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

The second (1892-94), follows the investigations of Faraday, Maxwell and Hertz
and shares its general basic features with Fourier’s theory of heat conduction.
"Like the latter it rests on the idea that the true origins of any changes
occurring in any point of the universe must be found in the immediate vicinity of
this point" 407.
It is thus interesting to analyse the reasons for this double almost
contemporary shift: from action-at-adistance to contiguous, from conservation of
energy to least action, and their possible connection. The analysis is here outlined
only with reference to the modification of the concept of energy and to the
expression of PLA in the ’86 paper.
Helmholtz recognizes the following points:
a) H=F-L and E=F+L (where H is the Hamilton function, F potential
energy and L vis viva); the name of kinetic potential should be attributed to H,
after F. Neumann’s work on mutual potential, and Clausius’ work on
electrodynamic potential. As already recalled the old restrictive hypothesis
according to which the velocity is solely part of the vis viva, as a homogeneous
quadratic function, has to be dropped. H can be any function whatsoever of
coordinates and velocities.
b) In the electrodynamic case such functions have been given by F.
Neumann, Clausius and Weber. But the dielectric has to be taken into account,
following recent results. In this case, the form of the function differs from that for
ponderable masses.
c) The starting point for the research of the ’86 paper was the search
for an expression of the kinetic potential in Maxwell’s theory.
d) In a great variety of cases, the conservation of energy holds, while
least action does not. Thus PLA expresses a particular feature of the conservative
forces not intrinsically connected with their conservative character. That is, it
says something else which is more specific 408. In view of these remarks, my
opinion is that Helmholtz’s shift towards Maxwell in the late seventies was due
to the possibility of localising the energy in the dielectric. This allowed for a
better explanation of optics, of experimental results and also for a distinction
between kinetic and potential energy of the dielectric medium to be maintained,
while in the other theories this distinction was lost. At the same time, most of the
407 Neumann, Carl. Die Elektrischen Kräfte. Part 2. Leipzig: Teubner, 1898. P.IV.
408 Helmholtz "kleinst Wirk" Introduction and par.2.