Vorlesungen über analytische Mechanik

Jacobi, Carl Gustav Jacob; Pulte, Helmut

doi:10.1007/978-3-322-80289-7

Cited by 26 publications

(8 citation statements)

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“…In that respect, at least as long as a Lagrangian density is the right place to start physical theorizing, the benefits of talking about uniformities of nature are retained by talking about symmetries. Nineteenth century results about the relationship between symmetries and conservation laws (Lagrange 1811 ; Hamilton 1834 ; Jacobi 1996 ) were generalized and synthesized by Max Born and then Emmy Noether during the 1910s (Born 1914 ; Noether 1918 ). Born noted that for theories of continua (field theories), conservation of energy and of momentum hold when the Lagrangian density is independent of time and of place, respectively.…”

Section: Symmetries Imply Conservation Laws and Vice Versamentioning

confidence: 99%

Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

Pitts

2020

Found Sci

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Misconceptions about energy conservation abound due to the gap between physics and secondary school chemistry. This paper surveys this difference and its relevance to the 1690s-2010s Leibnizian argument that mind-body interaction is impossible due to conservation laws. Justifications for energy conservation are partly empirical, such as Joule's paddle wheel experiment, and partly theoretical, such as Lagrange's statement in 1811 that energy is conserved if the potential energy does not depend on time. In 1918 Noether generalized results like Lagrange's and proved a converse: symmetries imply conservation laws and vice versa. Conservation holds if and only if nature is uniform. The rise of field physics during the 1860s-1920s implied that energy is located in particular places and conservation is primordially local: energy cannot disappear in Cambridge and reappear in Lincoln instantaneously or later; neither can it simply disappear in Cambridge or simply appear in Lincoln. A global conservation law can be inferred in some circumstances. Einstein's General Relativity, which stimulated Noether's work, is another source of difficulty for conservation laws. As is too rarely realized, the theory admits conserved quantities due to symmetries of the Lagrangian, like other theories. Indeed General Relativity has more symmetries and hence (at least formally) more conserved energies. An argument akin to Leibniz's finally gets some force. While the mathematics is too advanced for secondary school, the ideas that conservation is tied to uniformities of nature and that energy is in particular places, are accessible. Improved science teaching would serve the truth and enhance the social credibility of science.

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Section: Symmetries Imply Conservation Laws and Vice Versamentioning

confidence: 99%

Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

Pitts

2020

Found Sci

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“…The main principles of dynamics of systems with one-sided holonomic (geometrical) constraints are well-known (a worth mentioning exposition of the history of this problem can be found in [2]). In our paper, we take the first steps in studying the dynamics of systems with one-sided non-integrable constraints.…”

Section: ∑︁˙˙mentioning

confidence: 99%

On the dynamics of systems with one-sided non-integrable constraints

Козлов

2019

Theor appl mech (Belgr)

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In the paper we take the first steps in studying the dynamics of systems with one-sided differential constraints defined by inequalities in the phase space. We give a general definition of motion for systems with such constraints. Within the framework of the classical non-holonomic model, and also for systems with servoconstraints (according to Béghin), we present the conditions under which the system leaves two-sided differential constraints. As an example, we consider the Chaplygin sleigh with a one-sided constraint, which is realized by means of an anisotropic force of viscous friction. Variational principles for the determination of motion of systems with one-sided differential constraints are presented.

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“…The history of the discovery of the principle of least action is quite instructive. I shall present it in Jacobi's very vivid account (somewhat abridged) [9].…”

Section: This Argument Differs Only In Form From the Argument Of Eulementioning

confidence: 99%

Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler)

Козлов¹

2007

Russ. Math. Surv.

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This article concerns the life of Leonhard Euler and his achievements in theoretical mechanics. A number of topics are discussed related to the development of Euler's ideas and methods: divergent series and asymptotics of solutions of non-linear differential equations; the hydrodynamics of a perfect fluid and Hamiltonian systems; vortex theory for systems on Lie groups with left-invariant kinetic energy; energy criteria of stability; Euler's problem of two gravitating centres in curved spaces.

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Vorlesungen über analytische Mechanik

Cited by 26 publications

References 0 publications

Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

On the dynamics of systems with one-sided non-integrable constraints

Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler)

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