n-body problem

$$F=ma$$ $$a=\frac{F}{m}$$ $$F=\gamma \cdot \frac{m_1{m}_2}{r_{12}^2}$$ $$a_1=\gamma \cdot \frac{\frac{m_1{m}_2}{r_{12}^2}}{m_1}=\gamma \cdot \frac{m_2}{r_{12}^2}$$

Ez csak a nagysaga a gyorsulasnak, de $a$ egy vektor, ezert ki kell szamolnunk az iranyat is. A gyorsulas iranya az ero iranyaval egyezik meg, ami a masik test fele nez. Tehat az irany

$$\frac{{\vec{x}}_j-{\vec{x}}_i}{\Vert {\vec{x}}_j-{\vec{x}}_i\Vert }$$

Tehat a gyorsulas vektora a kovetkezo:

$$a_i=m_j\frac{{\vec{x}}_j-{\vec{x}}_i}{\Vert {\vec{x}}_j-{\vec{x}}_i{\Vert }^3}$$

Az n-test vonzomozgasat leiro differencial egyenlet a kovetkezo:

$$x_i^{\prime \prime }(t)=\sum _{i\ne j}{m}_j\frac{\overrightarrow{x_j}-\overrightarrow{x_i}}{\Vert \overrightarrow{x_j}-\overrightarrow{x_i}{\Vert }^3}$$

A fenti masodrendu differencial egyenletet at lehet irni kett elso rendu diff egyenletkent a kovetkezo modon:

$$\begin{array}{rl}{x}_i^{\prime }(t)& =v_i(t)\\ v_i^{\prime }(t)& =\sum _{i\ne j}{m}_j\frac{\overrightarrow{x_j}-\overrightarrow{x_i}}{\Vert \overrightarrow{x_j}-\overrightarrow{x_i}{\Vert }^3}\end{array}$$

Spec. eset: n = 3

$$\begin{array}{rlrlr}{u}_1& =x_1,& u_2=x_1^{\prime },& u_3=y_1,& u_4=y_1^{\prime }\\ u_5& =x_2,& u_6=x_2^{\prime },& u_7=y_2,& u_8=y_2^{\prime }\\ u_9& =x_3,& u_{10}=x_3^{\prime },& u_{11}=y_3,& u_{12}=y_3^{\prime }\end{array}$$ $$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_1^{\prime }\\ y_1\\ y_1^{\prime }\end{array}\right]$$

differencial egyenlet:
elso test

$$\begin{array}{rl}{p}_{1x}^{\prime }& =v_{1x}\\ v_{1x}^{\prime }& =m_2\frac{p_{2x}-p_{1x}}{r_{21}^3}+m_3\frac{p_{3x}-p_{1x}}{r_{31}^3}\\ p_{1y}^{\prime }& =v_{1y}\\ v_{1y}^{\prime }& =m_2\frac{p_{2y}-p_{1y}}{r_{21}^3}+m_3\frac{p_{3y}-p_{1y}}{r_{31}^3}\end{array}$$

masodik test

$$\begin{array}{rl}{p}_{2x}^{\prime }& =v_{2x}\\ v_{2x}^{\prime }& =m_1\frac{p_{1x}-p_{2x}}{r_{21}^3}+m_3\frac{p_{3x}-p_{2x}}{r_{32}^3}\\ p_{2y}^{\prime }& =v_{2y}\\ v_{2y}^{\prime }& =m_1\frac{p_{1y}-p_{2y}}{r_{21}^3}+m_3\frac{p_{3y}-p_{2y}}{r_{32}^3}\end{array}$$

harmadik test

$$\begin{array}{rl}{p}_{3x}^{\prime }& =v_{3x}\\ v_{3x}^{\prime }& =m_1\frac{p_{1x}-p_{3x}}{r_{31}^3}+m_2\frac{p_{2x}-p_{3x}}{r_{32}^3}\\ p_{3y}^{\prime }& =v_{3y}\\ v_{3y}^{\prime }& =m_1\frac{p_{1y}-p_{3y}}{r_{31}^3}+m_2\frac{p_{2y}-p_{3y}}{r_{32}^3}\end{array}$$

a vektormezot leiro matrix:

$$\left[\begin{array}{cccccccccccc}0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ a& & & & & & & & & & & \\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ a& & & & & & & & & & & \\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ a& & & & & & & & & & & \\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ a& & & & & & & & & & & \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ a& & & & & & & & & & & \\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$