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The Different Thermodynamics: What Ought to Be the Proper Mathematical Instrument?
Published in Evgeni B. Starikov, A Different Thermodynamics and Its True Heroes, 2019
But in the case of our interest we have a quite different situation: There are definitely energy supplies, energy stocks—this is namely the potential energy. The latter ought to serve as a source of the kinetic energy, the vis viva—that is, the basis for the driving force. here we won’t dwell on the ways of getting the kinetic energy from the potential one—to simplify our consideration, we assume the Hamiltonian dynamical picture, where our system ought to win the kinetic energy from the potential without hindrances. Sure, that this is in effect nothing more than a drastic idealization. But in our simple case we assume that the following functional relationship takes place, namely that x(t) ≡ −z(t), where x(t) is a kinetic and z(t) a potential energy.
Single Degree-of-Freedom Undamped Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
In 1686 Leibnitz published in Acta Eruditorum a paper dealing with the integral calculus with the first appearance in print of the ∫ $ \mathop \smallint \limits_{{}}^{{}} $ notation. Newton’s Principia appeared the following year. Newton’s “method of fluxions” was written in 1671, but Newton failed to get it published. It did not appear in print until John Colson produced an English translation from the Latin in 1736, resulting in a dispute with Leibnitz. Leibnitz’s vis viva (Latin for living force) was mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter. Here too his thinking gave rise to another regrettable dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes in France; hence academics in those countries tended to neglect Leibnitz’s idea. Eventually vis viva was found useful, and the two approaches were seen as complementary.
Mathematical formulas from the sciences
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Vis-viva equation (elliptic orbits) v2=G(m1+m2)2r1a $ v^{2} = G(m_{1} + m_{2} )\left( {\begin{array}{*{20}c} \frac{2}{r} & \frac{1}{a} \\ \end{array} } \right) $
Helmholtz, the conservation of force and the conservation of vis viva
Published in Annals of Science, 2019
Without linking it with an explicit conservation principle but rather by appealing to the impossibility of perpetual motion, Leibniz introduced the concept of vis viva as the proper measure of a body's motive force by identifying the latter with the work done in raising a weight to a certain height or with the effect that could be obtained when it fell through a certain distance. Even in the absence of an explicit concept of work, that understanding easily gave way over time to a generalization in terms of the action of a force over a distance, whether as a simple product or the integral of a varying force. Although that product, that integral, became a standard component of more or less standardized formulations in rational mechanics of the related but distinct principles of vis viva and its conservation, its original connection with the raising of a weight and its subsequent falling was largely lost from that later context, which interpreted the conservation of vis viva in particular largely in terms of an idealized system of massy particles.