Timeline for How to find the associated conservation law from a given symmetry
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 25, 2020 at 17:33 | history | edited | Sharik |
edited tags
|
|
| Apr 10, 2020 at 11:47 | vote | accept | Sharik | ||
| Apr 10, 2020 at 11:44 | comment | added | Igor Khavkine | I should have written "...applying the Euler-Lagrange operator to both sides...". The Euler-Lagrange operator starts off a sequence, in the same way that the exterior derivative starts off the de Rham sequence, where the Helmholtz operator is next in the sequence. | |
| Apr 6, 2020 at 14:36 | comment | added | Igor Khavkine | @BenMcKay Every conserved current is equivalent to one s.t. its conservation condition takes the form $\mathrm{d} j[\phi] = \xi \cdot E[\phi]$ (no derivatives of the equations $E[\phi]=0$). Applying the Helmhotz operator to both sizes gives $H(\xi \cdot E[\phi])=0$, implying differential conditions on $\xi$. On appropriate equivalence classes, $j \leftrightarrow \xi$ is a bijection. When $E[\phi]=0$ is variational, the condition on $\xi$ is the same as being a symmetry of the Lagrangian. | |
| Apr 6, 2020 at 13:55 | comment | added | Ben McKay | This theorem is known to hold for variational problems in very broad generality, but I don't think there is a theorem of this type for nonvariational problems in any generality. | |
| Apr 6, 2020 at 13:17 | answer | added | Igor Khavkine | timeline score: 9 | |
| Apr 6, 2020 at 0:00 | comment | added | LSpice | I don't know if it'll help to answer your specific question, but I hope you won't mind my taking the excuse to share Baez's exposition Noether's theorem in a nutshell, which I've always found second to none on this subject. | |
| Apr 5, 2020 at 23:21 | comment | added | Michael Engelhardt | Time-translation invariance implies the conservation of energy. | |
| Apr 5, 2020 at 23:12 | answer | added | Tom Price | timeline score: 1 | |
| Apr 3, 2020 at 22:05 | comment | added | Jonny Evans | Conceptually, Noether's theorem is saying that a 1-parameter family of diffeomorphisms of the space F of fields induces a Hamiltonian flow on T^*F, and the Hamiltonian (Noether charge) is then conserved under that flow. I wrote a blog post about this a couple of years ago: jde27.uk/blog/noether.html - in particular the conservation of field momentum associated with space translation is worked out. Admittedly, I find it easier to think about going from the conserved quantity to the symmetry (from Hamiltonian to its flow) but that's just because differentiating is easier than integrating. | |
| Apr 3, 2020 at 21:24 | answer | added | Carlo Beenakker | timeline score: 3 | |
| Apr 3, 2020 at 21:07 | comment | added | Willie Wong | The equation you wrote is variational. So you can use the general formalism there to make the computation starting from an appropriate Lagrangian. (Here the action is $$-u_t^2 + u_x^2 + F(u)$$ where $F$ is an antiderivative of the scalar function $f$.) Following en.wikipedia.org/wiki/Noether%27s_theorem#Field_theory_version you can get a conserved current $j$. Integrating $j^0$ along $\{t = const\}$ gives you the time-independent conserved quantity. | |
| Apr 3, 2020 at 20:54 | history | asked | Sharik | CC BY-SA 4.0 |