C. Relations between the forces and coordinates.

Finally, it follows from equation (9) that:

For the case of rest, where the right-hand side will be zero, this will yield the general law of conservative forces: (9i)

However, the same condition is also fulfilled when the motion proceeds temporarily in such a way that the right-hand side of (9h) is equal to zero.

Thus, we can also apply the law (9i) in order to define a force function for the forces of warm bodies (monocyclic systems, resp.), in the event that only one of the functions η in equation (6b ) remains constant during the motion. If we therefore neglect the vis viva L on the associated motions then from equation (6b ) then we will have simply:

so our equation (9i) will be fulfilled.

However, we will almost always be in this case when we are concerned with the mechanics of terrestrial bodies that contain more or less heat. Even when the bodies are in a state of violent internal motion, we can, e.g., define force functions for the molecular forces for their elastic effects by means of the law that was proved here, and apply them as if their state of equilibrium were one of stable equilibrium in absolute rest.

The reciprocal relationships that were expressed in the equations: (9a)

in conjunction with the fact that the Pa are linear functions of the q′ a , which one can write as:

and with the previously-given definitions:

are sufficient to prove that a kinetic potential exists such that the force Pa could be expressed in terms of the differential quotients of it in the way that Lagrange gave, and that the equations of motion could be reduced to the principle of least action. The relationships between the forces that were summarized here thus include a way of completely characterizing the motions that that are subject to the principle of least action.

The proof of this theorem can be given immediately with the previously-prepared tools of analysis for the case in which no more than three coordinates pa are present.

However, theorems from the theory of potentials functions in spaces of three dimensions will be used for that.

If one would wish to go on to more coordinates pa then one would need the corresponding theorems for a larger number of coordinates. They can be defined to the extent that they are necessary for our proof. However, since that is something that is interesting in its own right, it seems to me that it would not be suitable to go into that peripherally, and for that reason I would prefer to give the stated proof on another occasion.

Other general characteristics of the motions that take place under the principle of least action will be described in the next paragraphs.

On the physical meaning of the principle of least action. (Continuation of the paper on page 137 of this volume) (By H. von Helmholtz) Translated by D. H. Delphenich