n Body Playground

7/27/2019 n Body Playground

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nBodyPlayGround.m

A test program for the program 'nBodyWpar.m'. Before running this program

set the number of bodies 'N', the gravitational constant 'G', and the length of

the simulation 'simulationTime'. The program then generates random

masses and initial conditions and integrates the resulting system. The output

is several graphs illustrating the results. The program is fun to play with but

also demonstrates that 'nBodyWpar.m' works properly with any number of

bodies and parameter values that may occur in other work.

%N body simulation in inerital frame.

clear %clears workspace to get ready for run

%--------------------------------------------------------------------------%----------------------NOTES-----------------------------------------------%--------------------------------------------------------------------------

%USE:%User specifies the number of bodies 'N', the 'simulationTime', andthe%gravitationa constant 'G'.

%The program generates random masses and initial conditions for thebodies,%integrates them and plots the trajectories and some analysis.

%OUTPUT:%The output is several plots.

%Two three dimensional plots are given. One is a plot of all the%trajectories in the given reference frame, and the other incoordinates%with the center of mass at the orgin. Both are plots in R^3 and bothsets%of coordinates are inertial. Each trajectory has it's own coloralthough%for large N these will get harder to tell apart. Each trajectory is%marked with a circle at time zero and an astric at the final time tomark%the beginning and end of the trajectory.

%Several two dimensional plots are output as well. The most important

is

http://www.math.utexas.edu/users/jjames/MatLabCode/nBodyPlayGround.m





%the plot of the change in energy as a function of time. This is agood%indicator of how much integration errof is creeping into the problem.The%energy will usually jump when the system passes near a collision, butas

%long as the scale of the axis is less than the acceptable errortolerence%you have in mind, then the results are likely good (at least forshort%times). For example if you requair nine signigfigant figures, andthe%drift in the energy is on the order of 10^(-11), this is a goodindicator%that your requirements have been met.

%PURPOSE:%The program is fun to play with, but more importantly it tests thefile

%'nBodyWpar' (which is the N-Body vector field with parameters for G,N%and masses). The present program demonstrates that 'nBodyWpar'properly%adapts to any number of masses, with any values of the mass and%gravitational constant. It also demonstrates how to use it in aprogram.

%The code to compute the constants of motion could be put into theirown%m-files. This would greatly reduce the number of lined of code intheis%program but I wanted to keep the number of files down.

%--------------------------------------------------------------------------%----------------------USER SPECIFIED--------------------------------------%------------------------PARAMETERS----------------------------------------%--------------------------------------------------------------------------

%As the user you specify these. The Program does the rest.%Remember that the the system of ODEs is a 6*N system of first order%scalar ODEs and expect matlab to bog down accordingly.

%If G=1 then integration show interesting behavior for'simulationTime'%between 2 and 10. For such values the program run in a resonableamount%of time for 'N' between 2 and 12. (Run times often one minute ormuch%less, however if the randomly chosen initial conditions pass thesystem%near a singularity then the program will run slower even in thisrange).

%I've run the program with as many as 40 bodies on a home desk top and



%obtained results in an hour. I've had out of memory errore for morethan%45 bodies (with simulation times over 10).

N=12 %number of bodiessimulationTime=4 %How long to run the simulation

G=1 %gravitational constant

%The value of the gravitation constant in the metric system is%%G_metric=6.67300 × 10-11 (m3 kg-1 s-2)%%which is very small

%--------------------------------------------------------------------------%----------PROGRAM: mess with the following at your own risk...-----------%---------------------------------------------------------------------

-----

%----------------------DATA------------------------------------------------

%Random MassesMass =(1/G)*rand(1, N); %must be a row vector

%rand gives uniformly distributed random%values for the masses between 0 and 1.%if graviational constant is small (like%in the metric system) then multiplying

by%1/G makes the masses ''big'' enough to%interact strongly.

%Random Initial conditionsrandIc=((1)/(2*G))*randn(1, 6*N); %must be a row vector

%'randn' gives a Gaussiandistribution of

%random values centered around zero.%Then positions and velocities may

be%positive or negative.%The 1/2G gives ''big'' positions%and velocities if G is small.

There are

%6*N as each body has 3 position%components and 3 velocity

components and%there are N such bodies

%------------------INTEGRATION---------------------------------------------

%Integrate the Systemtspan = [0 simulationTime];y0=randIc;options=odeset('RelTol',1e-12,'AbsTol',1e-12);

[t,y] = ode113('nBodyWpar',tspan,y0,options,flag,N,G,Mass);



integrationDone=1

%--------Quantities neede for the constants of motion loops----------------

A=size(y) %vector containing (rows,columns)timeMax=A(1,1) %number of rows of y is number of timesteps

M=0; %initialize Mfor i=1:N

M=M+Mass(1,i); %sum of the masses is needed to%compute the constants of motion

end

%--------------------------------------------------------------------------

%-----------------Analysis of Integrals of Motion--------------------------%--------------------------------------------------------------------------

%The next section computes the 10 classical integrals of motion.

%First the components of the velociety of the center%of mass. Then the components of the position of the%center of mass (which are not constant, but linear in%time.) Then the appropriate difference which is a%constant.

%Next the three components of angular momentum are computed.

%And finally energy, moment of inertia, and it's derivatives.

%--------------------------------------------------------------------------%--------------------------------------------------------------------------%--------------------------------------------------------------------------

%Linear Momentum

%first integral; component one of V_cmI1=0;for i=1:timeMax

S=0; %Initialize Sfor j=1:N

S=S+Mass(1,j)*y(i,3*N+(3*j-2));end

I1(i)=S/M;end

%second integral; component two of V_cmI2=0;

for i=1:timeMaxS=0; %Initialize S



for j=1:NS=S+Mass(1,j)*y(i,3*N+(3*j-1));

endI2(i)=S/M;

end

%third integral; component three of V_cmI3=0;for i=1:timeMax


S=S+Mass(1,j)*y(i,3*N+(3*j));end

I3(i)=S/M;end

%Center of Mass as a function of time; component one of R_cm(t)CM1=0;for i=1:timeMax

S=0; %Initialize Sfor j=1:NS=S+Mass(1,j)*y(i,3*j-2);

endCM1(i)=S/M;

end

%Center of Mass as a function of time; component two of R_cm(t)CM2=0;for i=1:timeMax


S=S+Mass(1,j)*y(i,3*j-1);end

CM2(i)=S/M;end

%Center of Mass as a function of time; component three of R_cm(t)CM3=0;for i=1:timeMax


S=S+Mass(1,j)*y(i,3*j);end

CM3(i)=S/M;end

%Fourth integral: component one of R_cm(t)-[V_cm(t)]*tI4=0;for i=1:timeMax


S=CM1(i)-I1(i)*t(i);end

I4(i)=S;end

%Fifth integral: component two of R_cm(t)-[V_cm(t)]*tI5=0;for i=1:timeMax




S=CM2(i)-I2(i)*t(i);end

I5(i)=S;end

%Sixth integral: component three of R_cm(t)-[V_cm(t)]*t

I6=0;for i=1:timeMaxS=0; %Initialize S

for j=1:NS=CM3(i)-I3(i)*t(i);

endI6(i)=S;

end

%Angular Momentum

%seventh integral; component one of h

I7=0;for i=1:timeMaxS=0; %Initialize S

for j=1:NS=S+Mass(1,j)*[y(i,3*j-1)*y(i,3*N+3*j)-

y(i,3*j)*y(i,3*N+3*j-1)];end

I7(i)=S/M;end

%eighth integral; component two of hI8=0;for i=1:timeMax


S=S+Mass(1,j)*[y(i,3*j)*y(i,3*N+3*j-2)-y(i,3*j-2)*y(i,3*N+3*j)];

endI8(i)=S/M;

end

%nineth integral; component three of hI9=0;for i=1:timeMax


S=S+Mass(1,j)*[y(i,3*j-2)*y(i,3*N+3*j-1)-y(i,3*j-1)*y(i,3*N+3*j-2)];end

I9(i)=S/M;end

%Energy: (T-U)(t)

%Potential Energy: U(t)U=0;for i=1:timeMax


s=0;

for k=1:N



Rjk=(y(i,3*j-2)-y(i,3*k-2))^2+(y(i,3*j-1)-y(i,3*k-1))^2+(y(i,3*j)-y(i,3*k))^2;

if k~=js=s+Mass(1,k)/[sqrt(Rjk)];

elses=s+0;

endendS=S+Mass(1,j)*s;

endU(i)=(G/2)*S;

end

%Kenetic Energy; T(t)T=0;for i=1:timeMax


S=S+Mass(1,j)*[(y(i,3*N+(3*j-2)))^2+(y(i,3*N+(3*j-1)))^2+(y(i,3*N+(3*j)))^2];

endT(i)=S/2;

end

%Tenth integral; Energy E(t)E=0;for i=1:timeMax

E(i)=T(i)-U(i);end

%Aux Quanties

%I or moment of ineritaI=0;for i=1:timeMax


S=S+Mass(1,j)*[(y(i,3*j-2))^2+(y(i,3*j-1))^2+(y(i,3*j))^2];endI(i)=S;

end

%d/dt(I)Idot=0;for i=1:timeMax


S=S+Mass(1,j)*[y(i,3*N+(3*j-2))*y(i,3*j-2)+y(i,3*N+(3*j-1))*y(i,3*j-1)+y(i,3*N+(3*j))*y(i,3*j)];

endIdot(i)=S*2;

end

%d^2/(dt)^2(I)

Idotdot=0;for i=1:timeMax



Idotdot(i)=2*T(i)+2*E(i);end

%Change in Energy. If integration is perfect then this is alwayszero.%Its deviation from zero measures the solutions deviation from the

real%one.

%deltaEdeltaE=0;deltaE(1)=0;for i=2:timeMax

deltaE(i)=E(i)-E(1);end

%--------------------------------------------------------------------------

%--------------------------------Plotting----------------------------------%--------------------------------------------------------------------------

%COLORING:%Specify that the first six colors used will be red, blue, green,black,%yellow, and magenta.colorMatrix(1:6, 1:3)=[1, 0, 0;

0, 0, 1;0, 1, 0;0, 0, 0;1, 1, 0;1, 0, 1];

%For maore than six bodies the plot will be a mess anyway, but fill inthe%rest of the colors with the 'hot' commandcolorMatrix(7:N+2, 1:3)=hot((N+2-6));

%Plotting observables

plot(t(1:timeMax),I5(1:timeMax))grid ontitle('Moment of Inertia vs. time')xlabel('time axis'), ylabel ('I')

figureplot(t(1:timeMax),E(1:timeMax))grid ontitle('Energy vs. Time')xlabel('time axis'), ylabel ('Energy')

figureplot3(CM1(:,1), CM2(:,2), CM3(:,3))grid ontitle('Path of the center of Mass')

xlabel('x coor CM'), ylabel('y coor CM'), zlabel('z coor CM')



figureplot(t(1:timeMax), deltaE(1:timeMax))grid ontitle('delta energy as a function of time')xlabel('time'), ylabel('delta energy')

%PLOTS OF TRAJECTORIESfigurehold on

for k=1:Nplot3(y(:, 3*k-2), y(:, 3*k-1), y(:, 3*k), 'Color',

colorMatrix(k,:))plot3(y(1, 3*k-2), y(1, 3*k-1), y(1, 3*k), 'o' ,'Color',

colorMatrix(k,:))plot3(y(timeMax, 3*k-2), y(timeMax, 3*k-1), y(timeMax, 3*k),

'*','Color', colorMatrix(k,:))end

grid ontitle('The trajectories of the bodies (o is start, * is end)')xlabel('x axis'), ylabel('y axis'), zlabel('z axis')

figureCM= [CM1' CM2' CM3'];hold onfor k=1:Nplot3(y(:, 3*k-2)-CM(:,1), y(:, 3*k-1)-CM(:,2), y(:, 3*k)-

CM(:,3),'Color', colorMatrix(k,:))plot3(y(1, 3*k-2)-CM(1,1), y(1, 3*k-1)-CM(1,2), y(1, 3*k)-CM(1,3),

'o' ,'Color', colorMatrix(k,:))plot3(y(timeMax, 3*k-2)-CM(timeMax,1), y(timeMax, 3*k-1)-

CM(timeMax,2), y(timeMax, 3*k)-CM(timeMax,3), '*' ,'Color',colorMatrix(k,:))end

grid ontitle('The trajectories of the bodies in center of mass coordinates (ois start, * is end)')xlabel('x axis'), ylabel('y axis'), zlabel('z axis')

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