energy for this system that is conserved. One can define a potential energy for the system so that the sum of the kinetic and potential energy is conserved. When the kinetic energy of the system increases, the potential energy decreases, and vice versa, such that the sum of the two remains unchanged.

Terminology. Once appropriate energies have been defined for all systems of interest, one speaks of the law of energy conservation. This law consists of the principle of energy conservation combined with appropriate energy expressions.

The distinction between the terms principle of energy conservation and law of energy conservation that is made here is a slight one. The law of energy conservation is what one usually deals with in applications.