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Observers and Frames of Reference in Classical Continuum Theory

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An Expedition to Continuum Theory

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 210))

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Abstract

This chapter is dedicated to a relatively abstract question: Do balance laws and constitutive equations keep their form if we switch from an observer at rest to an arbitrarily moving one? In mathematical terms this problem can be analyzed by using so-called Euclidean transformations and establish an almost philosophical principle according to which true laws of nature must keep their form, independently of the frame of reference.

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Notes

  1. 1.

    For the determinant of Eq. (8.3.1) we could simply write det(O′) = ±1. This is a consequence of Eqs. (8.2.5/8.2.7), which follows from the fact that the inverse of a rotation matrix is given by its transposed. However, there are didactic reasons why we do not write it that way in Eq. (8.3.1): In Chap. 13 we will introduce so-called world tensors in complete analogy to Eq. (8.3.1/8.3.2). However, in contrast to the Euclidean transforms the corresponding world transforms are not normalized.

  2. 2.

    The Latin citations stem from the third edition of Newton’s book and can be found in the most carefully edited two volumes by Koyré et al. [9]. When compared to the first edition of the Principia (1687) we notice considerable differences in the wording, in terms of alterations as well as additional comments. This is an indication of Newton’s lifelong struggle with his findings, and it also shows how his understanding grew steadily over time. Moreover, note, that all but one translation of the original Latin text stem from the book of Chandrasekhar (1995). The translation of the “hypotheses non fingo” passage is from Cohen and Whitman (1999).

  3. 3.

    Strictly speaking Thirring’s analysis also allows the hollow sphere to rotate with an angular velocity different from ω.

References

  1. Becker E, Bürger W (1975) Kontinuumsmechanik. Teubner Studienbücher Mechanik Band 20. B.G. Teubner, Stuttgart

    Google Scholar 

  2. Bertram A (2008) Elasticity and plasticity of large deformations, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  3. Bertram A, Svendsen B (2004) Reply to Rivlins’s material symmetry revisited or much ado about nothing. GAMM-Mitt 1:88–93

    MathSciNet  Google Scholar 

  4. d’Alembert JL (1967) Traité de dynamique dans lequel les lois de l’équilibre & du mouvement des corps sont réduites au plus petit nombre possible, & démontrées d’une manière nouvelle, & ou l’on donne un principe général pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres, d’ une manière quelconque. David, Paris

    Google Scholar 

  5. Einstein A (1914) Die formale Grundlage der allgemeinen Relativitätstheorie. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), pp 1030–1085

    Google Scholar 

  6. Einstein A (1920) Relativity. The special and the general theory. A popular exposition, 3rd edn. Methuen & Co. Ltd., London, p. 68

    Google Scholar 

  7. Einstein A (1921) A brief outline of the development of the theory of relativity. Nature 106(2677):782–784

    Article  Google Scholar 

  8. Greve R (2003) Kontinuumsmechanik—Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin, New York

    MATH  Google Scholar 

  9. Koyré A, Cohen IB, Whitman A (1972) Isaac Newton’s Philosophiae Naturalis Principia Mathematica, 3rd edn (1726), vol I/II with variant readings. Cambridge University Press, Cambridge

    Google Scholar 

  10. Liu I-S (2010) Continuum mechanics. Springer, Berlin, New York

    Google Scholar 

  11. Mach E (1976) Die Mechanik—historisch-kritisch dargestellt. Wissenschaftliche Buchgesellschaft Darmstadt

    Google Scholar 

  12. Thirring H (1918) Über die Wirkungen rotierender ferner Massen in der Einsteinschen Gravitationstheorie. Physikalische Zeitschrift, pp 33–39

    Google Scholar 

  13. Thirring H (1921) Berichtigung zu meiner Arbeit: “Über die Wirkungen rotierender ferner Massen in der Einsteinschen Gravitationstheorie”. Physikalische Zeitschrift, pp 29–30

    Google Scholar 

  14. Truesdell C (1966) The elements of continuum mechanics. Springer, Berlin, New York

    MATH  Google Scholar 

  15. Truesdell C, Noll W (1965) The non-linear theories of mechanics. In: Flügge S (ed) Encyclopedia of physics, volume III/3. Springer, Berlin, Göttingen

    Google Scholar 

  16. Truesdell C, Toupin R (1960) The classical field theories. In: Flügge S (ed.) Encyclopedia of physics, vol III/1, Principles of classical mechanics and field theory. Springer, Berlin, Göttingen

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mechanics, Technical University of Berlin, Berlin, Germany

    Wolfgang H. Müller

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  1. Wolfgang H. Müller
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Correspondence to Wolfgang H. Müller .

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Müller, W.H. (2014). Observers and Frames of Reference in Classical Continuum Theory. In: An Expedition to Continuum Theory. Solid Mechanics and Its Applications, vol 210. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7799-6_8

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  • DOI: https://doi.org/10.1007/978-94-007-7799-6_8

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