By douady R.
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Additional info for Algebre et theories galoisiennes
Noether's theorem is one of the fundamental theorems of Hamiltonian mechanics, and has proven to be extremely fruitful in the analysis of such PDE. Of course, the field of Hamiltonian mechanics offers many more beautiful mathematical results than just Noether's theorem; it is of great interest to see how much else of this theory (which is still largely confined to ODE) can be extended to the PDE setting. See [Kuk3] for some further discussion. Noether's theorem can be phrased symplectically, in the context of Hamiltonian mechanics, or variationally, in the context of Lagrangian mechanics.
Abstractly, the principle works as follows. 21 (Abstract bootstrap principle). Let I be a time interval. and for each t E I suppose we have two statements, a "hypothesis" H(t) and a "conclusion" C(t). Suppose we can verify the following four assertions: (a) (Hypothesis implies conclusion) If H(t) is true for some time t E 1, then C(t) is also true for that time t. (b) (Conclusion is stronger than hypothesis) If C(t) is true for some t E I, then H(t') is true for all t' E I in a neighbourhood of t.
I D) to the ODE Otu(t) = F(u(t)) for some F E for all times t E I. D). 15. 3. PROOF. By a limiting argument (writing I as the union of compact intervals) it suffices to prove the claim for compact I. We can use time translation invariance to set to = 0. By splitting I into positive and negative components, and using the change of variables t --. -t if necessary, we may take I = 10, T] for some T > 0. Here, the relevant scalar quantity to analyze is the distance IIu(t) - v(t)IIv between u and v, where IIIIv is some arbitrary norm on D.
Algebre et theories galoisiennes by douady R.