{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:03:37Z","timestamp":1760231017260,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,25]],"date-time":"2022-08-25T00:00:00Z","timestamp":1661385600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National SKA Program of China","award":["2020SKA0110401","11988101","12171466"],"award-info":[{"award-number":["2020SKA0110401","11988101","12171466"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2020SKA0110401","11988101","12171466"],"award-info":[{"award-number":["2020SKA0110401","11988101","12171466"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The dynamic equation of a mass point in the circular restricted three-body problem is governed by Coriolis and centrifugal force, in addition to a co-rotating potential relative to the frame. In this paper, we provide an explicit, symmetric integrator for this problem. Such an integrator is more efficient than the symplectic Euler method and the Gauss Runge\u2013Kutta method as regards this problem. In addition, we proved the integrator is symplectic by the discrete Hamilton\u2019s principle. Several groups of numerical experiments demonstrated the precision and high efficiency of the integrator in the examples of the quadratic potential and the bounded orbits in the circular restricted three-body problem.<\/jats:p>","DOI":"10.3390\/sym14091769","type":"journal-article","created":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T02:09:36Z","timestamp":1661911776000},"page":"1769","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Highly Efficient Numerical Integrator for the Circular Restricted Three-Body Problem"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1091-6843","authenticated-orcid":false,"given":"Xiongbiao","family":"Tu","sequence":"first","affiliation":[{"name":"National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiao","family":"Wang","sequence":"additional","affiliation":[{"name":"National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China"},{"name":"School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifa","family":"Tang","sequence":"additional","affiliation":[{"name":"LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,25]]},"reference":[{"key":"ref_1","unstructured":"Feng, K. (1985). On difference schemes and symplectic geometry. Proceedings of 1984 Beijing Symposium on Differential Geometry and Differential Equations, Science Press."},{"key":"ref_2","first-page":"279","article-title":"Difference Schemes for Hamiltonian Formalism and Symplectic Geometry","volume":"4","author":"Feng","year":"1986","journal-title":"J. Comput. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0167-2789(90)90019-L","article-title":"Fourth-order symplectic integration","volume":"43","author":"Forest","year":"1990","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1088\/0951-7715\/3\/2\/001","article-title":"Symplectic integration of Hamiltonian systems","volume":"3","author":"Channell","year":"1990","journal-title":"Nonlinearity"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/0021-9991(91)90299-Z","article-title":"A symplectic integration algorithm for separable Hamiltonian functions","volume":"92","author":"Candy","year":"1991","journal-title":"J. Comput. Phys."},{"key":"ref_6","first-page":"4345278","article-title":"Compact Local Structure-Preserving Algorithms for the Nonlinear Schr\u00f6dinger Equation with Wave Operator","volume":"2020","author":"Huang","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.apnum.2018.09.005","article-title":"A novel kind of efficient symplectic scheme for Klein\u2013Gordon\u2013Schr\u00f6dinger equation","volume":"135","author":"Kong","year":"2019","journal-title":"Appl. Numer. Math."},{"key":"ref_8","unstructured":"Boris, J. (1970). Relativistic plasma simulation-optimization of a hybrid code. Proceeding of Fourth Conference on Numerical Simulations of Plasmas, Naval Research Laboratory."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"084503","DOI":"10.1063\/1.4818428","article-title":"Why is Boris algorithm so good?","volume":"20","author":"Qin","year":"2013","journal-title":"Phys. Plasmas"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.jcp.2015.09.007","article-title":"Comment on \u201cSymplectic integration of magnetic systems\u201d: A proof that the Boris algorithm is not variational","volume":"301","author":"Ellison","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.jcp.2014.10.032","article-title":"Volume-preserving algorithms for charged particle dynamics","volume":"281","author":"He","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"044501","DOI":"10.1063\/1.4916570","article-title":"Volume-preserving algorithm for secular relativistic dynamics of charged particles","volume":"22","author":"Zhang","year":"2015","journal-title":"Phys. Plasmas"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.jcp.2015.10.032","article-title":"Higher order volume-preserving schemes for charged particle dynamics","volume":"305","author":"He","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s10543-018-0713-1","article-title":"Energy behaviour of the Boris method for charged-particle dynamics","volume":"301","author":"Hairer","year":"2018","journal-title":"BIT Numer. Math."},{"key":"ref_15","unstructured":"Broucke, R.A. (1968). Periodic Orbits in the Restricted Three Body Problem with Earth-Moon Masses, Jet Propulsion Laboratory. Technical Report."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Murray, C.D., and Dermott, S.F. (1999). Solar System Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139174817"},{"key":"ref_17","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer. [2nd ed.]."},{"key":"ref_18","first-page":"31","article-title":"Formal energy of a symplectic scheme for hamiltonian systems and its applications (I)","volume":"27","author":"Tang","year":"1994","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1088\/0004-6256\/148\/6\/116","article-title":"A Study on Periodic Solutions for the Circular Restricted Three-body Problem","volume":"148","author":"Gao","year":"2014","journal-title":"Astron. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s10569-017-9766-8","article-title":"On the period of the periodic orbits of the restricted three body problem","volume":"129","author":"Perdomo","year":"2017","journal-title":"Celest. Mech. Dyn. Astron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"101319","DOI":"10.1016\/j.newast.2019.101319","article-title":"Periodic solution of the nonlinear Sitnikov restricted three-body problem","volume":"75","author":"Abouelmagd","year":"2020","journal-title":"New Astron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1007\/s10543-019-00785-0","article-title":"New high order symplectic integrators via generating functions with its application in many-body problem","volume":"129","author":"Tu","year":"2020","journal-title":"BIT Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1080\/10556780500140664","article-title":"Derivation of symmetric composition constants for symmetric integrators","volume":"20","author":"Sofroniou","year":"2005","journal-title":"Optim. Methods Softw."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1769\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:14:58Z","timestamp":1760141698000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1769"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,25]]},"references-count":23,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091769"],"URL":"https:\/\/doi.org\/10.3390\/sym14091769","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,25]]}}}