Jacobi coordinates

In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions,^[3] and in celestial mechanics.^[4] An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees.^[5] In words, the algorithm is described as follows:^[5]

Let m_j and m_k be the masses of two bodies that are replaced by a new body of virtual mass M = m_j + m_k. The position coordinates x_j and x_k are replaced by their relative position r_jk = x_j − x_k and by the vector to their center of mass R_jk = (m_j q_j + m_kq_k)/(m_j + m_k). The node in the binary tree corresponding to the virtual body has m_j as its right child and m_k as its left child. The order of children indicates the relative coordinate points from x_k to x_j. Repeat the above step for N − 1 bodies, that is, the N − 2 original bodies plus the new virtual body.

For the N-body problem the result is:^[2]

{\boldsymbol {r}}_{j}={\frac {1}{m_{0j}}}\sum _{k=1}^{j}m_{k}{\boldsymbol {x}}_{k}\ -\ {\boldsymbol {x}}_{j+1}\ ,\quad j\in \{1,2,\dots ,N-1\}

{\boldsymbol {r}}_{N}={\frac {1}{m_{0N}}}\sum _{k=1}^{N}m_{k}{\boldsymbol {x}}_{k}\ ,

with

m_{0j}=\sum _{k=1}^{j}\ m_{k}\ .

The vector ${\boldsymbol {r}}_{N}$ is the center of mass of all the bodies and ${\boldsymbol {r}}_{1}$ is the relative coordinate between the particles 1 and 2:

The result one is left with is thus a system of N-1 translationally invariant coordinates ${\boldsymbol {r}}_{1},\dots ,{\boldsymbol {r}}_{N-1}$ and a center of mass coordinate ${\boldsymbol {r}}_{N}$ , from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian equal to $1$ .

If one is interested in evaluating a free energy operator in these coordinates, one obtains

H_{0}=-\sum _{j=1}^{N}{\frac {\hbar ^{2}}{2m_{j}}}\,\partial _{{\boldsymbol {x}}_{j}}^{2}=-{\frac {\hbar ^{2}}{2m_{0N}}}\,\partial _{{\boldsymbol {r}}_{N}}^{2}\!-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N-1}\!\left({\frac {1}{m_{j+1}}}+{\frac {1}{m_{0j}}}\right)\partial _{{\boldsymbol {r}}_{j}}^{2}

In the calculations can be useful the following identity

\sum _{k=j+1}^{N}{\frac {m_{k}}{m_{0k}m_{0k-1}}}={\frac {1}{m_{0j}}}-{\frac {1}{m_{0N}}}

References

^ David Betounes (2001). Differential Equations. Springer. p. 58; Figure 2.15. ISBN 0-387-95140-7.
^ ^a ^b Patrick Cornille (2003). "Partition of forces using Jacobi coordinates". Advanced electromagnetism and vacuum physics. World Scientific. p. 102. ISBN 981-238-367-0.
^ John Z. H. Zhang (1999). Theory and application of quantum molecular dynamics. World Scientific. p. 104. ISBN 981-02-3388-4.
^ For example, see Edward Belbruno (2004). Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press. p. 9. ISBN 0-691-09480-2.
^ ^a ^b Hildeberto Cabral, Florin Diacu (2002). "Appendix A: Canonical transformations to Jacobi coordinates". Classical and celestial mechanics. Princeton University Press. p. 230. ISBN 0-691-05022-8.

Read other articles:

Wrestling at the 1998 Asian Games – Men's Greco-Roman 58 kg

Men's Greco-Roman 58 kg at the 1998 Asian GamesVenueThammasat Gymnasium 1Dates13–14 December 1998Competitors10 from 10 nationsMedalists Kim In-sub South Korea Asliddin Khudoyberdiev Uzbekistan Sheng Zetian China← 19942002 → Wrestling at the1998 Asian GamesFreestyleGreco-Roman54 kg54 kg58 kg58 kg63 kg63 kg69 kg69 kg76 kg76 kg85 kg85 kg97 kg97 kg130 kg130 kgvte Main article: Wrestling at the 1998 As...

Nieuw Amsterdam

Peta kota asli Nieuw Amsterdam, yang disebut Castello Plan, dari 1660 (sudut kiri bawah kira-kira selatan, sedangkan sudut kanan atas kira-kira utara) Bangunan peninggalan Belanda dari awal abad ke-20 di Pearl Street di daerah Manhattan sebelah bawah yang mengingatkan kehadiran awal orang Belanda di kota ini. Bangunan asli berarsitektur New Amsterdam abad ke-17 telah punah dan hanya meninggalkan jejak-jejak arkeologi. Nieuw Amsterdam (Inggris: New Amsterdamcode: en is deprecated , berarti Ams...

Parlamento del Regno di Sicilia

Parlamento del Regno di SiciliaPalazzo Reale o dei Normanni StatoRegno di Sicilia TipoMonocamerale/bicamerale Istituito1130 Soppresso1849 SuccessoreAssemblea Regionale Siciliana SedePalermo IndirizzoPalazzo Reale, Piazza del Parlamento Modifica dati su Wikidata · Manuale «Che benedica Iddio ed ispiri i voti del Parlamento indirizzati a questo santo fine; ch'Egli riguardi benigno la terra di Sicilia, e la congiunga ai grandi destini della nazione italiana, libera, indipendente ed unita....

Tour des Alpes 2017

Tour des Alpes 2017 GénéralitésCourse41e Tour des AlpesCompétitionUCI Europe Tour 2017 2.HCÉtapes5Dates17 – 21 avril 2017Distance787,7 kmPays Italie AutricheLieu de départKufsteinLieu d'arrivéeTrenteÉquipes18Partants139Arrivants109Vitesse moyenne37,821 km/hSite officielSite officielRésultatsVainqueur Geraint Thomas (Sky)Deuxième Thibaut Pinot (FDJ)Troisième Domenico Pozzovivo (AG2R La Mondiale)Meilleur grimpeur Alexander Foliforov (Gazprom-RusVelo)Meilleur sprinteur Pascal Ackerm...

Yeprem Khan

Armenian revolutionary (1868–1912) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Yeprem Khan – news · newspapers · books · scholar · JSTOR (April 2024) (Learn how and when to remove this message) Yeprem KhanBirth nameYeprem DavidianNickname(s)KhanBorn1868Barsum, Elisabethpol Governorate, Russian EmpireDi...

Kiss (gruppo musicale)

KissI Kiss in concerto alla O2 Arena nel 2019 Paese d'origine Stati Uniti GenereHard rock[1][2][3][4]Pop metal[5][6] Periodo di attività musicale1973 – 2023 EtichettaCasablanca (1973–1982)Mercury (1982–2003)Sanctuary (2003–2008)Roadrunner-Universal (2009–2023) Album pubblicati44 Studio20 Live7 Raccolte17 Logo ufficiale Sito ufficiale Modifica dati su Wikidata · Manuale I Kiss (spesso reso graficament...

Mancera Island

This article uses bare URLs, which are uninformative and vulnerable to link rot. Please consider converting them to full citations to ensure the article remains verifiable and maintains a consistent citation style. Several templates and tools are available to assist in formatting, such as reFill (documentation) and Citation bot (documentation). (September 2022) (Learn how and when to remove this message) Mancera IslandNative name: Isla ManceraMancera Island, view from Niebla, ChileRegional ma...

Западный округ (Краснодар)

У этого термина существуют и другие значения, см. Западный округ. Западный внутригородской округ город Краснодар Дата основания 1936 год Дата упразднения 1994 Прежние имена Кагановичский, Ленинский районы Микрорайоны Дубинка, Черёмушки, Покровка Площадь 22[1] км² Насе...

Seulimeum, Aceh Besar

SeulimeumKecamatanNegara IndonesiaProvinsiAcehKabupatenAceh BesarPemerintahan • CamatRahmad Sentosa, S.sos, MAPPopulasi • Total- jiwaKode Kemendagri11.06.04 Kode BPS1108040 Luas- km²Desa/kelurahan41/2 Rumah Hendrikus Colijn di Seulimeum (sekitar tahun 1900) Seulimeum adalah sebuah kecamatan di Kabupaten Aceh Besar, Provinsi Aceh, Indonesia. Daftar Desa/Kelurahan per Mukim Mukim Lamkabeu Ayon Batee Lhee Bayu Mangeu Meunasah Tunong Mukim Lampanah Beureunut Lampanah...

Russell Kirk

Conservative American political theorist and writer (1918–1994) Russell KirkKirk in 1962BornRussell Amos Kirk(1918-10-19)October 19, 1918Plymouth, Michigan, U.S.DiedApril 29, 1994(1994-04-29) (aged 75)Mecosta, Michigan, U.S.EducationMichigan State University (BA)Duke University (MA)University of St Andrews (DLitt)Notable workThe Conservative MindRoots of American OrderSpouse Annette Courtemanche (m. 1963)Children4Era20th-century philosophyRegionWe...

2013–14 Texas Longhorns men's basketball team

American college basketball season 2013–14 Texas Longhorns men's basketballNCAA tournament, Round of 32ConferenceBig 12Record24–11 (11–7 Big 12)Head coachRick Barnes (16th season)Assistant coaches Rob Lanier Russell Springmann Chris Ogden Home arenaFrank Erwin CenterSeasons← 2012–132014–15 → 2013–14 Big 12 men's basketball standings vte Conf Overall Team W L PCT W L PCT No. 10 Kansas 14 – 4 .778 25 – 10...

Kongres Bahasa Jawa

Kongres Bahasa Jawa IV Kongres Bahasa Jawa adalah kegiatan rutin 5 tahunan yang membahas mengenai Bahasa dan Budaya Jawa. Acara ini diselenggarakan oleh 3 Provinsi di Pulau Jawa secara bergantian yakni Jawa Tengah, Jawa Timur dan Daerah Istimewa Jogjakarta. Dihadiri oleh para praktisi budaya Jawa, Birokrat, Akademisi dan masyarakat pencinta Budaya Jawa serta undangan khusus baik dari dalam maupun luar negeri. Kongres Bahasa Jawa I diadakan di Kota Semarang pada 15-20 Juli 1991 dan hingga saat...

مجلس خبراء القيادة

35°41′16.82″N 51°23′58.72″E / 35.6880056°N 51.3996444°E / 35.6880056; 51.3996444 مجلس خبراء القيادة مجلس خبرگان رهبري النوع التأسيس 1982 النوع نصيحة البلد إيران القيادة رئيس أحمد جنتيمنذ 24 مايو 2016 Deputy Speaker محمد يزديمنذ 10 مارس 2015 الأعضاء 86 الجماعات السياسية جمعية علماء الدين المجاهدين�...

Introduction to Kant's Anthropology

2008 essay Introduction to Kant's Anthropology AuthorMichel FoucaultLanguageFrenchSubjectAnthropology from a Pragmatic Point of ViewPublisherSemiotext(e)Publication date2008Publication placeFranceMedia typePrint (Paperback)Pages124ISBN978-1-58435-054-5OCLC153578160 Introduction to Kant's Anthropology (French: Introduction à l'Anthropologie) is an introductory essay to Michel Foucault's translation of Immanuel Kant's 1798 book Anthropology from a Pragmatic Point of View[1] — a ...

Pulau Perak

Pulau PerakNegaraIndonesiaGugus kepulauanKepulauan SeribuProvinsiDKI JakartaKabupatenKepulauan SeribuLuas- km²Populasi- Pulau Perak adalah sebuah pulau yang terletak di Kepulauan Seribu di Daerah Khusus Ibukota Jakarta, Indonesia. Lihat pula Kabupaten Administratif Kepulauan Seribu Kepulauan Seribu Pranala luar Situs resmi Kabupaten Administratif Kepulauan Seribu Diarsipkan 2017-02-22 di Wayback Machine. lbsPulau di Kepulauan Seribu Pulau Air Besar Pulau Air Kecil Pulau Ayer Pulau Belan...

Kolkata Port Assembly constituency

Constituency of the West Bengal Legislative Assembly, in India Kolkata PortConstituency No. 158 for the West Bengal Legislative AssemblyInteractive Map Outlining Kolkata Port Assembly ConstituencyConstituency detailsCountryIndiaRegionEast IndiaStateWest BengalDistrictKolkataLS constituencyKolkata DakshinEstablished2011Total electors235,854ReservationNoneMember of Legislative Assembly17th West Bengal Legislative AssemblyIncumbent Firhad Hakim PartyAll India Trinamool CongressElected year2021 K...

Richard McNally

Professor of psychology Richard McNallyBorn (1954-04-17) April 17, 1954 (age 70)Detroit, Michigan, U.S.Alma materWayne State University (BS)University of Illinois at Chicago (PhD)Known forResearch into anxiety disordersScientific careerFieldsPsychology Pathology PsychopathologyInstitutionsHarvard University Richard McNally (born April 17, 1954) is an American psychologist and director of clinical training at Harvard University's department of psychology. As a clinical psycholog...

Mława railway station

Railway station in Mława, Poland MławaRail StationGeneral informationLocationMława, Mława, Masovian PolandCoordinates53°07′29″N 20°21′09″E / 53.1248374°N 20.3525378°E / 53.1248374; 20.3525378Owned byPolskie Koleje Państwowe S.A.Services Preceding station PKP Intercity Following station Ciechanówtowards Łódź Fabryczna IC Działdowotowards Kołobrzeg Ciechanówtowards Bielsko-Biała Główna or Racibórz Działdowotowards Olsztyn Główny Ciecha...

Leduc No. 1

Crude oil discovery in Alberta, Canada in Alberta, CanadaLeduc No. 1Leduc #1 wellTypecrude oil discovery in Alberta, CanadaLocationAlberta, CanadaNearest cityLeduc CountyEstablished1990Built1946Governing bodyParks CanadaWebsitehttp://www.canadianenergymuseum.ca/ National Historic Site of Canada Leduc No. 1 was a major crude oil discovery made near Leduc, Alberta, Canada, on February 13, 1947. It provided the geological key to Alberta's most prolific conventional oil reserves and resulted in ...

Sát thủ John Wick: Phần 4

John Wick: Chapter 4 Áp phích teaser của phimĐạo diễnChad StahelskiTác giảShay HattenMichael FinchSản xuấtChad StahelskiBasil IwanykErica LeeDiễn viênKeanu ReevesLaurence FishburneLance ReddickChân Tử ĐanRina SawayamaShamier AndersonBill SkarsgårdScott AdkinsHiroyuki SanadaClancy BrownIan McShaneQuay phimDan LaustsenDựng phimEvan SchiffÂm nhạcTyler BatesJoel J. RichardHãng sản xuấtSummit EntertainmentThunder Road Films87North Productions Phát hànhLionsgateC...