User talk:Zero Memory

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Équation du cylindre :

Failed to parse (Cannot write to or create math output directory): x^2 + y^2 = r^2 \,



-h \leq z  \leq h  \,

Équation du rayon :


    R(t) = R_0 + t\overrightarrow{R_d} \,

Ce rayon est substitué dans l'équation du cylindre :


(x_0 + tx_d)^2 + (y_0 + ty_d)^2 = 1 \,

Une fois développé :


 A^2t + Bt + C = 0 \,

   A = x_d^2 + y_d^2 \,

  B = 2(x_0x_d + y_oy_d)\,

  C = x_0^2 +y_0^2 -r^2 \,

On résoud ainsi avec l'équation de degrée 2


  \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} \,



  z_0 + tz_d = \pm 1 \,

 x^2 + y^2 \leq r^2



\Theta_1'' = \frac{-2gsin(\Theta_1) - \Theta_2''cos(\Theta_1 - \Theta_2) - (\Theta_2')^2sin(\Theta_1 - \Theta_2)}{2}

\Theta_2'' = -gsin(\Theta_2) - \Theta_1''cos(\Theta_1 - \Theta_2) + (\Theta_1')^2sin(\Theta_1 - \Theta_2) \,

\Theta_2'' + (\frac{-2gsin(\Theta_1) - \Theta_2''cos(\Theta_1 - \Theta_2) - (\Theta_2')^2sin(\Theta_1 - \Theta_2)}{2})cos(\Theta_1 - \Theta_2) - (\Theta_1')^2sin(\Theta_1-\Theta_2) = -gsin(\Theta_2)

\Theta_2'' =  \frac{-2gsin(\Theta_1) + 2(\Theta_1')^2sin(\Theta_1 - \Theta_2) + 2gsin(\Theta_1)cos(\Theta_1-\Theta_2) + (\Theta_2')^2sin(\Theta_1-\Theta_2)cos(\Theta_1-\theta_2)}{2-(cos^2(\Theta_1-\Theta_2))}

2\Theta_1'' + (-gsin(\Theta_2) - \Theta_1''cos(\Theta_1-\Theta_2) + (\Theta_1')^2sin(\Theta_1-\Theta_2))cos(\Theta_1-\Theta_2) + (\Theta_2')^2sin(\Theta_1-\Theta_2) = -2gsin(\Theta_1)\,

\Theta_1'' = \frac{-2gsin(\Theta_1) + gsin(\Theta_2)cos(\Theta_1-\Theta_2) - (\Theta_1')^2sin(\Theta_1-\Theta_2)cos(\Theta_1-\Theta_2) - (\Theta_2')^2sin(\Theta_1-\Theta_2)}{2-cos^2(\Theta_1-\Theta_2)}

m_1 = 1 \,
m_2 = 1 \,

L_1  = 1 \,
L_2 = 1 \,

(m_1 + m_2)L_1^2 \Theta_1'' + m_2L_1L_2\Theta_2''cos(\Theta_1 - \Theta_2)
+ m_2L_1L_2(\Theta_2')^2 sin(\Theta_1-\Theta_2) = -(m_1+m_2)L_1gsin(\Theta_1)

m_2L_2^2\Theta_2'' + m_2L_1L_2\Theta_1''cos(\Theta_1-\Theta_2) -
m_2L_1L_2(\Theta_1')^2sin(\Theta_1 - \Theta_2) = -(m_2)L_2gsin(\Theta_2)



2\Theta_1'' + \Theta_2''cos(\Theta_1 - \Theta_2)
+ (\Theta_2')^2 sin(\Theta_1-\Theta_2) = -2gsin(\Theta_1) \,

\Theta_2'' + \Theta_1''cos(\Theta_1-\Theta_2) -
(\Theta_1')^2sin(\Theta_1 - \Theta_2) = -gsin(\Theta_2) \,



\Theta_1(0) = \frac{\pi}{4}

\Theta_1'(0) = 0\,

\Theta_2(0) = 0\,

\Theta_2'(0) = 0\,
\int_0^{1} \! {dx\over {1+x+x^2}}
 F(x) = \frac{2}{3} \sqrt{3} arctan (\frac{1}{3}(2x + 1)\sqrt{3} )
 F(1) - F(0) = \frac{\pi}{9} \sqrt{3}  = 0,60459978807807261686469275254739

Simpson 1/3 :

 S_i = \frac{h}{3}( f_i + 4f_{i+1} + f_{i+2} )

Simpson 3/8 :

 S_i = \frac{3h}{8} ( f_i + 3f_{i+1} + 3f_{i+2} + f_{i+3} )


 
\begin{pmatrix}
x^{[n+1]} \\
y^{[n+1]} \end{pmatrix}
=
\begin{pmatrix}
x^{[n]} \\
y^{[n]} \end{pmatrix}
+
\begin{pmatrix}
\nabla x^{[n]} \\
\nabla y^{[n]} \end{pmatrix}

\begin{pmatrix}
\nabla x^{[n]} \\
\nabla y^{[n]} \end{pmatrix}
=
-
\begin{pmatrix}
f(x^{[n]},y^{n]}) \\
g(x^{[n]},y^{[n]}) \end{pmatrix}
\begin{pmatrix}
\frac{\partial f(x^{[n]},y^{[n]})}{\partial x} && \frac{\partial f(x^{[n]},y^{[n]})}{\partial y} \\
\frac{\partial g(x^{[n]},y^{[n]})}{\partial x} && \frac{\partial g(x^{[n]},y^{[n]})}{\partial y} \end{pmatrix}^{-1}



A^{-1} = \begin{bmatrix}
a & b \\ c & d \\
\end{bmatrix}^{-1} =
\frac{1}{ad - bc} \begin{bmatrix}
d & -b \\ -c & a \\
\end{bmatrix}