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

The principle of conservation of vis viva for a point mass m, moving with
velocity q, along the path r, under the action of a central force , can be written
as:
or if Q and q are the velocities at the distances R and r:
128
This is formally identical with the well known theorem of vis viva-work.
The first term is in fact the well known variation of the vis viva and the second
term has the dimension of work (force by elementary displacement in the
direction of the force integrated along a line). In the first chapter Helmholtz often
utilized the word "Arbeit" so we would expect it mentioned again here. Instead a
bold reinterpretation of the equation takes place, the second member of the
equation is in fact not defined as "Arbeit" but as:
"the sum of tension forces ( Spannkräfte) between the distances R and
r". 129
Helmholtz made an effort at clarifying the innovation: the tension force
was meant to be in explicit conceptual duality with the living force ("in contrast
to that which Mechanics calls living forces" 130), a "force" that tends to move the
point m, until movement actually takes place. A geometrical interpretation of the
concept is given, it represents: "the set of all the intensities of the force acting in
the distances between R and r". In fact if the intensities of correspond to
ordinates perpendicular to the line of abscissae connecting the point m and the
centre of force a, the integral represent an area given by the "sum of the infinite
abscissae (read: ordinates) lying on it".
The partly unsuccessful effort (the integral is not the sum of the abscissae
but of infinitesimal surfaces) to give a geometrical interpretation of the concept of
Spannkraft, stresses the fact that the tension forces dr are also very different
from the Newtonian forces : dimensionally they are represented by the product
of a force by a displacement; they exist when the material point is not in motion,
tend to put it in motion and are "consumed" by the acquired motion (compare
with the constant relation force-matter described as conceptual model for the
128 Helmholtz Erhaltung p.13
129 Helmholtz Erhaltung p.14
130 Ibid