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Euler, Newton, and Foundations for Mechanics

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... In fact, the conception of the laws of mechanics as dynamical laws, expressed by means of differential equations, was developed only during the eighteenth century. Furthermore, from the eighteenth century until the present, there have been various formulations of mechanics and various proposals for the basic mechanical principles, and attempts to reduce all of this to one well-defined and consistent theory with one basic set of laws have been only partially successful (Truesdell 1968;Stan 2017; Stan forthcoming-a; Stan forthcoming-b). Stan describes how during the eighteenth century, there was no general agreement on what exactly the laws of motion were, and various mechanical principles, laws of motion, and formulations of mechanics were developed: 'this diversity of foundational perspectives defies any attempt to show that post-Newtonian mechanics is a unified theory' (Stan forthcoming-a). ...
... The most pressing issue was that mechanics dealt with several distinct conceptions of matter (Stan 2017; see also Wilson 2013). The main conceptions of matter used in mechanics are mass points, rigid bodies and deformable continua. ...
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Determinism is generally regarded as one of the main characteristics of classical physics, that is, the physics of the eighteenth and nineteenth century. However, an inquiry into eighteenth and nineteenth century physics shows that the aim of accounting for all phenomena on the basis of deterministic equations of motion remained far out of reach. Famous statements of universal determinism, such as those of Laplace and Du Bois-Reymond, were made within a specific context and research program and did not represent a majority view. I argue that in this period, determinism was often an expectation rather than an established result, and that especially toward the late nineteenth and early twentieth century, it was often thought of as a presupposition of physics: physicists such as Mach, Poincaré and Boltzmann regarded determinism as a feature of scientific research, rather than as a claim about the world. It is only retrospectively that an image was created according to which classical physics was uniformly deterministic.
... These types of matter all have different properties and are irreducible to each other. They are also subjected to different dynamics: point particles only interact through forces which act between pairs of particles, depending on their distance 1 ; rigid bodies can rotate and can collide with each other; and deformable continua can additionally undergo internal stresses (Stan 2017). ...
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The idea that all physical phenomena should ultimately be reducible to matter and motion was influential throughout the nineteenth century, although this ideal was never realized and never without critics. But could the notion of matter itself be understood? A unified conception of matter was lacking in nineteenth century physics. Physicists used different conceptions of matter, and debated the question of the true nature of matter on the basis of philosophical as well as empirical arguments; it turned out to be very challenging to develop a conception of matter that was consistent with experimental findings as well as philosophically satisfactory. Towards the end of the nineteenth century, physicists increasingly rejected the question of the true nature of matter, arguing that this question was irrelevant for physics or altogether meaningless. This was sometimes seen as an emancipation of physics from philosophy, and sometimes as a result of philosophical reflection on physics.
... 362, 375).2 Wilson takes Newton's second law as F = ma(Wilson 2007, p. 174), which is not historically supported(Pourciau 2006;Stan 2017), but unlike Massin she does not make any historical claim. ...
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Philosophers have disagreed on the composition of force for decades. The main divergence concerns the fundamental question: given a certain motion that is observable, which force or forces are present in it, component or resultant forces? The present paper focuses on the conditions for dealing with this problem. I will argue that we are not able to infer force from the observation of a motion, as required by the problem. I will further argue that the validity of the Newtonian algorithm is not a sufficient condition for that inference because the Gaussian algorithm, which is equally valid, differs from the former with regard to the forces present in motions. Under these circumstances, I will make recourse to an experiment available in physics in which the force present in a motion is measured. Thus, we obtain a numerical value for this force. This result, playing the role of a counterexample, clarifies the composition of force issue significantly.
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The Executive Council of the International Society of Biomechanics has initiated and overseen the commemorations of the Society’s 50th Anniversary in 2023. This included multiple series of lectures at the ninth World Congress of Biomechanics in 2022 and XXIXth Congress of the International Society of Biomechanics in 2023, all linked to special issues of International Society of Biomechanics’ affiliated journals. This special issue of the Journal of Applied Biomechanics is dedicated to the biomechanics of the neuromusculoskeletal system. The reader is encouraged to explore this special issue which comprises 6 papers exploring the current state-of the-art, and future directions and roles for neuromusculoskeletal biomechanics. This editorial presents a very brief history of the science of the neuromusculoskeletal system’s 4 main components: the central nervous system, musculotendon units, the musculoskeletal system, and joints, and how they biomechanically integrate to enable an understanding of the generation and control of human movement. This also entails a quick exploration of contemporary neuromusculoskeletal biomechanics and its future with new fields of application.
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The early modern era produced the Scientific Revolution, which originated our present understanding of the natural world. Concurrently, philosophers established the conceptual foundations of modernity. This rich and comprehensive volume surveys and illuminates the numerous and complicated interconnections between philosophical and scientific thought as both were radically transformed from the late sixteenth to the mid-eighteenth century. The chapters explore reciprocal influences between philosophy and physics, astronomy, mathematics, medicine, and other disciplines, and show how thinkers responded to an immense range of intellectual, material, and institutional influences. The volume offers a unique perspicuity, viewing the entire landscape of early modern philosophy and science, and also marks an epoch in contemporary scholarship, surveying recent contributions and suggesting future investigations for the next generation of scholars and students.
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Having explained why I do not see Boscovich’s conception as belonging to the corpuscular-dynamical theories of matter that flourished in eighteenth-century Britain, and why he can hardly be associated with “dynamicists” of various kinds, I will now proceed to explore his theory of matter as it emerges from his writings and in its proper context.
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My thesis in this paper is: the modern concept of laws of motion—qua dynamical laws—emerges in 18th-century mechanics. The driving factor for it was the need to extend mechanics beyond the centroid theories of the late-1600s. The enabling result behind it was the rise of differential equations. In consequence, by the mid-1700s we see a deep shift in the form and status of laws of motion. The shift is among the critical inflection points where early modern mechanics turns into classical mechanics as we know it. Previously, laws of motion had been channels for truth and reference into mechanics. By 1750, the laws lose these features. Instead, now they just assert equalities between functions; and serve just to entail (differential) equations of motion for particular mechanical setups. This creates two philosophical problems. First, it’s unclear what counts as evidence for the laws of motion in the Enlightenment. Second, it’s a mystery whether these laws retain any notion of causality. That subverts the early-modern dictum that physics is a science of causes.
Metaphysical Foundations of Natural Science” (1786)
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La storia di F=ma. Le seconda legge del moto nel XVIII secolo
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Eulers Herleitung des Drehimpulssatzes
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