{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:31:50Z","timestamp":1775694710600,"version":"3.50.1"},"reference-count":58,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T00:00:00Z","timestamp":1692748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend the prescription to the first prolongation of the Noether vector. It is possible to show that the latter application provides a general constraint on the infinitesimal generator \u03be, related to the spacetime translations. This approach can be used for several applications. In the second part of the work, we consider a gravity theory, including the coupling between a scalar field \u03d5 and the Gauss\u2013Bonnet topological term G. In particular, we study a gravitational action containing the function F(G,\u03d5) and select viable models by the existence of symmetries. Finally, we evaluate the selected models in a spatially flat cosmological background and use symmetries to find exact solutions.<\/jats:p>","DOI":"10.3390\/sym15091625","type":"journal-article","created":{"date-parts":[[2023,8,23]],"date-time":"2023-08-23T07:25:24Z","timestamp":1692775524000},"page":"1625","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss\u2013Bonnet Gravity"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0333-6234","authenticated-orcid":false,"given":"Francesco","family":"Bajardi","sequence":"first","affiliation":[{"name":"Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare(INFN), Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4886-2024","authenticated-orcid":false,"given":"Salvatore","family":"Capozziello","sequence":"additional","affiliation":[{"name":"Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare(INFN), Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy"},{"name":"Dipartimento di Fisica \u201cE. Pancini\u201d, Universit\u00e1 di Napoli \u201cFederico II\u201d, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3220-6375","authenticated-orcid":false,"given":"Tiziana","family":"Di Salvo","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Chimica \u201cEmilio Segr\u00e9\u201d, Universit\u00e0 degli Studi di Palermo, 90123 Palermo, Italy"}]},{"given":"Francesca","family":"Spinnato","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Chimica \u201cEmilio Segr\u00e9\u201d, Universit\u00e0 degli Studi di Palermo, 90123 Palermo, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,23]]},"reference":[{"key":"ref_1","unstructured":"Misner, C.W., Thorne, K.S., Wheeler, J.A., and Freeman, W.H. (1973). 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