naa a22 4500 510774172 CHVBK 20180411083224.0 cr unu---uuuuu 180411e20131201xx s 000 0 eng 10.1007/s40295-015-0054-0 doi (NATIONALLICENCE)springer-10.1007/s40295-015-0054-0 On Noether's Theorem and the Various Integrals of the Damped Linear Oscillator [Elektronische Daten] [Andrew Sinclair, John Hurtado, Chris Bertinato, Peter Betsch] Noether's theorem provides deep insight into the connection between analytical mechanics and the integrals of dynamic systems, specifically, showing how symmetries of the action integral are connected to the integrals of motion. To demonstrate Noether's theorem, the harmonic oscillator is often used as a simple example problem. Presentations in the literature, however, often focus on the single absolutely-invariant symmetry for this problem. This paper presents a complete application of Noether's theorem to the damped harmonic oscillator, including general solutions of the divergence-invariant Killing equations and the associated integrals for all underdamped, critically-damped, and overdamped cases. This treatment brings forward several interesting issues. Five different symmetries produce independent solutions to the Killing equations, but of course, only two independent integrals exist for this second-order system. Also, integrals of a particular desired form may not be produced directly from Noether's theorem and are referred to as non-Noether or asymmetric integrals. For the damped oscillator, one such example is the time-independent integrals, referred to as motion constants. American Astronautical Society, 2015 Killing equations nationallicence Time-independent integrals nationallicence Time-dependent Lagrangian nationallicence Sinclair Andrew Aerospace Engineering Department, Auburn University, 211 Davis Hall, 36849, Auburn, AL, USA aut Hurtado John Aerospace Engineering Department, Texas A&M University, 3141 TAMU, 77843, College Station, TX, USA aut Bertinato Chris Aerospace Engineering Department, Texas A&M University, 3141 TAMU, 77843, College Station, TX, USA aut Betsch Peter Department of Mechanical Engineering, University of Siegen, Siegen, Germany aut The Journal of the Astronautical Sciences Springer US; http://www.springer-ny.com 60/3-4(2013-12-01), 396-407 0021-9142 60:3-4<396 2013 60 40295 https://doi.org/10.1007/s40295-015-0054-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s40295-015-0054-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Sinclair Andrew Aerospace Engineering Department, Auburn University, 211 Davis Hall, 36849, Auburn, AL, USA aut NATIONALLICENCE 700 1- Hurtado John Aerospace Engineering Department, Texas A&M University, 3141 TAMU, 77843, College Station, TX, USA aut NATIONALLICENCE 700 1- Bertinato Chris Aerospace Engineering Department, Texas A&M University, 3141 TAMU, 77843, College Station, TX, USA aut NATIONALLICENCE 700 1- Betsch Peter Department of Mechanical Engineering, University of Siegen, Siegen, Germany aut NATIONALLICENCE 773 0- The Journal of the Astronautical Sciences Springer US; http://www.springer-ny.com 60/3-4(2013-12-01), 396-407 0021-9142 60:3-4<396 2013 60 40295 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer