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ROCHESTER AST 570 - Lecture Notes - Mean Motion Resonances

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Mean%Mo'on%Resonances% Hamilton0Jacobi%equa'on%and%Hamiltonian%formula'on %for%20body%problem%in%2%dimensions% Cannonical%transforma'on%to%heliocentric%coordinates%for%N0body%problem%% Symplec'c%integrators% Canonical%transforma'on%with%resonant%angle% Mean%mo'on%resonances!Orbital%trajectories%of%Jupiter's%fou r%largest%moons%over%a%period%of%10%Earth%days,%illustra'ng%the%4:2:1%resonance%among%the%orbits%of%the%inner%three%Integrable%mo'on%• Integrable:%%n%degree%of%freedom%Hamiltonian%has%n%conserved%quan''es%• Examples:%– Hamiltonian%is%a%func'on%of%coordinates%only%– Hamiltonian%is%a%func'on%of%momenta%only%–%in%this%case%we%can%call%the%momenta%ac'on%variables%and%we%can%say%we%have%transformed%to%ac'on%angle%variables%– Hamiltonian%has%1%degree%of%freedom%and%is%'me%independent.%%H(x,v)%gives%level%contours%and%mo'on%is%along%level%contours.%• Arnold0Louiville%theorem%%0%integrable%implies%th at%the%Hamiltonian%can%be%transformed%to%depend%only%on%ac'ons%Hamilton%Jacobi%equa'on%• If%the%coordin ate%q%does%not%appear%in%the%Hamiltonian%then%the%correspondi ng%momentum%p%is%constant%• Try%to%find%a%Hamiltonian%that%vanishes%altogether,%then%everythi ng%is%conser ved%• Genera'ng%fun c'on%S2(q1,q2,…;P1,*P2,*…,t)%func'on%of%old%coordinates%and%new%momenta%which %we%would%like%conserved%• New%Hamiltonian%K = H(q1,q2, ..;Sq1,Sq2, ..., t)+St=0Finding%conserved%momenta%%%20body%problem%• Our%new%momenta%are%conserved,%constants%of%integra'on,%Pi*• 2%body%in%2%dimensi o n s%• Hamilton%Jacobi%equa'on%Qi=SPipi=SqiH(r, ; pr,L)=p2r2+L22r2µrHr,⇥S⇥r,⇥S⇥⇥+⇥S⇥t=0new%momenta%old%coordinates%%• Assume%separable%• Subs'tute%momenta%in%to%Hamilton%Jacobi%equa'on%• Since%separable%• Hamilton%Jacobi%equa'on%gives%S = Sr(r)+S()+St(t)12⇤⇥Sr⇥r⇥2+1r2⇥S⇥⇥2⌅µr= ⇥S⇥t⇤S⇤⇥= 2⇥St⇥t= 1⇤dSrdr⌅2=21+µr⇥22r2Because%we%choose%S*%separable%these%deriva'ves%must%be%constant,%as%they%are%constant%we%also%make%them%be%our%new%momenta%S = 1t + 2⇥ +rdr⇥2(1+µr) 22r2S%depends%on%new%monenta%so%α1α2%are%new%momenta%⇤S⇤⇥= 2= L = P2⇥S⇥1= Q1⇥S⇥2= Q2P1= 1S = 1t + 2⇥ +rdr⇥2(1+µr) 22r2Q1= t +⇤rdr2(1+µr) 22r2⇥1/2P2= L =µa(1  e2)K = H +⇥S⇥t= H  1=0P1= µ2aNew%coordinates%using%expression%for%L%is%energy%= ✓• Sub%in%for%constants%and%use%%Q1= t +⇤rdr2(1+µr) 22r2⇥1/2r = a(1  e cos E), dr = ae sin EdEQ1= t +a3/2⇥µ(1  e cos E)dE= t +a3/2⇥µ(E  e sin E)= t +MnQ1= Q1%is%'me%of%perihelion%• With%a%similar%integral %we%can%show%that%Q2%is%angle%of%perihelion%• 3D%problem%done%similarly%• These%coordinates%not%necessarily%ideal.%%Add%and%subtract%them%to%find%the%Delaunay,%modified%Delauney%and%Poin caré%coordinates%%K =0Q2=   f = ⇥(Hamilton%Jacobi%equa'on%and%2%body%problem%con'nued)%Q1= ⌧P1= µ2aP2= L =µa(1  e2)Cannonical%transforma'ons%Different%approaches%%1. Choose%desirable%genera'ng%func'ons%2. Solve%integrals%resul'ng%from%Hamilton0Jacobi%equa'on%%3. Choose%new%coordinates%and%momenta%and%show%they%sa'sfy%Poisson%brackets%4. Use%expansions%(e.g.,%Birkhoff%normal%form)%(can%lead%to%problems%with%small%divi sors)%%Example%(con'nuing%two%body%problem)%F2=(t   )g(P1)F2t= K = g(P1)⇥F2⇥  = g(P1)=p1= µ2a⇥F2⇥P1= g0(P1)(t   )=Mn = µ1/2a3/2= g0(P1)dgdP1=dgdadadP1dP1da=µ1/22a1/2dgda=µ2a2P1=pµaH =0,q1=  ,p1= µ2aDesirable%to%have%a%third%angle%as%a%coordinate%The%mean%anomaly%%M*=*n(t2τ)*K = g(P1)=µ2a= µ22P21as%expected%Harmonic%oscillator%• Use%Poisson%bracket%to%check%that%these%variables%are%cannonical%H =p22+ Kq22{x, y} =⇥x⇥⇥y⇥I⇥x⇥I⇥y⇥ =KH(I, ⇥)=Iyou%don’t%get%1%here%unless%there%is%the%factor%of%2%in%the%variable%%q =2Isin ⇥p =2I cos ⇥{q, p} =⇥2Icos ⇥22Icos ⇥ +2/2Isin ⇥2I sin ⇥=1Harmonic%oscillator%F2=p22tan ⇥F2⇥=p22sec2 = I⇥F2⇥p= p tan  = qp2(1 + tan2)=2I2I = p2+ q2p =2I cos q =2I sin pq= cot Using%a%genera'n g%fun c'on%Heliocentric%coordinates%F2(ri, Pi)=i>0(ri r0) · Pi+ r0· P0H =ip2i2mii=jGmimj2|ri rj|F2ri= Pi= pii =0F2Pi=(ri r0)=Qii ⇥=0F2P0= r0F2r0= P0i>0Pi= p0H =⇤i>0P2i2miGmim0|Qi|⇥⇤i=j i,j>0Gmimj2|ri rj|+12m0(P0⇤i>0Pi)2N0bodies%in%iner'al%frame%genera'ng%fu nc'on %new%Hamiltonian%Democra'c%heliocentric%coordinate%system%• If%P0=0%%%%%Hamiltonian%becomes%F2r0= P0i>0Pi= p0P0=pibarycenter%so%can%be%set%to%zero%This%is%equivalent%to%using%momenta%that%are%in%center%of%mass%coordinate%system%H =⇤i>0p2i2miGmim0|Qi|⇥⇤i=j i,j>0Gmimj2|ri rj|+12m0(⇤i>0Pi)2Keplerian%%term%Interac'on%term%Drie%term%Democra'c%heliocentric%coordinates%• Hamiltonian%is%nicely%separable%in%%– heliocentric%coordinates%and%%– barycentric%momenta%%H =⇤i>0P2i2miGmim0|Qi|⇥⇤i=j i,j>0Gmimj2|ri rj|+12m0(⇤i>0Pi)2Jacobi%%Coordinates%To%add%a%body:%work%with%respect%to%center%of%mass%of%all%previous%bodies.%%Coordinate%system%requires%a%tree%to%define.%%H =⇤i>0P2i2miGmim0|Qi|⇥⇤i=j i,j>0Gmimj2|ri rj|+12m0(⇤i>0Pi)2Symplec'c%Integrators%• Wisdom%and%Holman%used%a%Hamiltonian%in%Jacobi%coordinates%for%the%N0body%system%that%also%separated%into%Keplerian%and%Interac'on%terms.%• Hamiltonian%approximated%by%%– Integrated%all%bodies%with%f,g%func'ons%for%dt%=%1/Ω%%(that’s%Hkep)%– Veloci'es%given%a%kick%caused%by%Interac'ons%– Integrate%bodies%again%with%f,g%func'ons%dt%=%1/Ω%• Integrator%is%symplec'c%but%integrates%a%Hamiltonian%that%approximates%the%real%one.%• The%integrator%has%bounded%energy%error%and%allows%very%large%step%sizes%H = Hkep+ (t)Hint(Q)Second%order%Symplec'c%integrator%dzdt=zq˙q +zp˙p =zqHpzpHqdzdt= {z, H} = DHzz( ) = exp( DH)z(0)H = H0+ H1z( ) = exp[ (DH0+ DH1)]z(0)exp[ (A + B)] =ni=1exp(ci A) exp(di


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