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Question Number 217837 by SdC355 last updated on 22/Mar/25

$$\mathrm{do}\mathrm{you}\mathrm{guys}\mathrm{know}\mathrm{about}\mathrm{Three}\mathrm{Body}\mathrm{problem}??$$$$\mathrm{when}2-\mathrm{Body}\mathrm{problem}$$$$\mathrm{we}\mathrm{can}\mathrm{solve}\mathrm{equation}\mathrm{of}\mathrm{motions}{\overrightarrow{\mathbf{x}}}_1\left(t\right),{\overrightarrow{\mathbf{x}}}_2\left(t\right)$$$$\mathrm{but}\mathrm{why}\mathrm{we}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{solve}3-\mathrm{body}\mathrm{problem}?$$$$\mathrm{The}\mathrm{reason}\mathrm{why}\mathrm{we}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{solve}3-\mathrm{Body}\mathrm{problem}$$$$\mathrm{is}\mathrm{because}\mathrm{this}\mathrm{Equation}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{Linear}\mathrm{equation}???$$

Answered by MrGaster last updated on 22/Mar/25

$$\mathrm{For}\mathrm{the}2-\mathrm{body}\mathrm{problem},\mathrm{the}\mathrm{equations}\mathrm{of}$$$$\mathrm{motion}\mathrm{for}\mathrm{particles}\mathrm{can}\mathrm{expressed}\mathrm{as}:$$$$m_1\frac{d^2-{\mathit{x}}_1}{{dt}^2}={Gm}_1{m}_2\frac{{\mathit{x}}_2-{\mathit{x}}_1}{\mid {\mathit{x}}_2-{\mathit{x}}_1{\mid }^3}$$$$m_2\frac{d^2{\mathit{x}}_2}{{dt}^2}={Gm}_1{m}_2\frac{{\mathit{x}}_1-{\mathit{x}}_2}{\mid {\mathit{x}}_1-{\mathit{x}}_2{\mid }^3}$$$$\mathrm{where},{\mathit{x}}_1\left(t\right)\mathrm{and}{\mathit{x}}_2\left(t\right)\mathrm{represent}\mathrm{the}\mathrm{positions}\mathrm{of}$$$$\mathrm{thetwo}\mathrm{particles}\mathrm{at}\mathrm{time}t,m_1\mathrm{and}{m}_2\mathrm{are}\mathrm{their}\mathrm{masses}\mathrm{and}G\mathrm{is}\mathrm{theg}$$$$\mathrm{ravitational}\mathrm{constant}.$$$$\mathrm{By}\mathrm{introducing}\mathrm{center}\mathrm{ofs}$$$$\mathrm{mas}\mathrm{coordinates}\mathrm{and}$$$$\mathrm{relativecoordinates}\mathrm{thee}$$$$\mathrm{abov}\mathrm{equations}\mathrm{can}\mathrm{bef}$$$$\mathrm{simpliied}\mathrm{to}\mathrm{a}\mathrm{single}\mathrm{particleo}$$$$\mathrm{mtion}\mathrm{equation}\mathrm{which}\mathrm{cane}$$$$\mathrm{thn}\mathrm{be}\mathrm{solved}.$$$$\mathrm{For}\mathrm{the}3-\mathrm{body}\mathrm{problem}\mathrm{theu}$$$$\mathrm{eqations}\mathrm{of}\mathrm{motion}\mathrm{for}\mathrm{ther}$$$$\mathrm{paticles}\mathrm{are}:$$$$m_1\frac{d^2{\mathit{x}}_1}{{dt}^2}={Gm}_1{m}_2\frac{{\mathit{x}}_2-{\mathit{x}}_1}{\mid {\mathit{x}}_2-{\mathit{x}}_1{\mid }^3}+{Gm}_1{m}_3\frac{{\mathit{x}}_3-{\mathit{x}}_1}{\mid {\mathit{x}}_3-{\mathit{x}}_1{\mid }^3}$$$$m_2\frac{d^2{\mathit{x}}_2}{{dt}^2}={Gm}_2{m}_1\frac{{\mathit{x}}_1-{\mathit{x}}_2}{\mid {\mathit{x}}_1-{\mathit{x}}_2{\mid }^3}+{Gm}_2{m}_3\frac{{\mathit{x}}_3-{\mathit{x}}_2}{\mid {\mathit{x}}_3-{\mathit{x}}_2{\mid }^3}$$$$m_3\frac{d^2{\mathit{x}}_3}{{dt}^2}={Gm}_3{m}_1\frac{{\mathit{x}}_1-{\mathit{x}}_3}{\mid {\mathit{x}}_1-{\mathit{x}}_3{\mid }^3}+{Gm}_3{m}_2\frac{{\mathit{x}}_2-{\mathit{x}}_3}{\mid {\mathit{x}}_2-{\mathit{x}}_3{\mid }^3}$$$$\mathrm{where}{\mathit{x}}_1\left(t\right),{\mathit{x}}_2\left(t\right),\mathrm{and}{\mathit{x}}_3\left(t\right)$$$$\mathrm{represent}\mathrm{the}\mathrm{positions}\mathrm{of}$$$$\mathrm{thethree}\mathrm{particles}\mathrm{at}\mathrm{time}t$$$$\mathrm{and}{m}_1,m_2,\mathrm{and}{m}_3\mathrm{are}\mathrm{their}\mathrm{masses}.$$$$$$$$\mathrm{The}\mathrm{equations}\mathrm{of}\mathrm{motion}\mathrm{forh}$$$$\mathrm{te}3\mathrm{body}\mathrm{problem}\mathrm{are}\mathrm{a}\mathrm{set}$$$$\mathrm{ofnonlinear}\mathrm{differentialt}$$$$\mathrm{equaions}\mathrm{and}\mathrm{their}\mathrm{solutionsa}$$$$\mathrm{re}\mathrm{extremely}\mathrm{complex}\mathrm{andn}$$$$\mathrm{canot}\mathrm{be}\mathrm{expressed}\mathrm{in}\mathrm{termsf}$$$$\mathrm{o}\mathrm{elementary}\mathrm{functions}.\mathrm{Thes}$$$$\mathrm{pecific}\mathrm{reasons}\mathrm{are}\mathrm{as}\mathrm{follows}:$$$$\bullet \mathrm{Nonlinearity}:\mathrm{The}\mathrm{equations}\mathrm{of}\mathrm{the}3\mathrm{bodyr}$$$$\mathrm{poblem}\mathrm{are}\mathrm{nonlinearg}$$$$\mathrm{meanin}\mathrm{that}\mathrm{the}$$$$\mathrm{interactionsbetween}\mathrm{thei}$$$$\mathrm{partcles}\mathrm{are}\mathrm{not}\mathrm{simplyl}$$$$\mathrm{lineary}\mathrm{additive}\mathrm{but}\mathrm{arel}$$$$\mathrm{couped}\mathrm{with}\mathrm{each}\mathrm{other}.$$$$\bullet \mathrm{Chaos}:\mathrm{The}\mathrm{solutions}\mathrm{to}\mathrm{theo}$$$$3-\mathrm{bdy}\mathrm{problem}\mathrm{are}\mathrm{highlyi}$$$$\mathrm{senstive}\mathrm{to}\mathrm{initial}\mathrm{conditionsw}$$$$\mathrm{ith}\mathrm{small}\mathrm{changes}\mathrm{in}\mathrm{initialo}$$$$\mathrm{cnditions}\mathrm{leading}\mathrm{to}\mathrm{vastlyf}$$$$\mathrm{diferent}\mathrm{solutions}\mathrm{makingg}$$$$\mathrm{lonterm}\mathrm{predictionsl}$$$$\mathrm{impossibe}.$$$$\bullet \mathrm{Lack}\mathrm{of}\mathrm{General}:\mathrm{Solution}:$$$$\mathrm{Unlike}\mathrm{the}2-\mathrm{body}\mathrm{probleme}$$$$\mathrm{th}3-\mathrm{body}\mathrm{problem}\mathrm{has}\mathrm{now}$$$$\mathrm{knon}\mathrm{general}\mathrm{solution}\mathrm{i}.\mathrm{e}.,$$$$\mathrm{thee}\mathrm{is}\mathrm{no}\mathrm{universal}\mathrm{solutiono}$$$$\mathrm{frm}\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{expressed}\mathrm{int}$$$$\mathrm{perms}\mathrm{of}\mathrm{elementarys}$$$$\mathrm{function}.$$

Commented by SdC355 last updated on 22/Mar/25

$$\mathrm{wow}\mathrm{thx}\mathrm{a}\mathrm{lot}\mathrm{♡}$$

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