{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T03:28:26Z","timestamp":1773199706271,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,11]],"date-time":"2024-04-11T00:00:00Z","timestamp":1712793600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson\u2019s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark\u2019s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.<\/jats:p>","DOI":"10.3390\/axioms13040255","type":"journal-article","created":{"date-parts":[[2024,4,12]],"date-time":"2024-04-12T03:34:37Z","timestamp":1712892877000},"page":"255","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Simpson\u2019s Variational Integrator for Systems with Quadratic Lagrangians"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2311-6933","authenticated-orcid":false,"given":"Juan Antonio","family":"Rojas-Quintero","sequence":"first","affiliation":[{"name":"CONAHCYT, Tecnol\u00f3gico Nacional de M\u00e9xico, I. T. Ensenada, Ensenada 22780, B.C., Mexico"},{"name":"IMT Atlantique, LS2N, UMR CNRS 6004, 44307 Nantes, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4858-6234","authenticated-orcid":false,"given":"Fran\u00e7ois","family":"Dubois","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Saclay, Laboratoire de Math\u00e9matiques d\u2019Orsay, 91400 Orsay, France"},{"name":"Conservatoire National des Arts et M\u00e9tiers, Structural Mechanics and Coupled Systems Laboratory, 75141 Paris, France"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-0416-0376","authenticated-orcid":false,"given":"Jos\u00e9 Guadalupe","family":"Cabrera-D\u00edaz","sequence":"additional","affiliation":[{"name":"Tecnol\u00f3gico Nacional de M\u00e9xico, I. T. Ensenada, Ensenada 22780, B.C., Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Goldstine, H.H. (1977). 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