{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T03:37:30Z","timestamp":1775187450624,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T00:00:00Z","timestamp":1757030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2023YFA1407100"],"award-info":[{"award-number":["2023YFA1407100"]}]},{"name":"National Key Research and Development Program of China","award":["2021B0301030005"],"award-info":[{"award-number":["2021B0301030005"]}]},{"name":"National Key Research and Development Program of China","award":["2023A1515010871"],"award-info":[{"award-number":["2023A1515010871"]}]},{"name":"National Key Research and Development Program of China","award":["33620230100546CB"],"award-info":[{"award-number":["33620230100546CB"]}]},{"name":"Guangdong Province Science and Technology Major Project","award":["2023YFA1407100"],"award-info":[{"award-number":["2023YFA1407100"]}]},{"name":"Guangdong Province Science and Technology Major Project","award":["2021B0301030005"],"award-info":[{"award-number":["2021B0301030005"]}]},{"name":"Guangdong Province Science and Technology Major Project","award":["2023A1515010871"],"award-info":[{"award-number":["2023A1515010871"]}]},{"name":"Guangdong Province Science and Technology Major Project","award":["33620230100546CB"],"award-info":[{"award-number":["33620230100546CB"]}]},{"name":"Guangdong Natural Science Foundation","award":["2023YFA1407100"],"award-info":[{"award-number":["2023YFA1407100"]}]},{"name":"Guangdong Natural Science Foundation","award":["2021B0301030005"],"award-info":[{"award-number":["2021B0301030005"]}]},{"name":"Guangdong Natural Science Foundation","award":["2023A1515010871"],"award-info":[{"award-number":["2023A1515010871"]}]},{"name":"Guangdong Natural Science Foundation","award":["33620230100546CB"],"award-info":[{"award-number":["33620230100546CB"]}]},{"name":"SECyT, National University of Cordoba","award":["2023YFA1407100"],"award-info":[{"award-number":["2023YFA1407100"]}]},{"name":"SECyT, National University of Cordoba","award":["2021B0301030005"],"award-info":[{"award-number":["2021B0301030005"]}]},{"name":"SECyT, National University of Cordoba","award":["2023A1515010871"],"award-info":[{"award-number":["2023A1515010871"]}]},{"name":"SECyT, National University of Cordoba","award":["33620230100546CB"],"award-info":[{"award-number":["33620230100546CB"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The swinging sticks pendulum is an intriguing physical system that exemplifies the intersection of Lagrangian mechanics and chaos theory. It consists of a series of slender, interconnected metal rods, each with a counterweighted end that introduces an asymmetrical mass distribution. The rods are arranged to pivot freely about their attachment points, enabling both rotational and translational motion. Unlike a simple pendulum, this system exhibits complex and chaotic behavior due to the interplay between its degrees of freedom. The Lagrangian formalism provides a robust framework for modeling the system\u2019s dynamics, incorporating both rotational and translational components. The equations of motion are derived from the Euler\u2013Lagrange equations and lack closed-form analytical solutions, necessitating the use of numerical methods. In this work, we employ the Bulirsch\u2013Stoer method, a high-accuracy extrapolation technique based on the modified midpoint method, to solve the equations numerically. The system possesses four fixed points, each one associated with a different level of energy. The fixed point with the lowest energy level is a center, around which small perturbations are studied. The other three fixed points are unstable. The maximum energy used for the perturbations is 0.001% larger than the lowest equilibrium energy. When the system\u2019s total energy is low, nonlinear terms in the equations can be neglected, allowing for a linearized treatment based on small-angle approximations. Under these conditions, the pendulum oscillates with small amplitudes around a stable equilibrium point. The resulting motion is analyzed using tools from nonlinear dynamics and Fourier analysis. Several trajectories are generated and examined to reveal frequency interactions and the emergence of complex dynamical behavior. When a small initial perturbation is applied to one rod, its motion is characterized by a single frequency with significantly greater amplitude and angular velocity compared to the second rod. In contrast, the second rod displayed dynamics that involved two frequencies. The present study, to the best of our knowledge, is the first attempt to describe the dynamical behavior of this pendulum.<\/jats:p>","DOI":"10.3390\/sym17091467","type":"journal-article","created":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T12:18:18Z","timestamp":1757074698000},"page":"1467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Swinging Sticks Pendulum: Small Perturbations Analysis"],"prefix":"10.3390","volume":"17","author":[{"given":"Yundong","family":"Li","sequence":"first","affiliation":[{"name":"Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"},{"name":"Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel"},{"name":"Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rong","family":"Tang","sequence":"additional","affiliation":[{"name":"Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"},{"name":"Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel"},{"name":"Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3115-7655","authenticated-orcid":false,"given":"Bikash Kumar","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"},{"name":"Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel"},{"name":"Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1123-6460","authenticated-orcid":false,"given":"Marcelo F.","family":"Ciappina","sequence":"additional","affiliation":[{"name":"Department of Physics, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"},{"name":"Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel"},{"name":"Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion, Guangdong Technion-Israel Institute of Technology, 241 Daxue Road, Shantou 515063, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7250-0392","authenticated-orcid":false,"given":"Sergio","family":"Elaskar","sequence":"additional","affiliation":[{"name":"Department of Aerospace Engineering and Institute of Advanced Studies in Engineering and Technology (IDIT), FCEFyN, National University of Cordoba and CONICET, C\u00f3rdoba 5000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Schuster, H., and Just, W. 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