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

meaning left is: "There is something which is constant". But what is this
something? Poincarè distinguishes two cases: a universe whose evolution is
completely determined by the values of n parameters and of their derivatives and
a system in which there are p of the n parameters that vary independently (that is,
the system is a limited one, interacting with the exterior world). In the first case,
the n existing differential equations admit n-1 first integrals, i.e. constants. It
would be difficult to decide which deserves the name energy. Thus the principle
is nonsense. In the second case, the principle expresses a limitation: the n-p
relations admit a combination whose first member is a complete differential; if the
work of the external forces and the heat exchanged is zero, the integral of the
resulting zero differential is a constant, i.e. the energy. 430 Here conservation of
energy as existence of a complete differential, is clearly defined, but with the
constraint of admitting a limited system interacting with the outside. At this stage,
Poincarè does not discuss the Helmholtzian interpretation of energy as the sum of
T and U, but it will be seen that this was to be his choice, in order to avoid the
difficulties just expressed.
In 1902 Poincarè improves his analysis: now again, as in 1890, the distinction
between T and U is fundamental to give a specific meaning to energy. In an
analysis that became famous, Poincarè showed that, to give a specific meaning to
energy, its terms have to be of a particular form, one depending on square
velocities and one on positions 431. Weber’s case is explicitly quoted: a distinction
between kinetic and potential energy being impossible, how can we define
energy? 432 In this analysis, where PCE is now called Helmholtz’s and not
Mayer’s principle, the cases in which energy cannot clearly be divided into T and
U, (when for instance also an internal energy Q appears, not clearly distinguished
from the other two), are considered tautological expressions of conservation:
something is conserved, but the whole assertion is untestable due to its
unspecificity. Poincarè is thus led to what Planck called a substantialisation, and
the substantialisation is clearly expressed in the electromagnetic case. First of all
the distinction between T and U. Second their interpretation in electromagnetic
terms and their localisation:
"Et alors Maxwell s’est demandè s’il pouvait faire ce choix et celui des deux
ènergies T et U, de facon que les phènomenes èlectriques satisfassent à ce
430 Poincarè Thermod p.XI. Repr. in Poincarè La Science p.147.
431 Poincarè La Science pp.139-44.
432 Poincarè La Science p.141.