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The
N
{\displaystyle {\displaystyle N}}
-body problem is an inherent feature of a physical description of mechanical system of 

N
{\displaystyle {\displaystyle N}}
 interacting objects.
The system is described by Newton's Law

:   F
    →
    =

    m

    i

    d

    2

    r
    →

    i

    d

    t

    2
    ,
    i
    =
    1..
    N
    {\displaystyle {\displaystyle {\vec {F}}=m\_{i}{\frac {d^{2}{\vec {r}}\_{i}}{dt^{2}}},i=1..N}}

with continuous combined force from other bodies

:   F

    i
    =

    ∑

    j
    ≠
    i


    F

    i
    j
    ,
    i
    =
    1..
    N
    {\displaystyle {\displaystyle F\_{i}=\sum \_{j\neq i}^{}F\_{ij},i=1..N}}

Exact formulation of first mechanical problem is, given initial coordinates and velocities 

t
=
0
{\displaystyle {\displaystyle t=0}}
, to find coordinates and velocities at 

t
=

t

0
{\displaystyle {\displaystyle t=t\_{0}}}
 with accuracy more then given relative value 

ϵ
=

r
→
−

r
→
′


|

r
→

|
{\displaystyle {\displaystyle \epsilon ={\frac {{\vec {r}}-{\vec {r}}'}{|{\vec {r}}|}}}}

As well known from physical background, only the choice of 

N
=
2
{\displaystyle {\displaystyle N=2}}
 can be analytically solved.

Task

Write a simulation of three masses interacting under gravitation and show their evolution over a number of time-steps (at least 20).
assistant
(formerly Perl 6)

We'll try to simulate the Sun+Earth+Moon system, with plausible astrometric values.

We use a 18-dimension vector 

A
B
C
{\displaystyle {\displaystyle ABC}}
. The first nine dimensions are the positions of the three bodies. The other nine are the velocities. This allows us to write the dynamics as a first-temporal derivative equation, since

d

p
o
s
i
t
i
o
n

d
t
=

s
p
e
e
d
{\displaystyle {\displaystyle {\frac {d\mathrm {position} }{dt}}=\mathrm {speed} }}

and thus

d
A
B

C

1..9

d
t
=
A
B

C

10..18
{\displaystyle {\displaystyle {\frac {dABC\_{1..9}}{dt}}=ABC\_{10..18}}}

To keep things compact, we'll only display the first 20 lines of output.

```
# Simple Vector implementation
multi infix:<+>(@a, @b) { @a Z+ @b }
multi infix:<->(@a, @b) { @a Z- @b }
multi infix:<*>($r, @a) { $r X* @a }
multi infix:</>(@a, $r) { @a X/ $r }
sub norm { sqrt [+] @_ X** 2 }
 
# Runge-Kutta stuff
sub runge-kutta(&yp) {
    return -> \t, \y, \δt {
        my $a = δt * yp( t, y );
        my $b = δt * yp( t + δt/2, y + $a/2 );
        my $c = δt * yp( t + δt/2, y + $b/2 );
        my $d = δt * yp( t + δt, y + $c );
        ($a + 2*($b + $c) + $d) / 6;
    }
}
 
# gravitational constant
constant G = 6.674e-11;
# astronomical unit
constant au = 150e9;
 
# time constants in seconds
constant year = 365.25*24*60*60;
constant month = 21*24*60*60;
 
# masses in kg
constant $ma = 2e30;     # Sun
constant $mb = 6e24;     # Earth
constant $mc = 7.34e22;  # Moon

my &dABC = runge-kutta my &f = sub ( $t, @ABC ) {
    my @a = @ABC[0..2];
    my @b = @ABC[3..5];
    my @c = @ABC[6..8];
 
    my $ab = norm(@a - @b); 
    my $ac = norm(@a - @c);
    my $bc = norm(@b - @c);
 
    return [
        flat
        @ABC[@(9..17)],
        map G * *,
        $mb/$ab**3 * (@b - @a) + $mc/$ac**3 * (@c - @a),
        $ma/$ab**3 * (@a - @b) + $mc/$bc**3 * (@c - @b),
        $ma/$ac**3 * (@a - @c) + $mb/$bc**3 * (@b - @c);
    ];
}
 
loop (
    my ($t, @ABC) = 0,
        0, 0, 0,                                 # Sun position
        au, 0, 0,                                # Earth position
        0.998*au, 0, 0,                          # Moon position
        0, 0, 0,                                 # Sun speed
        0, 2*pi*au/year, 0,                      # Earth speed
        0, 2*pi*(au/year + 0.002*au/month), 0    # Moon speed
    ;
    $t < .2;
    ($t, @ABC) »+=« (.01, dABC($t, @ABC, .01))
) {
    printf "t = %.02f : %s\n", $t, @ABC.fmt("%+.3e");
}

```

Output:

```
t = 0.00 : +0.000e+00 +0.000e+00 +0.000e+00 +1.500e+11 +0.000e+00 +0.000e+00 +1.497e+11 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +2.987e+04 +0.000e+00 +0.000e+00 +3.090e+04 +0.000e+00
t = 0.01 : +9.008e-13 +5.981e-22 +0.000e+00 +1.500e+11 +2.987e+02 +0.000e+00 +1.497e+11 +3.090e+02 +0.000e+00 +1.802e-10 +1.794e-19 +0.000e+00 -5.987e-05 +2.987e+04 +0.000e+00 -1.507e-05 +3.090e+04 +0.000e+00
t = 0.02 : +3.603e-12 +4.785e-21 +0.000e+00 +1.500e+11 +5.973e+02 +0.000e+00 +1.497e+11 +6.181e+02 +0.000e+00 +3.603e-10 +7.177e-19 +0.000e+00 -1.197e-04 +2.987e+04 +0.000e+00 -3.014e-05 +3.090e+04 +0.000e+00
t = 0.03 : +8.107e-12 +1.615e-20 +0.000e+00 +1.500e+11 +8.960e+02 +0.000e+00 +1.497e+11 +9.271e+02 +0.000e+00 +5.405e-10 +1.615e-18 +0.000e+00 -1.796e-04 +2.987e+04 +0.000e+00 -4.521e-05 +3.090e+04 +0.000e+00
t = 0.04 : +1.441e-11 +3.828e-20 +0.000e+00 +1.500e+11 +1.195e+03 +0.000e+00 +1.497e+11 +1.236e+03 +0.000e+00 +7.206e-10 +2.871e-18 +0.000e+00 -2.395e-04 +2.987e+04 +0.000e+00 -6.028e-05 +3.090e+04 +0.000e+00
t = 0.05 : +2.252e-11 +7.476e-20 +0.000e+00 +1.500e+11 +1.493e+03 +0.000e+00 +1.497e+11 +1.545e+03 +0.000e+00 +9.008e-10 +4.486e-18 +0.000e+00 -2.993e-04 +2.987e+04 +0.000e+00 -7.535e-05 +3.090e+04 +0.000e+00
t = 0.06 : +3.243e-11 +1.292e-19 +0.000e+00 +1.500e+11 +1.792e+03 +0.000e+00 +1.497e+11 +1.854e+03 +0.000e+00 +1.081e-09 +6.460e-18 +0.000e+00 -3.592e-04 +2.987e+04 +0.000e+00 -9.041e-05 +3.090e+04 +0.000e+00
t = 0.07 : +4.414e-11 +2.051e-19 +0.000e+00 +1.500e+11 +2.091e+03 +0.000e+00 +1.497e+11 +2.163e+03 +0.000e+00 +1.261e-09 +8.792e-18 +0.000e+00 -4.191e-04 +2.987e+04 +0.000e+00 -1.055e-04 +3.090e+04 +0.000e+00
t = 0.08 : +5.765e-11 +3.062e-19 +0.000e+00 +1.500e+11 +2.389e+03 +0.000e+00 +1.497e+11 +2.472e+03 +0.000e+00 +1.441e-09 +1.148e-17 +0.000e+00 -4.789e-04 +2.987e+04 +0.000e+00 -1.206e-04 +3.090e+04 +0.000e+00
t = 0.09 : +7.296e-11 +4.360e-19 +0.000e+00 +1.500e+11 +2.688e+03 +0.000e+00 +1.497e+11 +2.781e+03 +0.000e+00 +1.621e-09 +1.453e-17 +0.000e+00 -5.388e-04 +2.987e+04 +0.000e+00 -1.356e-04 +3.090e+04 +0.000e+00
t = 0.10 : +9.008e-11 +5.981e-19 +0.000e+00 +1.500e+11 +2.987e+03 +0.000e+00 +1.497e+11 +3.090e+03 +0.000e+00 +1.802e-09 +1.794e-17 +0.000e+00 -5.987e-04 +2.987e+04 +0.000e+00 -1.507e-04 +3.090e+04 +0.000e+00
t = 0.11 : +1.090e-10 +7.961e-19 +0.000e+00 +1.500e+11 +3.285e+03 +0.000e+00 +1.497e+11 +3.399e+03 +0.000e+00 +1.982e-09 +2.171e-17 +0.000e+00 -6.586e-04 +2.987e+04 +0.000e+00 -1.658e-04 +3.090e+04 +0.000e+00
t = 0.12 : +1.297e-10 +1.034e-18 +0.000e+00 +1.500e+11 +3.584e+03 +0.000e+00 +1.497e+11 +3.709e+03 +0.000e+00 +2.162e-09 +2.584e-17 +0.000e+00 -7.184e-04 +2.987e+04 +0.000e+00 -1.808e-04 +3.090e+04 +0.000e+00
t = 0.13 : +1.522e-10 +1.314e-18 +0.000e+00 +1.500e+11 +3.882e+03 +0.000e+00 +1.497e+11 +4.018e+03 +0.000e+00 +2.342e-09 +3.032e-17 +0.000e+00 -7.783e-04 +2.987e+04 +0.000e+00 -1.959e-04 +3.090e+04 +0.000e+00
t = 0.14 : +1.766e-10 +1.641e-18 +0.000e+00 +1.500e+11 +4.181e+03 +0.000e+00 +1.497e+11 +4.327e+03 +0.000e+00 +2.522e-09 +3.517e-17 +0.000e+00 -8.382e-04 +2.987e+04 +0.000e+00 -2.110e-04 +3.090e+04 +0.000e+00
t = 0.15 : +2.027e-10 +2.019e-18 +0.000e+00 +1.500e+11 +4.480e+03 +0.000e+00 +1.497e+11 +4.636e+03 +0.000e+00 +2.702e-09 +4.037e-17 +0.000e+00 -8.980e-04 +2.987e+04 +0.000e+00 -2.260e-04 +3.090e+04 +0.000e+00
t = 0.16 : +2.306e-10 +2.450e-18 +0.000e+00 +1.500e+11 +4.778e+03 +0.000e+00 +1.497e+11 +4.945e+03 +0.000e+00 +2.883e-09 +4.593e-17 +0.000e+00 -9.579e-04 +2.987e+04 +0.000e+00 -2.411e-04 +3.090e+04 +0.000e+00
t = 0.17 : +2.603e-10 +2.938e-18 +0.000e+00 +1.500e+11 +5.077e+03 +0.000e+00 +1.497e+11 +5.254e+03 +0.000e+00 +3.063e-09 +5.186e-17 +0.000e+00 -1.018e-03 +2.987e+04 +0.000e+00 -2.562e-04 +3.090e+04 +0.000e+00
t = 0.18 : +2.919e-10 +3.488e-18 +0.000e+00 +1.500e+11 +5.376e+03 +0.000e+00 +1.497e+11 +5.563e+03 +0.000e+00 +3.243e-09 +5.814e-17 +0.000e+00 -1.078e-03 +2.987e+04 +0.000e+00 -2.712e-04 +3.090e+04 +0.000e+00
t = 0.19 : +3.252e-10 +4.102e-18 +0.000e+00 +1.500e+11 +5.674e+03 +0.000e+00 +1.497e+11 +5.872e+03 +0.000e+00 +3.423e-09 +6.477e-17 +0.000e+00 -1.138e-03 +2.987e+04 +0.000e+00 -2.863e-04 +3.090e+04 +0.000e+00
```