{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T04:40:17Z","timestamp":1760157617895,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,4]],"date-time":"2025-10-04T00:00:00Z","timestamp":1759536000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a novel numerical test to evaluate the reliability of numerical propagations, leveraging the fiber bundle structure of phase space typically induced by Lie symmetries, though not exclusively. This geometric test simultaneously verifies two properties: (i) preservation of conservation principles, and (ii) faithfulness to the symmetry-induced fiber bundle structure. To generalize the approach to systems lacking inherent symmetries, we construct an associated collective system endowed with an artificial G-symmetry. The original system then emerges as the G-reduced version of this collective system. By integrating the collective system and monitoring G-fiber bundle conservation, our test quantifies numerical precision loss and detects geometric structure violations more effectively than classical integral-based checks. Numerical experiments demonstrate the superior performance of this method, particularly in long-term simulations of rigid body dynamics and perturbed Keplerian systems.<\/jats:p>","DOI":"10.3390\/sym17101652","type":"journal-article","created":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T08:10:51Z","timestamp":1759738251000},"page":"1652","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Geometric Numerical Test via Collective Integrators: A Tool for Orbital and Attitude Propagation"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5930-8523","authenticated-orcid":false,"given":"Francisco","family":"Crespo","sequence":"first","affiliation":[{"name":"Department of Aerospace Engineering, Embry-Riddle Aeronautical University, 1 Aerospace Blvd, Daytona Beach, FL 32114, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jhon","family":"Vidarte","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica y F\u00edsica Aplicadas, Universidad Cat\u00f3lica de la Sant\u00edsima Concepci\u00f3n, Casilla 297, Concepci\u00f3n 4051381, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0438-3347","authenticated-orcid":false,"given":"Jersson Gerley","family":"Villafa\u00f1e","sequence":"additional","affiliation":[{"name":"Grupo GISDA, Departamento de Matem\u00e1tica, Facultad de Ciencias, Universidad del B\u00edo-B\u00edo, Collao 1202, Casilla 5-C, Concepci\u00f3n 4051381, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorge Luis","family":"Zapata","sequence":"additional","affiliation":[{"name":"Programa de Formaci\u00f3n Pedag\u00f3gica para Licenciados y\/o Profesionales, Facultad de Educaci\u00f3n, Universidad San Sebasti\u00e1n, Lientur 1457, Concepci\u00f3n 4080871, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2444","DOI":"10.1093\/mnras\/stw780","article-title":"Stability and chaos in Kustaanheimo\u2013Stiefel space induced by the Hopf fibration","volume":"459","author":"Roa","year":"2016","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_2","unstructured":"Valle, S.C.D., Urrutxua, H., and Solano-L\u00f3pez, P. (2024, January 14\u201318). Exploiting Gauge Freedom in KS Variables for High-Performance Numerical Orbital Propagation. Proceedings of the Proceedings of the International Astronautical Congress (IAC), Milan, Italy."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.geomphys.2015.02.016","article-title":"The Kepler system as a reduced 4D harmonic oscillator","volume":"92","year":"2015","journal-title":"J. Geom. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"359","DOI":"10.3934\/jgm.2018013","article-title":"Alternative Angle-Based Approach to the KS-Map. An Interpretation Through Symmetry and Reduction","volume":"10","author":"Ferrer","year":"2018","journal-title":"J. Geom. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1137\/19M1264060","article-title":"Alternative Reduction by Stages of Keplerian Systems. Positive, Negative, and Zero Energy","volume":"19","author":"Crespo","year":"2020","journal-title":"Siam J. Appl. Dyn. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1093\/imanum\/dru013","article-title":"Collective Lie\u2013Poisson integrators on R3","volume":"35","author":"McLachlan","year":"2014","journal-title":"Ima J. Numer. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1088\/0951-7715\/27\/6\/1525","article-title":"Collective symplectic integrators","volume":"27","author":"McLachlan","year":"2014","journal-title":"Nonlinearity"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"43","DOI":"10.3934\/jcd.2023015","article-title":"Geometric integration on symmetric spaces","volume":"11","author":"MuntheKaas","year":"2024","journal-title":"J. Comput. Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hairer, E., Wanner, G., and Lubich, C. (2006). Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer. [2nd ed.].","DOI":"10.4171\/owr\/2006\/14"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). Simulating Hamiltonian Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9780511614118"},{"key":"ref_11","unstructured":"Sanz-Serna, J.M. (2018). Numerical Hamiltonian Problems, Courier Dover Publications."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nakahara, M. (2003). Geometry, Topology and Physics, Institute of Physics Publishing. [2nd ed.].","DOI":"10.1201\/9781420056945"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Lee, J. (2012). Introduction to Smooth Manifolds, Springer. [2nd ed.]. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4419-9982-5"},{"key":"ref_14","unstructured":"Meyer, K. (1973). Symmetries and Integrals in Mechanics. Dynamical Systems, Academic Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Marsden, J., and Weinstein, A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 121\u2013130.","DOI":"10.1016\/0034-4877(74)90021-4"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/BF01208503","article-title":"Ueber die Theorie der algebraischen Formen","volume":"36","author":"Hilbert","year":"1890","journal-title":"Math. Ann."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Weyl, H. (1946). The Classical Groups, Their Invariants and Representations, Princenton Univ. Press.","DOI":"10.1515\/9781400883905"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0040-9383(75)90036-1","article-title":"Smooth function invariant under the action of a compact Lie group","volume":"14","author":"Schwarz","year":"1975","journal-title":"Topology"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Poenaru, V. (1976). Singularit\u00e9s C\u221e en Pr\u00e9sence de Sym\u00e9trie, Springer. LNM.","DOI":"10.1007\/BFb0079196"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1111\/j.1365-2966.2009.15437.x","article-title":"Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics","volume":"400","author":"Saha","year":"2009","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3934\/jgm.2015.7.151","article-title":"On the extended Euler system and the Jacobi and Weierstrass elliptic functions","volume":"7","author":"Crespo","year":"2015","journal-title":"J. Geom. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ortega, J.P., and Ratiu, T.S. (2004). Momentum Maps and Hamiltonian Reduction, Birkhauser.","DOI":"10.1007\/978-1-4757-3811-7"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bluman, G.W.K.S. (1989). Symmetries and Differential Equations, Springer.","DOI":"10.1007\/978-1-4757-4307-4"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Olver, P.J. (1993). Applications of Lie Groups to Differential Equations, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Marsden, J., and Ratiu, T. (1999). Introduction to Mechanics and Symmetry, Springer. [2nd ed.]. 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