Noether's theorem: Difference between revisions

Revision as of 12:45, 16 August 2024

Revision as of 08:19, 13 September 2024

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:<math>p_k = \frac{\partial L}{\partial \dot{q_k}}.</math>

In [[special relativity|special]] and [[general relativity]], these two conservation laws can be expressed either ''globally'' (as it is done above), or ''locally'' as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined ''locally'' at the space-time point: the [[stress–energy tensor]]<ref name="Goldstein1980">{{cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein |year=1980 |title= [[Classical Mechanics (textbookGoldstein)|Classical Mechanics]] |edition=2nd |publisher=Addison-Wesley |___location=Reading, MA |isbn= 0-201-02918-9 }}</ref>{{rp|592}}(this will be derived in the next section).

'''III. Rotational invariance'''