Uch jism muammosi: Versiyalar orasidagi farq

Kontent oʻchirildi Kontent qoʻshildi

Inline

12-Avgust 2023, 15:47 dagi koʻrinishi

Skalenli uchburchakning uchlarida joylashgan va boshlang'ich tezligi nolga teng bo'lgan uchta bir xil jismning taxminiy traektoriyalari. Ko'rinib turibdiki, massa markazi impulsning saqlanish qonuniga muvofiq, o'z joyida qoladi.

Fizika va klassik mexanikada uch jism muammosi uch nuqtali massalarning boshlang'ich pozitsiyalari va tezliklarini (yoki momentlarini ) olish va ularning keyingi harakatini Nyutonning harakat qonunlari va Nyutonning butun dunyo tortishish qonuniga muvofiq hal qilish masalasidir^[1]. Uch tana muammosi <span about="#mwt12" class="texhtml mvar" data-cx="[{"adapted":true,"partial":false,"targetExists":true,"mandatoryTargetParams":[],"optionalTargetParams":[]}]" data-mw="{"parts":[{"template":{"target":{"wt":"Mvar","href":"./Andoza:Mvar"},"params":{"1":{"wt":"n"}},"i":0}}]}" data-ve-no-generated-contents="true" id="mwFg" style="font-style:italic;" typeof="mw:Transclusion">n -tana muammosining alohida holatidir. Ikki tanali masalalardan farqli o'laroq, umumiy yopiq shakldagi yechim mavjud emas ^[1], chunki natijada paydo bo'lgan dinamik tizim ko'pchilik boshlang'ich sharoitlar uchun xaotikdir va odatda raqamli usullar talab qilinadi.

Tarixiy jihatdan, kengaytirilgan o'rganish uchun birinchi maxsus uch tana muammosi Oy, Yer va Quyoshni o'z ichiga olgan muammo edi^[2]. Kengaytirilgan zamonaviy ma'noda uch jismli muammo klassik mexanika yoki kvant mexanikasidagi uchta zarrachaning harakatini modellashtiradigan har qanday muammodir.

Matematik tavsifi

Uch jismli masalaning matematik bayoni vektor pozitsiyalari uchun Nyuton harakat tenglamalari nuqtai nazaridan berilishi mumkin. $\mathbf {r_{i}} =(x_{i},y_{i},z_{i})$ massalari bilan gravitatsiyaviy ta'sir qiluvchi uchta jism $m_{i}$ :

{\begin{aligned}{\ddot {\mathbf {r} }}_{\mathbf {1} }&=-Gm_{2}{\frac {\mathbf {r_{1}} -\mathbf {r_{2}} }{|\mathbf {r_{1}} -\mathbf {r_{2}} |^{3}}}-Gm_{3}{\frac {\mathbf {r_{1}} -\mathbf {r_{3}} }{|\mathbf {r_{1}} -\mathbf {r_{3}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {2} }&=-Gm_{3}{\frac {\mathbf {r_{2}} -\mathbf {r_{3}} }{|\mathbf {r_{2}} -\mathbf {r_{3}} |^{3}}}-Gm_{1}{\frac {\mathbf {r_{2}} -\mathbf {r_{1}} }{|\mathbf {r_{2}} -\mathbf {r_{1}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {3} }&=-Gm_{1}{\frac {\mathbf {r_{3}} -\mathbf {r_{1}} }{|\mathbf {r_{3}} -\mathbf {r_{1}} |^{3}}}-Gm_{2}{\frac {\mathbf {r_{3}} -\mathbf {r_{2}} }{|\mathbf {r_{3}} -\mathbf {r_{2}} |^{3}}}.\end{aligned}}

bu yerda $G$ tortishish doimiysi hisoblanadi^[3] ^[4]. Bu to'qqizta ikkinchi darajali differentsial tenglamalar to'plamidir. Muammoni Gamilton formalizmida ham ekvivalent tarzda ifodalash mumkin, bu holda u pozitsiyalarning har bir komponenti uchun bittadan bo'lgan 18 ta birinchi tartibli differensial tenglamalar to'plami bilan tavsiflanadi. $\mathbf {r_{i}}$ va momenta $\mathbf {p_{i}}$ :

{\frac {d\mathbf {r_{i}} }{dt}}={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p_{i}} }},\qquad {\frac {d\mathbf {p_{i}} }{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {r_{i}} }},

bu yerda ${\mathcal {H}}$ Gamiltonian :

{\mathcal {H}}=-{\frac {Gm_{1}m_{2}}{|\mathbf {r_{1}} -\mathbf {r_{2}} |}}-{\frac {Gm_{2}m_{3}}{|\mathbf {r_{3}} -\mathbf {r_{2}} |}}-{\frac {Gm_{3}m_{1}}{|\mathbf {r_{3}} -\mathbf {r_{1}} |}}+{\frac {\mathbf {p_{1}} ^{2}}{2m_{1}}}+{\frac {\mathbf {p_{2}} ^{2}}{2m_{2}}}+{\frac {\mathbf {p_{3}} ^{2}}{2m_{3}}}.

Ushbu holatda ${\mathcal {H}}$ oddiygina tizimning umumiy energiyasi, tortishish va kinetik.

Cheklangan uch tana muammosi

Cheklangan u^{[ iqtibos kerak ]}ch jismli muammoda arzimas massali jism ("planetoid") ikkita massiv jismning ta'siri ostida harakat qiladi. Arzimas massaga ega bo'lgan holda, planetoidning ikkita massiv jismga ta'sir qiladigan kuchini e'tiborsiz qoldirish mumkin va tizimni tahlil qilish va shuning uchun ikki jismli harakat nuqtai nazaridan tasvirlash mumkin^[3]. Odatda bu ikki jismli harakat massa markazi atrofida aylana orbitalardan iborat deb qabul qilinadi va planetoid dumaloq orbitalar bilan belgilangan tekislikda harakat qiladi deb taxmin qilinadi.

Cheklangan uch tanali muammoni nazariy jihatdan tahlil qilish to'liq muammodan ko'ra osonroqdir. Bu amaliy qiziqish uyg'otadi, chunki u ko'plab real muammolarni aniq tasvirlaydi, eng muhim misol Yer-Oy-Quyosh tizimidir. Shu sabablarga ko'ra, u uch tana muammosining tarixiy rivojlanishida muhim rol o'ynadi.

Matematik jihatdan muammo quyidagicha ifodalanadi. Mayli $m_{1,2}$ (tekislik) koordinatali ikkita massiv jismning massalari bo'lsin $(x_{1},y_{1})$ va $(x_{2},y_{2})$ , va ruxsat bering $(x,y)$ planetoidning koordinatalari bo'lsin. Oddiylik uchun ikkita massiv jism orasidagi masofa va tortishish doimiysi ikkalasi $1$ ga teng bo'ladigan birliklarni tanlanadi. Keyin, planetoidning harakati bilan beriladi

{\begin{aligned}{\frac {d^{2}x}{dt^{2}}}=-m_{1}{\frac {x-x_{1}}{r_{1}^{3}}}-m_{2}{\frac {x-x_{2}}{r_{2}^{3}}},\\{\frac {d^{2}y}{dt^{2}}}=-m_{1}{\frac {y-y_{1}}{r_{1}^{3}}}-m_{2}{\frac {y-y_{2}}{r_{2}^{3}}},\end{aligned}}

bu yerda $r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}$ . Ushbu shaklda harakat tenglamalari koordinatalar orqali aniq vaqtga bog'liqlikni olib boradi $x_{i}(t),y_{i}(t)$ . Biroq, bu vaqtga bog'liqlikni aylanadigan mos yozuvlar tizimiga aylantirish orqali olib tashlash mumkin, bu esa har qanday keyingi tahlilni soddalashtiradi.

Yechimlari

Umumiy yechimi

Gravitatsiyaviy o'zaro ta'sir qiluvchi 3 ta jismdan iborat tizim xaotik bo'lsa-da, elastik tarzda o'zaro ta'sir qiluvchi 3 ta jismdan iborat sistema xaotik emas.

Uch tanali masalaning umumiy yopiq shakldagi yechimi yo‘q ^[1], ya’ni chekli sonli standart matematik amallar bilan ifodalanadigan umumiy yechim yo‘q. Bundan tashqari, uchta jismning harakati odatda takrorlanmaydi, maxsus holatlar bundan mustasno^[5].

Biroq, 1912 yilda Finlyandiya matematigi Karl Fritiof Sundman $t 1/3$ kuchlari bo'yicha darajalar qatori ko'rinishidagi uch tanali muammoning analitik yechimi mavjudligini isbotladi^[6]. Bu qator nol burchak momentumiga mos keladigan boshlang'ich shartlar bundan mustasno, barcha haqiqiy $t$ uchun birlashadi. Amalda, oxirgi cheklov ahamiyatsiz, chunki nol burchak momentiga ega bo'lgan dastlabki shartlar kamdan-kam uchraydi, Lebesg o'lchovi nolga teng.

Bu natijani isbotlashda muhim masala shundaki, bu qator uchun yaqinlashish radiusi eng yaqin singulyarlikgacha bo'lgan masofa bilan belgilanadi. Shuning uchun uch tanali masalalarning mumkin bo'lgan yagonaligini o'rganish kerak. Quyida qisqacha muhokama qilinganidek, uch jismli muammoning yagona o'ziga xosligi ikkilik to'qnashuvlar (bir lahzada ikkita zarracha o'rtasidagi to'qnashuv) va uch marta to'qnashuv (bir lahzada uchta zarracha o'rtasidagi to'qnashuv).

To'qnashuvlar ikkilik yoki uchlik (aslida har qanday raqam) bo'lsin, bir oz ehtimoldan yiroq emas, chunki ular nol o'lchovining boshlang'ich shartlari to'plamiga mos kelishi ko'rsatilgan. Biroq, mos keladigan yechim uchun to'qnashuvlarni oldini olish uchun dastlabki holatga qo'yiladigan hech qanday mezon yo'q. Shunday qilib, Sundmanning strategiyasi quyidagi bosqichlardan iborat edi:

Regulyatsiya deb nomlanuvchi jarayonda ikkilik to'qnashuvdan tashqari yechimni tahlil qilishni davom ettirish uchun o'zgaruvchilarning tegishli o'zgarishidan foydalanish.
Uch marta to'qnashuvlar faqat burchak momenti $L$ yo'qolganda sodir bo'lishini isbotlash. Dastlabki ma'lumotlarni $L \neq 0$ ga cheklab, u uch tanali muammo uchun o'zgartirilgan tenglamalardan barcha haqiqiy yagonaliklarni olib tashladi.
Agar $L \neq 0$ bo'lsa, u holda nafaqat uch karra to'qnashuv bo'lishi mumkin emas, balki tizim uch karra to'qnashuvdan qat'iy chegaralanganligini ko'rsatadi. Bu shuni anglatadiki, differensial tenglamalar uchun Koshining mavjudlik teoremasidan foydalanib, haqiqiy o'q ( Kovalevskaya soyalari) atrofida joylashgan kompleks tekislikdagi chiziqda ( $L$ qiymatiga bog'liq holda) murakkab birliklar mavjud emas.
Ushbu chiziqni birlik diskiga tushiradigan konformal transformatsiyani toping. Misol uchun, agar $s = t 1/3$ (regulyatsiyadan keyingi yangi o'zgaruvchi) va agar $Andoza:Abs \leq β$ , keyin bu xarita tomonidan berilgan $\sigma ={\frac {e^{\frac {\pi s}{2\beta }}-1}{e^{\frac {\pi s}{2\beta }}+1}}.$

Bu Sundman teoremasining isbotini tugatadi.

Biroq, mos keladigan seriyalar juda sekin yaqinlashadi. Ya'ni, ma'noli aniqlik qiymatini olish uchun juda ko'p atamalar talab qilinadi, bu yechim juda kam amaliy ahamiyatga ega. Darhaqiqat, 1930 yilda Devid Beloriski hisoblab chiqdiki, agar Sundman seriyasi astronomik kuzatishlar uchun ishlatilsa, hisob-kitoblar kamida 10 ^{7006800000000000000♠8000000} atamani o'z ichiga oladi^[7].

Maxsus holatlar uchun echimlar

1767 yilda Leonhard Eyler davriy eritmalarning uchta turkumini topdi, ularda uchta massa har bir lahzada kollinear bo'ladi. Eylerning uch tana muammosiga qarang.

1772 yilda Lagrange yechimlar oilasini topdi, unda uchta massa har bir lahzada teng qirrali uchburchak hosil qiladi. Eylerning kollinear yechimlari bilan birgalikda bu yechimlar uch jismli muammoning markaziy konfiguratsiyasini tashkil qiladi. Ushbu yechimlar har qanday massa nisbati uchun amal qiladi va massalar Kepler ellipslari bo'ylab harakatlanadi. Ushbu to'rtta oila aniq analitik formulalar mavjud bo'lgan yagona ma'lum echimlardir. Dumaloq cheklangan uch tanali muammoning maxsus holatida, birlamchi nuqtalar bilan aylanadigan ramkada ko'rib chiqilgan bu yechimlar L₁, L₂, L₃, L₄ va L₅ deb ataladigan nuqtalarga aylanadi va Lagranj nuqtalari deb ataladi., L₄ va L₅ bilan Lagranj yechimining simmetrik misollari.

1892-1899 yillarda umumlashtirilgan ishida Anri Puankare cheklangan uch tanali muammoning cheksiz ko'p davriy yechimlari mavjudligini va ushbu echimlarni umumiy uch tanali muammoga aylantirish usullarini aniqladi.

1893 yilda Meissel Pifagorning uch jismli muammosi deb ataladigan narsani aytdi: 3:4:5 nisbatda uchta massa 3:4:5 o'lchamdagi to'g'ri burchakli uchburchakning uchlarida tinch holatda joylashgan. Burrau ^[8] bu muammoni 1913 yilda qo'shimcha ravishda o'rganib chiqdi. 1967 yilda Viktor Szebeheli va C. Frederik Peters raqamli integratsiyadan foydalangan holda ushbu muammoni hal qilishning yakuniy yo'llarini o'rnatdilar va shu bilan birga yaqin davriy yechim topdilar^[9].

1970-yillarda Mishel Xenon va Rojer A. Broukning har biri bir xil yechimlar oilasining bir qismini tashkil etuvchi yechimlar to‘plamini topdilar: Brouk-Xenon-Xadjidemetriu oilasi. Ushbu oilada uchta ob'ektning barchasi bir xil massaga ega va retrograd va to'g'ridan-to'g'ri shakllarni namoyish qilishi mumkin. Broukning ba'zi yechimlarida ikkita jism bir xil yo'ldan boradi^[11].

1993-yilda Santa-Fe institutida fizik Kris Mur tomonidan 1993-yilda uchta teng massali sakkizta shakl atrofida harakatlanadigan nolga teng burchak momentum eritmasi aniqlangan^[12]. Uning rasmiy mavjudligi keyinchalik 2000 yilda matematiklar Alen Chenciner va Richard Montgomery tomonidan isbotlangan^[13] ^[14] Eritma massa va orbital parametrlarning kichik buzilishlari uchun barqaror ekanligi ko'rsatilgan, bu esa bunday orbitalarni fizik koinotda kuzatish imkonini beradi. Biroq, barqarorlik doirasi kichik bo'lganligi sababli, bu hodisaning mumkin emasligi ta'kidlangan. Masalan, ikkilik-ikkilik tarqalish hodisasining ehtimoli natijada raqam-8 orbitasi foizning kichik bir qismi sifatida baholandi^[15].

2013-yilda Belgraddagi Fizika instituti fiziklari Milovan Šuvakov va Velko Dmitrashinovich teng massali nol burchakli momentli uch jism muammosi yechimlarining 13 ta yangi oilasini topdilar^[5] ^[11].

2015-yilda fizik Ana Hudomal teng massali nol burchakli impulsli uch jism muammosi uchun 14 ta yangi yechimlar turkumini topdi^[16].

2017 yilda tadqiqotchilar Xiaoming Li va Shijun Liao teng massali nol burchakli impulsli uch jism muammosining 669 ta yangi davriy orbitalarini topdilar^[17]. Buning ortidan 2018 yilda teng bo'lmagan massalarning nol burchakli impuls tizimi uchun qo'shimcha 1223 ta yangi yechim paydo bo'ldi^[18].

2018 yilda Li va Liao teng bo'lmagan massali "erkin tushish" uchta tana muammosiga 234 ta yechim haqida xabar berishdi^[19]. Uchta jism muammosining erkin yiqilish formulasi barcha uch jism dam olishdan boshlanadi. Shu sababli, erkin tushish konfiguratsiyasidagi massalar yopiq "halqa" bo'ylab aylanmaydi, balki ochiq "iz" bo'ylab oldinga va orqaga harakat qiladi.

Raqamli yondashuvlar

Kompyuterdan foydalanib, muammoni raqamli integratsiya yordamida o'zboshimchalik bilan yuqori aniqlik bilan hal qilish mumkin, ammo yuqori aniqlik uchun katta miqdordagi CPU vaqtini talab qiladi. Elektromagnit va tortishish o'zaro ta'sirini o'z ichiga olgan uch tana muammosini (va kengaytmasi, n-tana muammosini) raqamli hal qiladigan va maxsus nisbiylik kabi zamonaviy fizika nazariyalarini o'z ichiga olgan kompyuter dasturlarini yaratishga urinishlar mavjud^[20]. Bundan tashqari, tasodifiy yurishlar nazariyasidan foydalanib, turli natijalarning taxminiy ehtimolini hisoblash mumkin^[21] ^[22].

Tarixi

An'anaviy ma'noda uchta jismning tortishish muammosi 1687 yilda Isaak Nyuton o'zining "Philosophiæ Naturalis Principia Mathematica" asarini nashr etganida, Nyuton uzoq muddatli barqarorlik mumkinmi yoki yo'qligini aniqlashga harakat qilgan paytda, xususan, bizning Yerimiz, Oy tizimi bilan bog'liq., va Quyosh. U Uyg'onish davrining yirik astronomlari Nikolay Kopernik, Tyxo Brahe va Yoxannes Kepler qo'l ostida gravitatsiyaviy uch jism muammosining boshlanishiga rahbarlik qilgan^[23]. Prinsipiyaning 1-kitobining 66-taklifida va uning 22 ta xulosasida Nyuton uchta massiv jismning bir-birini bezovta qiluvchi tortishish kuchiga bog'liq bo'lgan harakati muammosini aniqlash va o'rganishda birinchi qadamlarni qo'ydi. 3-kitobning 25 dan 35 gacha bo'lgan takliflarida Nyuton, shuningdek, 66-taklif natijalarini Oy nazariyasiga, Yer va Quyoshning tortishish ta'siri ostida Oyning harakatiga tatbiq etishda birinchi qadamlarni qo'ydi^[24]. Keyinchalik bu muammo boshqa sayyoralarning Yer va Quyosh bilan o'zaro ta'sirida ham qo'llanildi^[23].

Jismoniy muammoni birinchi bo'lib Amerigo Vespuchchi, keyin esa Galileo Galiley, shuningdek, Simon Stevin ko'rib chiqdi, ammo ular nima hissa qo'shganini tushunmadilar. Galiley barcha jismlarning yiqilish tezligi bir xil va bir xil tarzda o'zgarishini aniqlagan bo'lsa-da, uni sayyoralar harakatlariga qo'llamagan^[23]. 1499 yilda Vespuchchi Braziliyadagi o'rnini aniqlash uchun Oyning holati haqidagi bilimlaridan foydalangan^[25]. Bu 1720-yillarda texnik ahamiyatga ega bo'ldi, chunki aniq yechim navigatsiyada qo'llanilishi mumkin edi, ayniqsa dengizda uzunlikni aniqlash uchun, amalda Jon Xarrisonning dengiz xronometrini ixtiro qilish orqali hal qilindi. Biroq, Oy nazariyasining aniqligi Quyosh va sayyoralarning Oyning Yer atrofidagi harakatiga bezovta qiluvchi ta'siri tufayli past edi.

Uzoq muddatli raqobatni rivojlantirgan Jan le Rond d'Alember va Aleksis Kler muammoni qandaydir umumiylik darajasida tahlil qilishga harakat qildilar; ular 1747 yilda Fanlar Akademiyasi Royale des Fansga o'zlarining raqobatbardosh birinchi tahlillarini taqdim etishdi^[26]. 1740-yillarda Parijda olib borilgan tadqiqotlari tufayli "uch tana muammosi" nomi paydo bo'ldi ( fransuzcha: Problème des trois Corps) keng foydalanila boshlandi. Jan le Rond d'Alember tomonidan 1761 yilda nashr etilgan hisobda bu nom birinchi marta 1747 yilda ishlatilganligini ko'rsatadi ^[27].

19-asrning oxiridan 20-asrning boshlariga qadar, PF Bedaque, H.-V.ni taklif qilgan olimlar tomonidan qisqa masofali jozibali ikki tanali kuchlardan foydalangan holda uch tana muammosini hal qilish yondashuvi ishlab chiqilgan. Hammer va U. van Kolk qisqa masofali uch tana muammosini qayta normallashtirish g'oyasini ishlab chiqdilar, bu olimlarga 21-asr boshidagi renormalizatsiya guruhining chegara tsiklining noyob namunasini taqdim etdi^[28]. Jorj Uilyam Xill 19-asr oxirida Venera va Merkuriyning harakatini qo'llash orqali cheklangan muammo ustida ishladi^[29].

20-asrning boshlarida Karl Sundman vaqtning barcha qiymatlari uchun amal qiladigan masalaga funktsiya nazariy isbotini taqdim etish orqali muammoga matematik va tizimli yondashdi. Bu olimlar birinchi marta nazariy jihatdan uch tana muammosini hal qilishdi. Biroq, bu tizimning sifatli yechimi yetarli boʻlmagani va olimlar uni amalda qoʻllashlari juda sekin boʻlganligi sababli, bu yechim haligacha baʼzi masalalarni yechilmay qoldi^[30]. O'tgan asrning 70-yillarida V. Efimov tomonidan ikki tanali kuchlarning uch tanaga ta'siri aniqlangan va u Efimov effekti deb nomlangan^[31].

2017-yilda Shijun Liao va Xiaoming Li tartibsiz tizimlar uchun raqamli simulyatsiyaning yangi strategiyasini milliy superkompyuterdan foydalangan holda toza raqamli simulyatsiya (CNS) yordamida uch jismli tizimning davriy yechimlarining 695 oilasini teng massasini muvaffaqiyatli qo'lga kiritdilar.^[32].

2019 yilda Breen va boshqalar. raqamli integrator yordamida o'qitilgan uch tanali muammo uchun tezkor neyron tarmoq hal qiluvchi e'lon qildi^[33].

Uch organ bilan bog'liq boshqa muammolar

"Uch jism muammosi" atamasi ba'zan umumiy ma'noda uchta jismning o'zaro ta'sirini o'z ichiga olgan har qanday jismoniy muammoga nisbatan qo'llaniladi.

Klassik va kvant mexanikasida, teskari kvadrat kuchdan tashqari, aniq analitik uch tanali yechimlarga olib keladigan noan'anaviy o'zaro ta'sir qonunlari mavjud. Bunday modellardan biri garmonik tortishish va itaruvchi teskari kub kuchining kombinatsiyasidan iborat^[34]. Ushbu model notrivial deb hisoblanadi, chunki u o'z ichiga o'ziga xosliklarni o'z ichiga olgan chiziqli bo'lmagan differentsial tenglamalar to'plami bilan bog'liq (masalan, chiziqli differensial tenglamalarning oson yechiladigan tizimini keltirib chiqaradigan harmonik o'zaro ta'sirlar bilan solishtirganda). Bu ikki jihatdan u Coulomb o'zaro ta'siriga ega (erimaydigan) modellarga o'xshaydi va natijada geliy atomi kabi jismoniy tizimlarni intuitiv tushunish uchun vosita sifatida taklif qilingan^[34] ^[35].

Nuqtali vorteks modeli doirasida ikki o'lchovli ideal suyuqlikdagi girdoblarning harakati faqat birinchi tartibli vaqt hosilalarini o'z ichiga olgan harakat tenglamalari bilan tavsiflanadi. Ya'ni Nyuton mexanikasidan farqli o'laroq, ularning nisbiy pozitsiyalari bilan tezlanish emas, balki tezlik aniqlanadi. Natijada, uch vorteks muammosi hali ham integral bo'lib qolmoqda ^[36], xaotik xatti-harakatni olish uchun kamida to'rtta vorteks kerak^[37]. Passiv kuzatuvchi zarrachaning uchta vorteksning tezlik maydonidagi harakati va Nyuton mexanikasining cheklangan uch jismli muammosi o'rtasida parallellik o'tkazish mumkin^[38].

Uch jismli tortishish muammosi ham umumiy nisbiylik nazariyasi yordamida o'rganilgan. Jismoniy jihatdan, juda kuchli tortishish maydonlari bo'lgan tizimlarda, masalan, qora tuynukning hodisa gorizonti yaqinida relativistik davolash zarur bo'ladi. Biroq, relyativistik muammo Nyuton mexanikasiga qaraganda ancha qiyin va murakkab raqamli texnikalar talab qilinadi. Hatto to'liq ikki tana muammosi (ya'ni massalarning ixtiyoriy nisbati uchun) umumiy nisbiylik nazariyasida qat'iy analitik yechimga ega emas^[39].

$n$ ta jism masalasi

Uch jism muammosi <span about="#mwt273" class="texhtml mvar" data-cx="[{"adapted":true,"partial":false,"targetExists":true,"mandatoryTargetParams":[],"optionalTargetParams":[]}]" data-mw="{"parts":[{"template":{"target":{"wt":"Mvar","href":"./Andoza:Mvar"},"params":{"1":{"wt":"n"}},"i":0}}]}" data-ve-no-generated-contents="true" id="mwAVw" style="font-style:italic;" typeof="mw:Transclusion">n -jism muammosining alohida holati bo'lib, u $n$ jismning tortishish kabi jismoniy kuchlardan biri ostida qanday harakatlanishini tavsiflaydi. Karl F. Sundman tomonidan $n = 3$ uchun va Qiudong Vang tomonidan $n > 3$ uchun isbotlanganidek, bu muammolar konvergent quvvat qatori ko'rinishidagi global analitik yechimga ega (batafsil ma'lumot uchun <span about="#mwt274" class="texhtml mvar" data-cx="[{"adapted":true,"partial":false,"targetExists":true,"mandatoryTargetParams":[],"optionalTargetParams":[]}]" data-mw="{"parts":[{"template":{"target":{"wt":"Mvar","href":"./Andoza:Mvar"},"params":{"1":{"wt":"n"}},"i":0}}]}" data-ve-no-generated-contents="true" id="mwAWQ" style="font-style:italic;" typeof="mw:Transclusion">n -tana muammosiga qarang). Biroq, Sundman va Vang seriyalari shunchalik sekin birlashadiki, ular amaliy maqsadlar uchun foydasizdir^[40]; shuning uchun hozirgi vaqtda raqamli integratsiya yoki ba'zi hollarda klassik trigonometrik qatorlar yaqinlashuvi ko'rinishidagi raqamli tahlil orqali yechimlarni taxminiy aniqlash kerak (qarang : <span about="#mwt275" class="texhtml mvar" data-cx="[{"adapted":true,"partial":false,"targetExists":true,"mandatoryTargetParams":[],"optionalTargetParams":[]}]" data-mw="{"parts":[{"template":{"target":{"wt":"Mvar","href":"./Andoza:Mvar"},"params":{"1":{"wt":"n"}},"i":0}}]}" data-ve-no-generated-contents="true" id="mwAWs" style="font-style:italic;" typeof="mw:Transclusion">n -tana simulyatsiyasi ). Atom tizimlari, masalan, atomlar, ionlar va molekulalar kvant $n$ -tana muammosi nuqtai nazaridan ko'rib chiqilishi mumkin. Klassik fizik tizimlar orasida $n$ -tana muammosi odatda galaktika yoki galaktikalar klasteriga ishora qiladi; yulduzlar, sayyoralar va ularning sun'iy yo'ldoshlari kabi sayyora tizimlarini ham $n$ - tana tizimlari sifatida ko'rib chiqish mumkin. Ba'zi ilovalar tebranish nazariyasi bilan qulay tarzda ko'rib chiqiladi, bunda tizim ikki tanali muammo va gipotetik bezovtalanmagan ikki tanali trayektoriyadan chetga chiqishga olib keladigan qo'shimcha kuchlar sifatida qaraladi.

Zamonaviy qarashda

1951 yilgi klassik ilmiy-fantastik filmda "Yer tik turgan kun" filmida musofir Klaatu janob duradgor taxallusidan foydalanib, prof. Barnhardtning doskasi. Bu tenglamalar uch tanali muammoning muayyan shaklining aniq tavsifidir.

Xitoylik yozuvchi Lyu Tsisinning “Yerning o‘tmishini eslash” trilogiyasining birinchi jildi “Uch jism muammosi” deb nomlanadi va markaziy syujet qurilmasi sifatida uch jism muammosini aks ettiradi^[41].

↑ ^1,0 ^1,1 ^1,2 Barrow-Green, June (2008). "The Three-Body Problem". in Gowers. The Princeton Companion to Mathematics. Princeton University Press. 726–728 b. Manba xatosi: Invalid <ref> tag; name "PrincetonCompanion" defined multiple times with different content
↑ „Historical Notes: Three-Body Problem“. Qaraldi: 2017-yil 19-iyul.
↑ ^3,0 ^3,1 Barrow-Green, June. Poincaré and the Three Body Problem. American Mathematical Society, 1997 — 8–12 bet. ISBN 978-0-8218-0367-7. Manba xatosi: Invalid <ref> tag; name "Barrow-Green1997" defined multiple times with different content
↑ „The Three-Body Problem“.
↑ ^5,0 ^5,1 Cartwright, Jon (8 March 2013). "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem". Science Now. https://www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem. Qaraldi: 2013-04-04. Uch jism muammosi]] Manba xatosi: Invalid <ref> tag; name "13solutions" defined multiple times with different content
↑ Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.
↑ Beloriszky, D. (1930). "Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps". Bulletin Astronomique. Série 2 6: 417–434.
↑ Burrau (1913). "Numerische Berechnung eines Spezialfalles des Dreikörperproblems". Astronomische Nachrichten 195 (6): 113–118. doi:10.1002/asna.19131950602. https://zenodo.org/record/1424886.
↑ Victor Szebehely; C. Frederick Peters (1967). "Complete Solution of a General Problem of Three Bodies". Astronomical Journal 72: 876. doi:10.1086/110355.
↑ Here the gravitational constant G has been set to 1, and the initial conditions are r₁(0) = -r₃(0) = (-0.97000436, 0.24308753); r₂(0) = (0,0); v₁(0) = v₃(0) = (0.4662036850, 0.4323657300); v₂(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000).
↑ ^11,0 ^11,1 Šuvakov, M. „Three-body Gallery“. Qaraldi: 2015-yil 12-avgust. Manba xatosi: Invalid <ref> tag; name "TBG" defined multiple times with different content
↑ Moore, Cristopher (1993). "Braids in classical dynamics". Physical Review Letters 70 (24): 3675–3679. doi:10.1103/PhysRevLett.70.3675. PMID 10053934. Archived from the original on 2018-10-08. https://web.archive.org/web/20181008213647/http://tuvalu.santafe.edu/~moore/braids-prl.pdf. Qaraldi: 2016-01-01. Uch jism muammosi]]
↑ Chenciner, Alain; Montgomery, Richard (2000). "A remarkable periodic solution of the three-body problem in the case of equal masses". Annals of Mathematics. Second Series 152 (3): 881–902. doi:10.2307/2661357.
↑ Montgomery, Richard (2001). "A new solution to the three-body problem". Notices of the American Mathematical Society 48: 471–481. https://www.ams.org/notices/200105/fea-montgomery.pdf.
↑ Heggie, Douglas C. (2000). "A new outcome of binary–binary scattering". Monthly Notices of the Royal Astronomical Society 318 (4): L61–L63. doi:10.1046/j.1365-8711.2000.04027.x.
↑ Hudomal, Ana (October 2015). "New periodic solutions to the three-body problem and gravitational waves". Master of Science Thesis at the Faculty of Physics, Belgrade University. http://www.scl.rs/theses/msc_ahudomal.pdf. Qaraldi: 5 February 2019. Uch jism muammosi]]
↑ Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits". Science China Physics, Mechanics & Astronomy 60 (12): 129511. doi:10.1007/s11433-017-9078-5. ISSN 1674-7348.
↑ Li, Xiaoming; Jing, Yipeng; Liao, Shijun (August 2018). "The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum". Publications of the Astronomical Society of Japan 70 (4). doi:10.1093/pasj/psy057.
↑ Li, Xiaoming; Liao, Shijun (2019). "Collisionless periodic orbits in the free-fall three-body problem". New Astronomy 70: 22–26. doi:10.1016/j.newast.2019.01.003.
↑ „3body simulator“ (en). 3body simulator. Qaraldi: 2022-yil 17-noyabr.
↑ Technion. „A Centuries-Old Physics Mystery? Solved“. SciTechDaily. SciTech (2021-yil 6-oktyabr). Qaraldi: 2021-yil 12-oktyabr.
↑ Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021). "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters". Physical Review 11 (3): 031020. doi:10.1103/PhysRevX.11.031020. https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031020. Qaraldi: 12 October 2021. Uch jism muammosi]]
↑ ^23,0 ^23,1 ^23,2 Valtonen, Mauri. The Three-body Problem from Pythagoras to Hawking. Springer, 3 May 2016. ISBN 978-3-319-22726-9. OCLC 1171227640.
↑ Newton, Isaac. Philosophiæ naturalis principia mathematica. London: G. & J. Innys, 1726. DOI:10.14711/spcol/b706487. 2022-yil 5-oktyabrda qaraldi.
↑ „Amerigo Vespucci“ (en-us). Biography (2021-yil 23-iyun). Qaraldi: 2022-yil 5-oktyabr.
↑ The 1747 memoirs of both parties can be read in the volume of Histoires (including Mémoires) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):
↑ Jean le Rond d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol. 2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.
↑ Mohr, R.F.; Furnstahl, R.J.; Hammer, H.-W.; Perry, R.J.; Wilson, K.G. (January 2006). "Precise numerical results for limit cycles in the quantum three-body problem". Annals of Physics 321 (1): 225–259. doi:10.1016/j.aop.2005.10.002. ISSN 0003-4916.
↑ "Coplanar Motion of Two Planets, One Having a Zero Mass". Annals of Mathematics, Vol. III, pp. 65–73, 1887.
↑ Barrow-Green, June. Poincaré and the Three Body Problem, History of Mathematics. Providence, Rhode Island: American Mathematical Society, 1996-10-29. DOI:10.1090/hmath/011. ISBN 978-0-8218-0367-7.
↑ Efimov, V. (1970-12-21). "Energy levels arising from resonant two-body forces in a three-body system" (en). Physics Letters B 33 (8): 563–564. doi:10.1016/0370-2693(70)90349-7. ISSN 0370-2693.
↑ Liao, Shijun; Li, Xiaoming (2019-11-01). "On the periodic solutions of the three-body problem" (en). National Science Review 6 (6): 1070–1071. doi:10.1093/nsr/nwz102. ISSN 2095-5138. PMID 34691975. PMC 8291409. https://academic.oup.com/nsr/article/6/6/1070/5537324.
↑ Breen, Philip G.; Foley, Christopher N.; Boekholt, Tjarda; Portegies Zwart, Simon (2020). "Newton versus the machine: Solving the chaotic three-body problem using deep neural networks". Monthly Notices of the Royal Astronomical Society 494 (2): 2465–2470. doi:10.1093/mnras/staa713.
↑ ^34,0 ^34,1 Crandall, R.; Whitnell, R.; Bettega, R. (1984). "Exactly soluble two-electron atomic model". American Journal of Physics 52 (5): 438–442. doi:10.1119/1.13650.
↑ Calogero, F. (1969). "Solution of a Three-Body Problem in One Dimension". Journal of Mathematical Physics 10 (12): 2191–2196. doi:10.1063/1.1664820.
↑ Aref, Hassan (1979-03-01). "Motion of three vortices". The Physics of Fluids 22 (3): 393–400. doi:10.1063/1.862605. ISSN 0031-9171. https://aip.scitation.org/doi/10.1063/1.862605.
↑ Aref, Hassan; Pomphrey, Neil (1980-08-18). "Integrable and chaotic motions of four vortices" (en). Physics Letters A 78 (4): 297–300. doi:10.1016/0375-9601(80)90375-8. ISSN 0375-9601.
↑ Neufeld, Z; Tél, T (1997-03-21). "The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices". Journal of Physics A: Mathematical and General 30 (6): 2263–2280. doi:10.1088/0305-4470/30/6/043. ISSN 0305-4470. https://iopscience.iop.org/article/10.1088/0305-4470/30/6/043.
↑ Musielak, Z. E.; Quarles, B. (2014). "The three-body problem". Reports on Progress in Physics 77 (6): 065901. doi:10.1088/0034-4885/77/6/065901. ISSN 0034-4885. PMID 24913140.
↑ Florin Diacu. "The Solution of the n-body Problem", The Mathematical Intelligencer, 1996.
↑ Qin. „In a Topsy-Turvy World, China Warms to Sci-Fi“. The New York Times (2014-yil 10-noyabr). 2019-yil 9-dekabrda asl nusxadan arxivlangan. Qaraldi: 2020-yil 5-fevral.

[PrincetonCompanion-1] 1,0 ^1,1 ^1,2 Barrow-Green, June (2008). "The Three-Body Problem". in Gowers. The Princeton Companion to Mathematics. Princeton University Press. 726–728 b. Manba xatosi: Invalid <ref> tag; name "PrincetonCompanion" defined multiple times with different content

[first-2] „Historical Notes: Three-Body Problem“. Qaraldi: 2017-yil 19-iyul.

[Barrow-Green1997-3] 3,0 ^3,1 Barrow-Green, June. Poincaré and the Three Body Problem. American Mathematical Society, 1997 — 8–12 bet. ISBN 978-0-8218-0367-7. Manba xatosi: Invalid <ref> tag; name "Barrow-Green1997" defined multiple times with different content

[4] „The Three-Body Problem“.

[13solutions-5] 5,0 ^5,1 Cartwright, Jon (8 March 2013). "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem". Science Now. https://www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem. Qaraldi: 2013-04-04. Uch jism muammosi]] Manba xatosi: Invalid <ref> tag; name "13solutions" defined multiple times with different content

[6] Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.

[7] Beloriszky, D. (1930). "Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps". Bulletin Astronomique. Série 2 6: 417–434.

[Burrau-8] Burrau (1913). "Numerische Berechnung eines Spezialfalles des Dreikörperproblems". Astronomische Nachrichten 195 (6): 113–118. doi:10.1002/asna.19131950602. https://zenodo.org/record/1424886.

[9] Victor Szebehely; C. Frederick Peters (1967). "Complete Solution of a General Problem of Three Bodies". Astronomical Journal 72: 876. doi:10.1086/110355.

[10] Here the gravitational constant G has been set to 1, and the initial conditions are r₁(0) = -r₃(0) = (-0.97000436, 0.24308753); r₂(0) = (0,0); v₁(0) = v₃(0) = (0.4662036850, 0.4323657300); v₂(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000).

[TBG-11] 11,0 ^11,1 Šuvakov, M. „Three-body Gallery“. Qaraldi: 2015-yil 12-avgust. Manba xatosi: Invalid <ref> tag; name "TBG" defined multiple times with different content

[12] Moore, Cristopher (1993). "Braids in classical dynamics". Physical Review Letters 70 (24): 3675–3679. doi:10.1103/PhysRevLett.70.3675. PMID 10053934. Archived from the original on 2018-10-08. https://web.archive.org/web/20181008213647/http://tuvalu.santafe.edu/~moore/braids-prl.pdf. Qaraldi: 2016-01-01. Uch jism muammosi]]

[13] Chenciner, Alain; Montgomery, Richard (2000). "A remarkable periodic solution of the three-body problem in the case of equal masses". Annals of Mathematics. Second Series 152 (3): 881–902. doi:10.2307/2661357.

[14] Montgomery, Richard (2001). "A new solution to the three-body problem". Notices of the American Mathematical Society 48: 471–481. https://www.ams.org/notices/200105/fea-montgomery.pdf.

[15] Heggie, Douglas C. (2000). "A new outcome of binary–binary scattering". Monthly Notices of the Royal Astronomical Society 318 (4): L61–L63. doi:10.1046/j.1365-8711.2000.04027.x.

[16] Hudomal, Ana (October 2015). "New periodic solutions to the three-body problem and gravitational waves". Master of Science Thesis at the Faculty of Physics, Belgrade University. http://www.scl.rs/theses/msc_ahudomal.pdf. Qaraldi: 5 February 2019. Uch jism muammosi]]

[17] Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits". Science China Physics, Mechanics & Astronomy 60 (12): 129511. doi:10.1007/s11433-017-9078-5. ISSN 1674-7348.

[18] Li, Xiaoming; Jing, Yipeng; Liao, Shijun (August 2018). "The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum". Publications of the Astronomical Society of Japan 70 (4). doi:10.1093/pasj/psy057.

[19] Li, Xiaoming; Liao, Shijun (2019). "Collisionless periodic orbits in the free-fall three-body problem". New Astronomy 70: 22–26. doi:10.1016/j.newast.2019.01.003.

[20] „3body simulator“ (en). 3body simulator. Qaraldi: 2022-yil 17-noyabr.

[21] Technion. „A Centuries-Old Physics Mystery? Solved“. SciTechDaily. SciTech (2021-yil 6-oktyabr). Qaraldi: 2021-yil 12-oktyabr.

[22] Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021). "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters". Physical Review 11 (3): 031020. doi:10.1103/PhysRevX.11.031020. https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031020. Qaraldi: 12 October 2021. Uch jism muammosi]]

[:1-23] 23,0 ^23,1 ^23,2 Valtonen, Mauri. The Three-body Problem from Pythagoras to Hawking. Springer, 3 May 2016. ISBN 978-3-319-22726-9. OCLC 1171227640.

[24] Newton, Isaac. Philosophiæ naturalis principia mathematica. London: G. & J. Innys, 1726. DOI:10.14711/spcol/b706487. 2022-yil 5-oktyabrda qaraldi.

[25] „Amerigo Vespucci“ (en-us). Biography (2021-yil 23-iyun). Qaraldi: 2022-yil 5-oktyabr.

[26] The 1747 memoirs of both parties can be read in the volume of Histoires (including Mémoires) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):

[27] Jean le Rond d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol. 2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.

[:0-28] Mohr, R.F.; Furnstahl, R.J.; Hammer, H.-W.; Perry, R.J.; Wilson, K.G. (January 2006). "Precise numerical results for limit cycles in the quantum three-body problem". Annals of Physics 321 (1): 225–259. doi:10.1016/j.aop.2005.10.002. ISSN 0003-4916.

[29] "Coplanar Motion of Two Planets, One Having a Zero Mass". Annals of Mathematics, Vol. III, pp. 65–73, 1887.

[30] Barrow-Green, June. Poincaré and the Three Body Problem, History of Mathematics. Providence, Rhode Island: American Mathematical Society, 1996-10-29. DOI:10.1090/hmath/011. ISBN 978-0-8218-0367-7.

[31] Efimov, V. (1970-12-21). "Energy levels arising from resonant two-body forces in a three-body system" (en). Physics Letters B 33 (8): 563–564. doi:10.1016/0370-2693(70)90349-7. ISSN 0370-2693.

[32] Liao, Shijun; Li, Xiaoming (2019-11-01). "On the periodic solutions of the three-body problem" (en). National Science Review 6 (6): 1070–1071. doi:10.1093/nsr/nwz102. ISSN 2095-5138. PMID 34691975. PMC 8291409. https://academic.oup.com/nsr/article/6/6/1070/5537324.

[33] Breen, Philip G.; Foley, Christopher N.; Boekholt, Tjarda; Portegies Zwart, Simon (2020). "Newton versus the machine: Solving the chaotic three-body problem using deep neural networks". Monthly Notices of the Royal Astronomical Society 494 (2): 2465–2470. doi:10.1093/mnras/staa713.

[Crandall1984-34] 34,0 ^34,1 Crandall, R.; Whitnell, R.; Bettega, R. (1984). "Exactly soluble two-electron atomic model". American Journal of Physics 52 (5): 438–442. doi:10.1119/1.13650.

[35] Calogero, F. (1969). "Solution of a Three-Body Problem in One Dimension". Journal of Mathematical Physics 10 (12): 2191–2196. doi:10.1063/1.1664820.

[36] Aref, Hassan (1979-03-01). "Motion of three vortices". The Physics of Fluids 22 (3): 393–400. doi:10.1063/1.862605. ISSN 0031-9171. https://aip.scitation.org/doi/10.1063/1.862605.

[37] Aref, Hassan; Pomphrey, Neil (1980-08-18). "Integrable and chaotic motions of four vortices" (en). Physics Letters A 78 (4): 297–300. doi:10.1016/0375-9601(80)90375-8. ISSN 0375-9601.

[38] Neufeld, Z; Tél, T (1997-03-21). "The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices". Journal of Physics A: Mathematical and General 30 (6): 2263–2280. doi:10.1088/0305-4470/30/6/043. ISSN 0305-4470. https://iopscience.iop.org/article/10.1088/0305-4470/30/6/043.

[39] Musielak, Z. E.; Quarles, B. (2014). "The three-body problem". Reports on Progress in Physics 77 (6): 065901. doi:10.1088/0034-4885/77/6/065901. ISSN 0034-4885. PMID 24913140.

[40] Florin Diacu. "The Solution of the n-body Problem", The Mathematical Intelligencer, 1996.

[41] Qin. „In a Topsy-Turvy World, China Warms to Sci-Fi“. The New York Times (2014-yil 10-noyabr). 2019-yil 9-dekabrda asl nusxadan arxivlangan. Qaraldi: 2020-yil 5-fevral.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]