Conservation and Innovation : Helmholtz's Struggle with Energy ...

Needless to say the plan was to be carefully carried out, but not without
the introduction of subtle innovations of extraordinary relevance, which are not
limited to overcoming the caloric model for heat still adopted by Carnot and
Clapeyron. The assertion "in the same way" refers to the methodology, but by
no means indicates that the same expression of the principle was to be applied.
Helmholtz in fact reformulates the principle of impossibility of perpetual motion,
utilising the term "work" ("Arbeit"), and mechanical terms as "force" and
"velocity":
"...the quantity of work obtained when a system of bodies moves from
one position to another under the action of specific forces must be the same as
that needed to carry the system back to the original position, independent of the
way, the trajectory or the velocity of the change" 97.
Thus, the term "Arbeit" is now a function of the state (position) of the
system, it is a total differential : in a closed path work cannot be created, but
cannot be destroyed. This is a first great innovation. The new concept of work,
being now a function of the positions, can be equated to another function of the
position: the vis viva.
In fact from Galileo's relation
v=√2gh,
where v is the final velocity acquired by a body of mass m in a fall from
height h under the acceleration g, we derive that
the work mgh equates the expression
,
which is a function of the position too. Here Helmholtz uses again the
word "Arbeit" for work, and following explicitly the French engineers' definition
of"travail", in equating work and vis viva gives priority to the concept of work. In
fact he defines
and not as the measure of vis viva; in this way it
"becomes identical with the quantity of work" 98.
97 Helmholtz: Erh .P.8
98 Compare this awareness of the role of the work concept with the difficulties in
chapter 5 to adopt a definition of "potential in itself" equivalent to work. An indication of
Helmholtz's theoretical rather than mathematical approach. See below.