Summary
Noether's theorem and Noether's inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamilton's type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.
Zusammenfassung
Der Noethersche Satz und seine Umkehrung werden für mechanische Systeme mit nichtkonservativen Kräften aufgestellt. Die Existenz von Erstintegralen hängt von der Existenz von Lösungen der verallgemeinerten Noether-Bessel-Hagen-Gleichung oder, gleichbedeutend, von der von Lösungen des Killingschen Systems partieller Differentialgleichungen ab. Die Theorie fußt auf der Idee, daß Transformationen von Zeit, verallgemeinerten Koordinaten und dissipativen Kräften die Transformation der verallgemeinerten Geschwindigkeiten bestimmen; wie im Fall von Variationen in einem Variationsprinzip von Hamiltonscher Art für rein nichtkonservative Systeme [17], [18]. Unter Verwendung dieser Theorie werden einige neue Erstintegrale nichtkonservativer Probleme erhalten.
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Djukic, D.S., Vujanovic, B.D. Noether's theory in classical nonconservative mechanics. Acta Mechanica 23, 17–27 (1975). https://doi.org/10.1007/BF01177666
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DOI: https://doi.org/10.1007/BF01177666