User talk:Zero Memory

Équation du cylindre :

$x^{2}+y^{2}=r^{2}\,$

$-h\leq z\leq h\,$

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 :

$A^{2}t+Bt+C=0\,$

$A=x_{d}^{2}+y_{d}^{2}\,$

$B=2(x_{0}x_{d}+y_{o}y_{d})\,$

$C=x_{0}^{2}+y_{0}^{2}-r^{2}\,$

On résoud ainsi avec l'équation de degrée 2

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

$z_{0}+tz_{d}=\pm 1\,$

$x^{2}+y^{2}\leq r^{2}$

$\Theta _{1}''={\frac {-2gsin(\Theta _{1})-\Theta _{2}''cos(\Theta _{1}-\Theta _{2})-(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})}{2}}$

$\Theta _{2}''=-gsin(\Theta _{2})-\Theta _{1}''cos(\Theta _{1}-\Theta _{2})+(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})\,$

$\Theta _{2}''+({\frac {-2gsin(\Theta _{1})-\Theta _{2}''cos(\Theta _{1}-\Theta _{2})-(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})}{2}})cos(\Theta _{1}-\Theta _{2})-(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})=-gsin(\Theta _{2})$

$\Theta _{2}''={\frac {-2gsin(\Theta _{1})+2(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})+2gsin(\Theta _{1})cos(\Theta _{1}-\Theta _{2})+(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})cos(\Theta _{1}-\theta _{2})}{2-(cos^{2}(\Theta _{1}-\Theta _{2}))}}$

$2\Theta _{1}''+(-gsin(\Theta _{2})-\Theta _{1}''cos(\Theta _{1}-\Theta _{2})+(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2}))cos(\Theta _{1}-\Theta _{2})+(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})=-2gsin(\Theta _{1})\,$

$\Theta _{1}''={\frac {-2gsin(\Theta _{1})+gsin(\Theta _{2})cos(\Theta _{1}-\Theta _{2})-(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})cos(\Theta _{1}-\Theta _{2})-(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})}{2-cos^{2}(\Theta _{1}-\Theta _{2})}}$

$m_{1}=1\,m_{2}=1\,$

$L_{1}=1\,L_{2}=1\,$

$(m_{1}+m_{2})L_{1}^{2}\Theta _{1}''+m_{2}L_{1}L_{2}\Theta _{2}''cos(\Theta _{1}-\Theta _{2})+m_{2}L_{1}L_{2}(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})=-(m_{1}+m_{2})L_{1}gsin(\Theta _{1})$

$m_{2}L_{2}^{2}\Theta _{2}''+m_{2}L_{1}L_{2}\Theta _{1}''cos(\Theta _{1}-\Theta _{2})-m_{2}L_{1}L_{2}(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})=-(m_{2})L_{2}gsin(\Theta _{2})$

$2\Theta _{1}''+\Theta _{2}''cos(\Theta _{1}-\Theta _{2})+(\Theta _{2}')^{2}sin(\Theta _{1}-\Theta _{2})=-2gsin(\Theta _{1})\,$

$\Theta _{2}''+\Theta _{1}''cos(\Theta _{1}-\Theta _{2})-(\Theta _{1}')^{2}sin(\Theta _{1}-\Theta _{2})=-gsin(\Theta _{2})\,$

$\Theta _{1}(0)={\frac {\pi }{4}}$

$\Theta _{1}'(0)=0\,$

$\Theta _{2}(0)=0\,$

$\Theta _{2}'(0)=0\,$

$\int _{0}^{1}\!{dx \over {1+x+x^{2}}}$

$F(x)={\frac {2}{3}}{\sqrt {3}}arctan({\frac {1}{3}}(2x+1){\sqrt {3}})$

$F(1)-F(0)={\frac {\pi }{9}}{\sqrt {3}}=0,60459978807807261686469275254739$

Simpson 1/3 :

$S_{i}={\frac {h}{3}}(f_{i}+4f_{i+1}+f_{i+2})$

Simpson 3/8 :

$S_{i}={\frac {3h}{8}}(f_{i}+3f_{i+1}+3f_{i+2}+f_{i+3})$

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

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

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