In most books on classical mechanics, Noether's Theorem is only formulated in conservative systems with an action principle. Therefore I was wondering if it is possible to also do that in non-conservative systems? It would be pretty cool if you could at least get the conservation of momentum (which is valid also in non-conservative systems if there are no external forces because of Newton 3) from translation symmetry and conservation of angular momentum (which is also valid in non-conservative systems as long as there are only central forces) as a consequence of rotation symmetry in non-conservative systems. If that is not the case, is there any way to extend Hamilton's principle or the generelization of it ($\int (\delta T + \delta W) dt = 0$, see for example Sommerfeld) so that it also covers Newton 3 (also maybe that's a topic worthy of its own question)?
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4$\begingroup$ One article that might be interesting is Galley's 2014 paper The principle of stationary nonconservative action for classical mechanics and field theories. IT develops a way to approach arbitrary non-conservative effects withint leaving Lagrangian mechanics. This is likely to help. Obviously you wont find a conservation law when there is no conservation, but if the non-conservative elements are unaffected by your symmetry, it might work. $\endgroup$– Cort AmmonCommented Jan 21, 2023 at 23:37
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1$\begingroup$ Related: Does Noether's Theorem Apply in Galley's Extension to Lagrangian Mechanics? $\endgroup$– Qmechanic ♦Commented Jan 22, 2023 at 3:00
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