Volume 15, Issue 1, March 2008, Pages 9 - 21
Noetherian first integrals
P.G.L. Leach, G.P. Flessas
Received 14 February 2007, Accepted 21 November 2007, Available Online 1 March 2008.
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- From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noethe- rian symmetries associated with the integrals which have been termed nonnoetherian. To further emphasise our point we construct the noetherian first integrals/invariants, which are associated with symmetries linear in the velocities, for the two-dimensional autonomous isotropic harmonic oscillator and the autonomous anisotropic oscillator and illustrate the roles which the invariants can play in the description of the clas- sical motion. We relate these symmetries to the corresponding problem in quantum mechanics. Further we show that the complete symmetry group of this anisotropic harmonic oscillator has the same representation as that of the corresponding isotropic oscillator. As a concluding example we show that a symmetry claimed to be non- noetherian is trivially Noetherian.
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TY - JOUR AU - P.G.L. Leach AU - G.P. Flessas PY - 2008 DA - 2008/03 TI - Noetherian first integrals JO - Journal of Nonlinear Mathematical Physics SP - 9 EP - 21 VL - 15 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2008.15.1.2 DO - https://doi.org/10.2991/jnmp.2008.15.1.2 ID - Leach2008 ER -