C++模擬三體運動

受力計算

對于三個物體 $b_1$, $b_2$, $b_3$, 其質量為 $m_1$, $m_2$, $m_3$, $t$時坐標為 $p_1$, $p_2$, $p_3$, 加在$b_1$上的力為 $F=\frac{G{m}_1{m}_2}{|p_1?p_2{|}^2}?(p_2?p_1)+\frac{G{m}_1{m}_3}{|p_1?p_3{|}^2}?(p_3?p_1)$.

其$x$ 分量為$F_x=\frac{G{m}_1{m}_2}{|p_1?p_2{|}^2}?\frac{p_{1x}?p_{2x}}{|p_1?p_2|}$. 對于其$y$, $z$ 分量同理.

因此, 可用以下函數計算$b_1$受到的重力:

Vector g_force(float m1, Vector p1, float m2, Vector p2, float m3, Vector p3) {float r = sqrt((p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y) + (p1.z - p2.z) * (p1.z - p2.z));Vector force;float f = G * (m1 * m2) / (r * r);force.x = f / r * (p2.x - p1.x);force.y = f / r * (p2.y - p1.y);force.z = f / r * (p2.z - p1.z);r = sqrt((p1.x - p3.x) * (p1.x - p3.x) + (p1.y - p3.y) * (p1.y - p3.y) + (p1.z - p3.z) * (p1.z - p3.z));f = G * (m1 * m3) / (r * r);force.x += f / r * (p3.x - p1.x);force.y += f / r * (p3.y - p1.y);force.z += f / r * (p3.z - p1.z);return force; }

運動計算

根據受力可以算出加速度, 乘上$\Delta t$得到速度增量;
速度乘上$\Delta t$得到坐標增量.

void Body::move(Vector force, float time) {Vector accel;accel.x = force.x / _mass;accel.y = force.y / _mass;accel.z = force.z / _mass;_speed.x += accel.x * time;_speed.y += accel.y * time;_speed.z += accel.z * time;_coordinate.x += _speed.x * time;_coordinate.y += _speed.y * time;_coordinate.z += _speed.z * time; }

數值模擬

對每一個body執行move, 用很小的時間步長, 和實時的重力, 并且輸出結果.

for (int i = 0; i < 100; i++){b1.move(g_force(m1, b1.get_coord(), m2, b2.get_coord(), m3, b3.get_coord()), 0.1);b2.move(g_force(m2, b2.get_coord(), m1, b1.get_coord(), m3, b3.get_coord()), 0.1);b3.move(g_force(m3, b3.get_coord(), m2, b2.get_coord(), m1, b1.get_coord()), 0.1);printf("Time: %f\n", (float)i / 10.0);output_vector(b1.get_coord(), "b1 coord");output_vector(b2.get_coord(), "b2 coord");output_vector(b3.get_coord(), "b3 coord");printf("\n");}

全部代碼

body.h

#include <cstdio> #include <cstdlib>struct Vector {float x;float y;float z; };class Body { private:Vector _coordinate;Vector _speed;float _mass; public:Body(float mass, Vector coord);void move(Vector force, float time);Vector get_speed();Vector get_coord(); };

body.cpp

#include "body.h"Body::Body(float mass, Vector coord) {_coordinate = coord;_mass = mass;_speed.x = 0;_speed.y = 0;_speed.z = 0; }void Body::move(Vector force, float time) {Vector accel;accel.x = force.x / _mass;accel.y = force.y / _mass;accel.z = force.z / _mass;_speed.x += accel.x * time;_speed.y += accel.y * time;_speed.z += accel.z * time;_coordinate.x += _speed.x * time;_coordinate.y += _speed.y * time;_coordinate.z += _speed.z * time; }Vector Body::get_speed() {return _speed; }Vector Body::get_coord() {return _coordinate; }

main.cpp

#include <cstdio> #include <cstdlib> #include <cmath>#include "body.h"const float G = 6.67e-11;void output_vector(Vector vec, char *comment) {printf("%s: ", comment);printf("x: %f, y: %f, z: %f\n", vec.x, vec.y, vec.z); }Vector g_force(float m1, Vector p1, float m2, Vector p2, float m3, Vector p3) {float r = sqrt((p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y) + (p1.z - p2.z) * (p1.z - p2.z));Vector force;float f = G * (m1 * m2) / (r * r);force.x = f / r * (p2.x - p1.x);force.y = f / r * (p2.y - p1.y);force.z = f / r * (p2.z - p1.z);r = sqrt((p1.x - p3.x) * (p1.x - p3.x) + (p1.y - p3.y) * (p1.y - p3.y) + (p1.z - p3.z) * (p1.z - p3.z));f = G * (m1 * m3) / (r * r);force.x += f / r * (p3.x - p1.x);force.y += f / r * (p3.y - p1.y);force.z += f / r * (p3.z - p1.z);return force; }int main() {float m1 = 1e10;float m2 = 1e10;float m3 = 1e10;Vector p1;p1.x = 0;p1.y = 1;p1.z = 0;Body b1 = Body(m1, p1);Vector p2;p2.x = 0.866;p2.y = -0.5;p2.z = 0;Body b2 = Body(m2, p2);Vector p3;p3.x = -0.866;p3.y = -0.5;p3.z = 0;Body b3 = Body(m3, p3);for (int i = 0; i < 100; i++){b1.move(g_force(m1, b1.get_coord(), m2, b2.get_coord(), m3, b3.get_coord()), 0.1);b2.move(g_force(m2, b2.get_coord(), m1, b1.get_coord(), m3, b3.get_coord()), 0.1);b3.move(g_force(m3, b3.get_coord(), m2, b2.get_coord(), m1, b1.get_coord()), 0.1);printf("Time: %f\n", (float)i / 10.0);output_vector(b1.get_coord(), "b1 coord");output_vector(b2.get_coord(), "b2 coord");output_vector(b3.get_coord(), "b3 coord");printf("\n");}return 0; }

模擬結果

Time: 0.000000 b1 coord: x: 0.000000, y: 0.996149, z: 0.000000 b2 coord: x: 0.862658, y: -0.498068, z: 0.000000 b3 coord: x: -0.862650, y: -0.498066, z: 0.000000Time: 0.100000 b1 coord: x: -0.000000, y: 0.988417, z: 0.000000 b2 coord: x: 0.855943, y: -0.494184, z: 0.000000 b3 coord: x: -0.855908, y: -0.494173, z: 0.000000Time: 0.200000 b1 coord: x: -0.000000, y: 0.976743, z: 0.000000 b2 coord: x: 0.845793, y: -0.488308, z: 0.000000 b3 coord: x: -0.845705, y: -0.488283, z: 0.000000Time: 0.300000 b1 coord: x: -0.000000, y: 0.961032, z: 0.000000 b2 coord: x: 0.832117, y: -0.480387, z: 0.000000 b3 coord: x: -0.831938, y: -0.480335, z: 0.000000Time: 0.400000 b1 coord: x: -0.000000, y: 0.941150, z: 0.000000 b2 coord: x: 0.814792, y: -0.470345, z: 0.000000 b3 coord: x: -0.814470, y: -0.470250, z: 0.000000Time: 0.500000 b1 coord: x: -0.000000, y: 0.916917, z: 0.000000 b2 coord: x: 0.793648, y: -0.458080, z: 0.000000 b3 coord: x: -0.793117, y: -0.457922, z: 0.000000Time: 0.600000 b1 coord: x: 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可視化: 略.

結論

通過數值模擬, 可以得出三體運動是混沌的.

總結

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