Document Zbl 1305.70024

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Dyn. Differ. Equations</a> <a href="/?q=in%3A340183" title="Articles in this Issue">26, No. 4, 817-841 (2014)</a>. </div> <div class="abstract">Summary: We consider the $N$-body problem with interaction potential $U_\alpha =\frac{1}{| x_i-x_j| ^\alpha }$ for $\alpha >1$. We assume that the particles have all the same mass and that $N$ is the order $| \mathcal R| $ of the rotation group $\mathcal {R}$ of one of the five Platonic polyhedra. We study motions that, up to a relabeling of the $N$ particles, are invariant under $\mathcal R$. By variational techniques we prove the existence of periodic and chaotic motions.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6389976">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A70F10" title="MSC2020">70F10</a> </td> <td class="space"> $n$-body problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52B15" title="MSC2020">52B15</a> </td> <td class="space"> Symmetry properties of polytopes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70K42" title="MSC2020">70K42</a> </td> <td class="space"> Equilibria and periodic trajectories for nonlinear problems in mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34C25" title="MSC2020">34C25</a> </td> <td class="space"> Periodic solutions to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49J99" title="MSC2020">49J99</a> </td> <td class="space"> Existence theories in calculus of variations and optimal control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37D45" title="MSC2020">37D45</a> </td> <td class="space"> Strange attractors, chaotic dynamics of systems with hyperbolic behavior </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3A%5C%28N%5C%29-body+problem">$N$-body problem</a>; <a href="/?q=ut%3Aperiodic+orbit">periodic orbit</a>; <a href="/?q=ut%3Achaotic+motions">chaotic motions</a>; <a href="/?q=ut%3Avariational+methods">variational methods</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1305.70024" data-ciurl="/ci/06389976" data-biburl="/bibtex/06389976.bib" data-amsurl="/amsrefs/06389976.bib" data-xmlurl="/xml/06389976.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06389976.pdf" title="Zbl 1305.70024 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10884-014-9401-2" aria-label="DOI for “Platonic polyhedra, periodic orbits and chaotic motions in the $N$-body problem with non-Newtonian forces”" title="10.1007/s10884-014-9401-2">DOI</a> <a class="btn btn-default btn-xs" type="button" href="http://hdl.handle.net/11568/649263" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Fusco, G., Gronchi, G.F., Negrini, P.: Platonic polyhedra, topological constraints and periodic solutions of the classical \[NN\]-body problem. Invent. Math. 185, 283-332 (2011) · <a href="/1305.70023" class="nowrap">Zbl 1305.70023</a> · <a href="https://doi.org/10.1007/s00222-010-0306-3" class="nowrap">doi:10.1007/s00222-010-0306-3</a></td> </tr><tr> <td>[2]</td> <td class="space">Cundy, H., Rollett, A.P.: Mathematical Models. Clarendon, Oxford (1954) · <a href="/0047.38807" class="nowrap">Zbl 0047.38807</a></td> </tr><tr> <td>[3]</td> <td class="space">http://adams.dm.unipi.it/ gronchi/nbody.html. Accessed 24 Sept 2014 · <a href="/1036.70007" class="nowrap">Zbl 1036.70007</a></td> </tr><tr> <td>[4]</td> <td class="space">Ferrario, D.L.: Transitive decomposition of symmetry groups for the \[n\] n-body problem. Adv. Math. 213(2), 763-784 (2007) · <a href="/1114.70013" class="nowrap">Zbl 1114.70013</a> · <a href="https://doi.org/10.1016/j.aim.2007.01.009" class="nowrap">doi:10.1016/j.aim.2007.01.009</a></td> </tr><tr> <td>[5]</td> <td class="space">Moore, C., Nauenberg, M.: New Periodic Orbits for the \[n\] n-Body Problem. arXiv:math/0511219v1 (2008) · <a href="/0984.70009" class="nowrap">Zbl 0984.70009</a></td> </tr><tr> <td>[6]</td> <td class="space">Chen, K.C.: Binary decomposition for planar \[NN\]-body problems and symmetric periodic solutions. Arch. Rational Mech. Anal. 170, 247-276 (2003) · <a href="/1036.70007" class="nowrap">Zbl 1036.70007</a> · <a href="https://doi.org/10.1007/s00205-003-0277-2" class="nowrap">doi:10.1007/s00205-003-0277-2</a></td> </tr><tr> <td>[7]</td> <td class="space">Chenciner, A.: Action minimizing solutions of the Newtonian \[n\] n-body problem: from homology to symmetry. In: Proceedings of the ICM 2002, Beijing (2002) · <a href="/1136.70310" class="nowrap">Zbl 1136.70310</a></td> </tr><tr> <td>[8]</td> <td class="space">Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du Problème newtonien de 4 corps de masses égales dans \[\mathbb{R}^3\] R3: orbites “hip-hop”. Cel. Mech. Dyn. Ast. 77, 139-152 (2000) · <a href="/0984.70009" class="nowrap">Zbl 0984.70009</a> · <a href="https://doi.org/10.1023/A:1008381001328" class="nowrap">doi:10.1023/A:1008381001328</a></td> </tr><tr> <td>[9]</td> <td class="space">Ferrario, D., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical \[n\] n-body problem. Invent. Math. 155, 305-362 (2004) · <a href="/1068.70013" class="nowrap">Zbl 1068.70013</a> · <a href="https://doi.org/10.1007/s00222-003-0322-7" class="nowrap">doi:10.1007/s00222-003-0322-7</a></td> </tr><tr> <td>[10]</td> <td class="space">Montgomery, R.: The \[n\] n-body problem, the braid group and action minimizing periodic solutions. Nonlinearity 11, 363-376 (1998) · <a href="/1076.70503" class="nowrap">Zbl 1076.70503</a> · <a href="https://doi.org/10.1088/0951-7715/11/2/011" class="nowrap">doi:10.1088/0951-7715/11/2/011</a></td> </tr><tr> <td>[11]</td> <td class="space">Ouyang, T., Xie, Z.: A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4 body problem, preprint (2013) · <a href="/1114.70013" class="nowrap">Zbl 1114.70013</a></td> </tr><tr> <td>[12]</td> <td class="space">Simó, C.: New families of solutions in \[NN\]-body problems. In: Proceedings of the Third European Congress of Mathematics, Casacuberta et al. edits, Progress in Mathematics vol. 201, pp. 101-115 (2001) · <a href="/1101.70009" class="nowrap">Zbl 1101.70009</a></td> </tr><tr> <td>[13]</td> <td class="space">Terracini, S., Venturelli, A.: Symmetric trajectories for the \[2N2N\]-body problem with equal masses. Arch. Rational Mech. Anal. 184, 465-493 (2007) · <a href="/1111.70010" class="nowrap">Zbl 1111.70010</a> · <a href="https://doi.org/10.1007/s00205-006-0030-8" class="nowrap">doi:10.1007/s00205-006-0030-8</a></td> </tr><tr> <td>[14]</td> <td class="space">Venturelli, A.: Une caracterisation variationelle des solutions de Lagrange du problème plan des trois corps. Comp. Rend. Acad. Sci. Paris, Série I 332, 641-644 (2001) · <a href="/1034.70007" class="nowrap">Zbl 1034.70007</a></td> </tr><tr> <td>[15]</td> <td class="space">Barutello, V., Ferrario, D. L., Terracini, S.: On the singularities of generalized solutions to \[n\] n-body-type problems. Int. Math. Res. Not., rnn 069, 1-78 (2008) · <a href="/1143.70005" class="nowrap">Zbl 1143.70005</a></td> </tr><tr> <td>[16]</td> <td class="space">Chenciner, A.: Symmetries and “simple” solutions of the classical \[NN\]-body problem. In: Proceedings of the ICMP03 (2003) · <a href="/1209.70011" class="nowrap">Zbl 1209.70011</a></td> </tr><tr> <td>[17]</td> <td class="space">Braides, A.: \[ \Gamma\] Γ-convergence for Beginners. Oxford University Press, Oxford (2005) · <a href="/1198.49001" class="nowrap">Zbl 1198.49001</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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