Summary
Both the «direct» and the «inverse» Noether's theorems are generalized to allow for infinitesimal transformations that add to the action functional an integral over a 4-divergence and an integral over a function vanishing «on the orbit» (Noether transformations). It is then shown that: i) to every Noether transformation there corresponds a«weak» continuity equation and a family of Nother transformations (Noether family) defining the same continuity equation; ii) every Noether family contains an invariance transformation; iii) to every «weak» continuity equation there corresponds a Noether family; iv) every Noether family contains a subset of Noether transformations equivalent to a 4-divergence translation of the Lagrangian density.
Riassunto
Sia il teorema «diretto» che quello «inverso» di Noether vengono generalizzati, in modo da tener conto di trasformazioni infinitesime che aggiungono al funzionale d'azione un integrale su una quadridivergenza ed un integrale su di una funzione che si annulla «sull'orbita» (trasformazioni di Noether). Si dimostra chea) ad ogni trasformazione di Noether corrisponde unaequazione di continuità «debole» e una famiglia di trasformazioni di Noether (famiglia di Noether) che definiscono la stessa equazione di continuità;b) ogni famiglia di Noether contiene una trasformazione di invarianza;c) ad ogni equazione di continuità «debole» corrisponde una famiglia di Noether;d) ogni famiglia di Noether contiene una sottofamiglia di trasformazioni di Noether equivalenti a una traslazione, per mezzo di una quadridivergenza, della densità Lagrangiana.
Резюме
Обобщаются и «прямая» и «обратная» теоремы Ноэтера, чтобы учесть бесконечно малые преобразования, которые добавляют к функционалу действия интеграл от 4-мерной дивергенции и интеграл от функции, обращающейся в нуль «на орбите» (преобразования Ноэтера). Затем показывается, что 1) каждому преобразованию Ноэтера соответствует«слабое» уравнение неирерывности и семейство преобразований Ноэтера (семейство Ноэтера), определяющее то же уравнение непрерывности; 2) каждое семейство Ноэтера содержит инвариантное преобразование; 3) каждое «слабое» уравнение непрерывности соответствует семейству Ноэтера; 4) каждое семейство Ноэтера содержит подсистему преобразований Ноэтера, эквивалентных трансляции 4-мерной дивергенции для плотности лагранжиана.
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——(7)
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Candotti, E., Palmieri, C. & Vitale, B. On the inversion of Noether's theorem in the Lagrangian formalism. Nuovo Cimento A (1965-1970) 70, 233–246 (1970). https://doi.org/10.1007/BF02758981
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DOI: https://doi.org/10.1007/BF02758981