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Question Number 217837 by SdC355 last updated on 22/Mar/25
do you guys know about Three Body problem?? when 2−Body problem we can solve equation of motions x_1 ^→ (t) , x_2 ^→ (t) but why we can′t solve 3−body problem? The reason why we can′t solve 3−Body problem is because this Equation isn′t Linear equation???
$$\mathrm{do}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{know}\:\mathrm{about}\:\mathrm{Three}\:\mathrm{Body}\:\mathrm{problem}?? \\ $$$$\mathrm{when}\:\mathrm{2}−\mathrm{Body}\:\mathrm{problem}\: \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{motions}\:\overset{\rightarrow} {\boldsymbol{\mathrm{x}}}_{\mathrm{1}} \left({t}\right)\:,\:\overset{\rightarrow} {\boldsymbol{\mathrm{x}}}_{\mathrm{2}} \left({t}\right) \\ $$$$\mathrm{but}\:\mathrm{why}\:\mathrm{we}\:\mathrm{can}'\mathrm{t}\:\mathrm{solve}\:\mathrm{3}−\mathrm{body}\:\mathrm{problem}? \\ $$$$\mathrm{The}\:\mathrm{reason}\:\mathrm{why}\:\mathrm{we}\:\mathrm{can}'\mathrm{t}\:\mathrm{solve}\:\mathrm{3}−\mathrm{Body}\:\mathrm{problem} \\ $$$$\mathrm{is}\:\mathrm{because}\:\mathrm{this}\:\mathrm{Equation}\:\mathrm{isn}'\mathrm{t}\:\:\mathrm{Linear}\:\mathrm{equation}??? \\ $$
Answered by MrGaster last updated on 22/Mar/25
For the 2-body problem,the equations of motion for particles can expressed as: m_1 ((d^2 −x_1 )/dt^2 )=Gm_1 m_2 ((x_2 −x_1 )/(∣x_2 −x_1 ∣^3 )) m_2 (d^2 x_2 /dt^2 )=Gm_1 m_2 ((x_1 −x_2 )/(∣x_1 −x_2 ∣^3 )) where,x_1 (t)and x_2 (t)represent the positions of thetwo particles at time t,m_1 and m_2 are their masses and G is theg ravitational constant. By introducing center ofs mas coordinates and relativecoordinates thee abov equations can bef simpliied to a single particleo mtion equation which cane thn be solved. For the 3-body problem theu eqations of motion for ther paticles are: m_1 (d^2 x_1 /dt^2 )=Gm_1 m_2 ((x_2 −x_1 )/(∣x_2 −x_1 ∣^3 ))+Gm_1 m_3 ((x_3 −x_1 )/(∣x_3 −x_1 ∣^3 )) m_2 (d^2 x_2 /dt^2 )=Gm_2 m_1 ((x_1 −x_2 )/(∣x_1 −x_2 ∣^3 ))+Gm_2 m_3 ((x_3 −x_2 )/(∣x_3 −x_2 ∣^3 )) m_3 (d^2 x_3 /dt^2 )=Gm_3 m_1 ((x_1 −x_3 )/(∣x_1 −x_3 ∣^3 ))+Gm_3 m_2 ((x_2 −x_3 )/(∣x_2 −x_3 ∣^3 )) where x_1 (t),x_2 (t),and x_3 (t) represent the positions of thethree particles at time t and m_1 ,m_2 ,and m_3 are their masses. The equations of motion forh te 3body problem are a set ofnonlinear differentialt equaions and their solutionsa re extremely complex andn canot be expressed in termsf o elementary functions. Thes pecific reasons are as follows: •Nonlinearity:The equations of the 3bodyr poblem are nonlinearg meanin that the interactionsbetween thei partcles are not simplyl lineary additive but arel couped with each other. •Chaos: The solutions to theo 3-bdy problem are highlyi senstive to initial conditionsw ith small changes in initialo cnditions leading to vastlyf diferent solutions makingg lonterm predictionsl impossibe. •Lack of General:Solution: Unlike the 2-body probleme th 3-body problem has now knon general solution i.e., thee is no universal solutiono frm that can be expressed int perms of elementarys function.
$$\mathrm{For}\:\mathrm{the}\:\mathrm{2}-\mathrm{body}\:\mathrm{problem},\mathrm{the}\:\mathrm{equations}\:\mathrm{of} \\ $$$$\mathrm{motion}\:\mathrm{for}\:\mathrm{particles}\:\mathrm{can}\:\mathrm{expressed}\:\mathrm{as}: \\ $$$${m}_{\mathrm{1}} \frac{{d}^{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{1}} }{{dt}^{\mathrm{2}} }={Gm}_{\mathrm{1}} {m}_{\mathrm{2}} \frac{\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{1}} }{\mid\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{1}} \mid^{\mathrm{3}} } \\ $$$${m}_{\mathrm{2}} \frac{{d}^{\mathrm{2}} \boldsymbol{{x}}_{\mathrm{2}} }{{dt}^{\mathrm{2}} }={Gm}_{\mathrm{1}} {m}_{\mathrm{2}} \frac{\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{2}} }{\mid\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{2}} \mid^{\mathrm{3}} } \\ $$$$\mathrm{where},\boldsymbol{{x}}_{\mathrm{1}} \left({t}\right)\mathrm{and}\:\boldsymbol{{x}}_{\mathrm{2}} \left({t}\right)\mathrm{represent}\:\mathrm{the}\:\mathrm{positions}\:\mathrm{of}\: \\ $$$$\mathrm{thetwo}\:\mathrm{particles}\:\mathrm{at}\:\mathrm{time}\:{t},{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \mathrm{are}\:\mathrm{their}\:\mathrm{masses}\:\mathrm{and}\:{G}\:\mathrm{is}\:\mathrm{theg} \\ $$$$\mathrm{ravitational}\:\mathrm{constant}. \\ $$$$\mathrm{By}\:\mathrm{introducing}\:\mathrm{center}\:\mathrm{ofs} \\ $$$$\mathrm{mas}\:\mathrm{coordinates}\:\mathrm{and}\: \\ $$$$\mathrm{relativecoordinates}\:\mathrm{thee} \\ $$$$\mathrm{abov}\:\mathrm{equations}\:\mathrm{can}\:\mathrm{bef} \\ $$$$\mathrm{simpliied}\:\mathrm{to}\:\mathrm{a}\:\mathrm{single}\:\mathrm{particleo} \\ $$$$\mathrm{mtion}\:\mathrm{equation}\:\mathrm{which}\:\mathrm{cane} \\ $$$$\mathrm{thn}\:\mathrm{be}\:\mathrm{solved}. \\ $$$$\mathrm{For}\:\mathrm{the}\:\mathrm{3}-\mathrm{body}\:\mathrm{problem}\:\mathrm{theu} \\ $$$$\mathrm{eqations}\:\mathrm{of}\:\mathrm{motion}\:\mathrm{for}\:\mathrm{ther} \\ $$$$\mathrm{paticles}\:\mathrm{are}: \\ $$$${m}_{\mathrm{1}} \frac{{d}^{\mathrm{2}} \boldsymbol{{x}}_{\mathrm{1}} }{{dt}^{\mathrm{2}} }={Gm}_{\mathrm{1}} {m}_{\mathrm{2}} \frac{\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{1}} }{\mid\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{1}} \mid^{\mathrm{3}} }+{Gm}_{\mathrm{1}} {m}_{\mathrm{3}} \frac{\boldsymbol{{x}}_{\mathrm{3}} −\boldsymbol{{x}}_{\mathrm{1}} }{\mid\boldsymbol{{x}}_{\mathrm{3}} −\boldsymbol{{x}}_{\mathrm{1}} \mid^{\mathrm{3}} } \\ $$$${m}_{\mathrm{2}} \frac{{d}^{\mathrm{2}} \boldsymbol{{x}}_{\mathrm{2}} }{{dt}^{\mathrm{2}} }={Gm}_{\mathrm{2}} {m}_{\mathrm{1}} \frac{\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{2}} }{\mid\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{2}} \mid^{\mathrm{3}} }+{Gm}_{\mathrm{2}} {m}_{\mathrm{3}} \frac{\boldsymbol{{x}}_{\mathrm{3}} −\boldsymbol{{x}}_{\mathrm{2}} }{\mid\boldsymbol{{x}}_{\mathrm{3}} −\boldsymbol{{x}}_{\mathrm{2}} \mid^{\mathrm{3}} } \\ $$$${m}_{\mathrm{3}} \frac{{d}^{\mathrm{2}} \boldsymbol{{x}}_{\mathrm{3}} }{{dt}^{\mathrm{2}} }={Gm}_{\mathrm{3}} {m}_{\mathrm{1}} \frac{\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{3}} }{\mid\boldsymbol{{x}}_{\mathrm{1}} −\boldsymbol{{x}}_{\mathrm{3}} \mid^{\mathrm{3}} }+{Gm}_{\mathrm{3}} {m}_{\mathrm{2}} \frac{\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{3}} }{\mid\boldsymbol{{x}}_{\mathrm{2}} −\boldsymbol{{x}}_{\mathrm{3}} \mid^{\mathrm{3}} } \\ $$$$\mathrm{where}\:\boldsymbol{{x}}_{\mathrm{1}} \left({t}\right),\boldsymbol{{x}}_{\mathrm{2}} \left({t}\right),\mathrm{and}\:\boldsymbol{{x}}_{\mathrm{3}} \left({t}\right) \\ $$$$\mathrm{represent}\:\mathrm{the}\:\mathrm{positions}\:\mathrm{of}\: \\ $$$$\mathrm{thethree}\:\mathrm{particles}\:\mathrm{at}\:\mathrm{time}\:{t} \\ $$$$\mathrm{and}\:{m}_{\mathrm{1}} ,{m}_{\mathrm{2}} ,\mathrm{and}\:{m}_{\mathrm{3}} \mathrm{are}\:\mathrm{their}\:\mathrm{masses}. \\ $$$$ \\ $$$$\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{motion}\:\mathrm{forh} \\ $$$$\mathrm{te}\:\mathrm{3body}\:\mathrm{problem}\:\mathrm{are}\:\mathrm{a}\:\mathrm{set}\: \\ $$$$\mathrm{ofnonlinear}\:\mathrm{differentialt} \\ $$$$\mathrm{equaions}\:\mathrm{and}\:\mathrm{their}\:\mathrm{solutionsa} \\ $$$$\mathrm{re}\:\mathrm{extremely}\:\mathrm{complex}\:\mathrm{andn} \\ $$$$\mathrm{canot}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{termsf} \\ $$$$\mathrm{o}\:\mathrm{elementary}\:\mathrm{functions}.\:\mathrm{Thes} \\ $$$$\mathrm{pecific}\:\mathrm{reasons}\:\mathrm{are}\:\mathrm{as}\:\mathrm{follows}: \\ $$$$\bullet\mathrm{Nonlinearity}:\mathrm{The}\:\mathrm{equations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{3bodyr} \\ $$$$\mathrm{poblem}\:\mathrm{are}\:\mathrm{nonlinearg} \\ $$$$\mathrm{meanin}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{interactionsbetween}\:\mathrm{thei} \\ $$$$\mathrm{partcles}\:\mathrm{are}\:\mathrm{not}\:\mathrm{simplyl} \\ $$$$\mathrm{lineary}\:\mathrm{additive}\:\mathrm{but}\:\mathrm{arel} \\ $$$$\mathrm{couped}\:\mathrm{with}\:\mathrm{each}\:\mathrm{other}. \\ $$$$\bullet\mathrm{Chaos}:\:\mathrm{The}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{theo} \\ $$$$\mathrm{3}-\mathrm{bdy}\:\mathrm{problem}\:\mathrm{are}\:\mathrm{highlyi} \\ $$$$\mathrm{senstive}\:\mathrm{to}\:\mathrm{initial}\:\mathrm{conditionsw} \\ $$$$\mathrm{ith}\:\mathrm{small}\:\mathrm{changes}\:\mathrm{in}\:\mathrm{initialo} \\ $$$$\mathrm{cnditions}\:\mathrm{leading}\:\mathrm{to}\:\mathrm{vastlyf} \\ $$$$\mathrm{diferent}\:\mathrm{solutions}\:\mathrm{makingg} \\ $$$$\mathrm{lonterm}\:\mathrm{predictionsl} \\ $$$$\mathrm{impossibe}. \\ $$$$\bullet\mathrm{Lack}\:\mathrm{of}\:\mathrm{General}:\mathrm{Solution}: \\ $$$$\mathrm{Unlike}\:\mathrm{the}\:\mathrm{2}-\mathrm{body}\:\mathrm{probleme} \\ $$$$\mathrm{th}\:\mathrm{3}-\mathrm{body}\:\mathrm{problem}\:\mathrm{has}\:\mathrm{now} \\ $$$$\mathrm{knon}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{i}.\mathrm{e}., \\ $$$$\mathrm{thee}\:\mathrm{is}\:\mathrm{no}\:\mathrm{universal}\:\mathrm{solutiono} \\ $$$$\mathrm{frm}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{int} \\ $$$$\mathrm{perms}\:\mathrm{of}\:\mathrm{elementarys} \\ $$$$\mathrm{function}. \\ $$
Commented by SdC355 last updated on 22/Mar/25
wow thx a lot♥
$$\mathrm{wow}\:\mathrm{thx}\:\mathrm{a}\:\mathrm{lot}\heartsuit \\ $$

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