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

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The real starting point for the 86 paper was the search for an expression of the kinetic potential in Maxwell's theory. The old restrictive hypothesis according to which velocity is solely part of the vis viva, as a homogeneous quadratic function, has to be dropped. Helmholtz, at the very moment when the contiguous action theory was receiving its own non mechanistic local principle of conservation, gives up with the requirement so strongly defended for almost forty years: he recognizes that energy can be any function whatever of the coordinates and velocities. The retreat from the 1847 positions is now almost complete: a lasting element is that the new interpretation of energy is still in the mechanical framework. But Helmholtz, now sixty-four and at the height of an extraordinary career, is again ready to start working towards new unifying horizons. The new regulative principle, PLA, should not be confined to mechanics and optics; Helmholtz applied it to electrodynamics and attempted to extend its range to thermodynamics: "From these facts we may even now draw the conclusion that the domain of validity of the principle of Least Action has reached far beyond the boundaries of the mechanics of ponderable bodies. Maupertuis’ high hopes for the absolute general validity of his principle appear to be approaching their fulfilment, however slender the mechanical proofs and however contradictory the metaphysical speculations which the author himself could at the time adduce in support of his new principle. Even at this stage, it can be considered as highly probable that it is the universal law pertaining to all processes in nature ..... In any case, the general validity of the Principle of Least Action seems to me assured, since it may claim a higher place as a heuristic and guiding principle in our endeavour to formulate the laws governing new classes of phenomena." 406 These statements are part of the 1886 paper on PLA. A second paper was to appear in 1887. C. Neumann analysing Helmholtz’s work in a long treatise of 1898, remarks that Helmholtz's electrical investigations can be divided into two groups: the first (1870-75) has its roots in the general conceptions of Newton’s gravitational theory, i.e. in the principle of direct actionat-a-distance. 406 Helmholtz, Hermann. "Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung." In Crelle's Jour 100 (1886): 137-66 and 213-22. Repr. in WA 3. Pp.203- . Quot. from; Yourgrau and Mandelstam. Variational Principles in Dynamics and Quantum Theory. 2nd ed. London: Pitman, 1960. P.143.
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.
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Fabio Bevilacqua Conservation and I
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different formulations 4 and priori
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and interesting results, despite th
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principle of correlation is express
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principle to the existing empirical
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the origin and meaning of the princ
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Whether physiological research, apa
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In a discussion on the origins of a
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his "Wärme" that one of the differ
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generate a determined equivalent of
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"Bericht", despite the application
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experiments in the most disparate b
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The premise reveals that the struct
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intelligible if it can be referred
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elevant consequence of the deductio
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Needless to say the plan was to be
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and from here that the direction an
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since Leonardo 113; the "ad nihilum
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But unfortunately for the whole str
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The principle of conservation of vi
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different from the one joining the
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"The equality that exists between e
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In virtue of its limitations to New
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Helmholtz, while asserting that lig
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the temperature of 1 Kg of water ca
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In his search for a mechanical equi
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had already been achieved in the Be
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approach is discussed at length and
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and unaware of Green's results190,
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potential function 201. It is worth
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and judged unreliable, despite prov
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