Abstract
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.
Similar content being viewed by others
References
Sinclair, A.J., Hurtado, J.E.: The motion constants of linear autonomous dynamical systems. Appl. Mech. Rev. 65(4), 041002–1 – 041002–9 (2013)
Noether, E.: Invariante variationsprobleme, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Kl. II, 235–257 (1918)
Noether, E.: Invariant variation problems. Transp. Theory Stat. Phys. 1(3), 186–207 (1971). Translation by M. A. Tavel of the original article
Logan, J.D.: Invariant Variational Principles. Academic, New York (1977)
González-Acosta, E., Corona-Galindo, M.G.: Noether’s theorem and the invariants for dissipative and driven dissipative like systems. Rev. Mex. Fis. 38(4), 511–517 (1992)
Choudhuri, A., Ghosh, S., Talukdar, B.: Symmetries and conservation laws of the damped harmonic oscillator. Pramana 70(4), 657–667 (2008)
Burns, S.A., Palmore, J.I.: The Newton transform: an operational method for constructing integrals of dynamical systems. Phys. D 37, 83–90 (1989)
Bateman, H.: On dissipative systems and related variational principles. Phys. Rev. 38(4), 815–819 (1931)
Dekker, H.: Classical and quantum mechanics of the damped harmonic oscillator. Phys. Rep. 80, 1–112 (1981)
Silva, M.R.M.C.d.: A transformation approach for finding first integrals of motion of dynamical systems. Int. J. Non Linear Mech. 9(4), 241–250 (1974)
Badrakhan, F.: Lagrangian formulation and first integrals of piecewise linear dissipative systems. J. Sound Vib. 82, 227–234 (1982)
Ghosh, S., Shamanna, J., Talukdar, B.: Inequivalent Lagrangians for the damped harmonic oscillator. Can. J. Phys. 82, 561–567 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sinclair, A.J., Hurtado, J.E., Bertinato, C. et al. On Noether’s Theorem and the Various Integrals of the Damped Linear Oscillator. J of Astronaut Sci 60, 396–407 (2013). https://doi.org/10.1007/s40295-015-0054-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40295-015-0054-0