Page de test mathjax "HTML+CSS"

Cette page utilise MathJax avec une sortie HTML+CSS des formules, ce document long contient une liste de formules de tests, parfois des cas d'école, parfois des données réeles... Elle peut être longue à afficher.

INPG

Petit échantillon extrait de formules produites avec mathtype par l'INPG

$ G = 20\log (T) = 20\log (\frac{1}{{\sqrt {1 + \left( {\frac{\omega }{{\omega _0 }}} \right)^2 } }}) $
$G \approx - 10\log (\left( {\frac{\omega }{{\omega _0 }}} \right)^2 ) = 20\log (\omega _0 ) - 20\log (\omega )$
$\mathop {\lim }\limits_{\omega \to \infty } \phi = - \tan ^{ - 1} (\infty ) = - 90^\circ$
$\underline T = \prod\limits_{i = 1}^N {\underline T _i }$
$G = 20\log (\left| {A_0 } \right|)$
$\underline T = \frac{{100}}{{\left( {1 + j\frac{\omega }{{\omega _1 }}} \right)\left( {1 + j\frac{\omega }{{\omega _2 }}} \right)}}$
$ \underline T = \underline T _0 \underline T _1 \underline T _2 $

wikipedia

Page Displaying_a_formula (liste de symboles et formules)

$ \nabla \, \partial x \, dx \, \dot x \, \ddot y\, dy/dx\, \frac{dy}{dx}\, \frac{\partial^2 y}{\partial x_1\,\partial x_2}$
$\cap \bigcap \cup \bigcup \biguplus \setminus \smallsetminus$
$\forall \exists \empty \emptyset \varnothing$
$\times \otimes \bigotimes \cdot \circ \bullet \bigodot$
$\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup$
$\star * / \div \frac{1}{2}$
$\land (or \and) \wedge \bigwedge \bar{q} \to p$
$\lor \vee \bigvee \lnot \neg q \And$
$\sqrt{2} \sqrt[n]{x}$
$\sim \approx \simeq \cong \dot= \overset{\underset{\mathrm{def}}{}}{=}$
$\le < \ll \gg \ge > \equiv \not\equiv \ne \mbox{or} \neq \propto$
$\Diamond \Box \triangle \angle \perp \| 45^\circ$
$\leftarrow (or \gets) \rightarrow (or \to) \nleftarrow \nrightarrow \leftrightarrow \nleftrightarrow \longleftarrow \longrightarrow \longleftrightarrow$
$\Leftarrow \Rightarrow \nLeftarrow \nRightarrow \Leftrightarrow \nLeftrightarrow \Longleftarrow \Longrightarrow \Longleftrightarrow (or \iff)$
$\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow \nearrow \searrow \swarrow \nwarrow$
$\rightharpoonup \rightharpoondown \leftharpoonup \leftharpoondown \upharpoonleft \upharpoonright \downharpoonleft \downharpoonright \rightleftharpoons \leftrightharpoons$
$\curvearrowleft \circlearrowleft \Lsh \upuparrows \rightrightarrows \rightleftarrows \Rrightarrow \rightarrowtail \looparrowright$
$\curvearrowright \circlearrowright \Rsh \downdownarrows \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail \looparrowleft$
$\mapsto \longmapsto \hookrightarrow \hookleftarrow \multimap \leftrightsquigarrow \rightsquigarrow \twoheadrightarrow \twoheadleftarrow$
$\And \eth \S \P \% \dagger \ddagger \ldots \cdots$
$\smile \frown \wr \triangleleft \triangleright \infty \bot \top$
$\vdash \vDash \Vdash \models \lVert \rVert \imath \hbar$
$\ell \mho \Finv \Re \Im \wp \complement$
$\diamondsuit \heartsuit \clubsuit \spadesuit \Game \flat \natural \sharp$
$\vartriangle \triangledown \lozenge \circledS \measuredangle \nexists \Bbbk \backprime \blacktriangle \blacktriangledown$
$\blacksquare \blacklozenge \bigstar \sphericalangle \diagup \diagdown \dotplus \Cap \Cup \barwedge$
$\veebar \doublebarwedge \boxminus \boxtimes \boxdot \boxplus \divideontimes \ltimes \rtimes \leftthreetimes$
$\rightthreetimes \curlywedge \curlyvee \circleddash \circledast \circledcirc \centerdot \intercal \leqq \leqslant$
$\eqslantless \lessapprox \approxeq \lessdot \lll \lessgtr \lesseqgtr \lesseqqgtr \doteqdot \risingdotseq$
$\fallingdotseq \backsim \backsimeq \subseteqq \Subset \preccurlyeq \curlyeqprec \precsim \precapprox \vartriangleleft$
$\Vvdash \bumpeq \Bumpeq \geqq \geqslant \eqslantgtr \gtrsim \gtrapprox \eqsim \gtrdot$
$\ggg \gtrless \gtreqless \gtreqqless \eqcirc \circeq \triangleq \thicksim \thickapprox \supseteqq$
$\Supset \succcurlyeq \curlyeqsucc \succsim \succapprox \vartriangleright \shortmid \shortparallel \between \pitchfork$
$\varpropto \blacktriangleleft \therefore \backepsilon \blacktriangleright \because \nleqslant \nleqq \lneq \lneqq$
$\lvertneqq \lnsim \lnapprox \nprec \npreceq \precneqq \precnsim \precnapprox \nsim \nshortmid$
$\nvdash \nVdash \ntriangleleft \ntrianglelefteq \nsubseteq \nsubseteqq \varsubsetneq \subsetneqq \varsubsetneqq \ngtr$
$\subsetneq$
$\ngeqslant \ngeqq \gneq \gneqq \gvertneqq \gnsim \gnapprox \nsucc \nsucceq \succneqq$
$\succnsim \succnapprox \ncong \nshortparallel \nparallel \nvDash \nVDash \ntriangleright \ntrianglerighteq \nsupseteq$
$\nsupseteqq \varsupsetneq \supsetneqq \varsupsetneqq$
$\jmath \surd \ast \uplus \diamond \bigtriangleup \bigtriangledown \ominus$
$\oslash \odot \bigcirc \amalg \prec \succ \preceq \succeq$
$\dashv \asymp \doteq \parallel$
$\ulcorner \urcorner \llcorner \lrcorner$
$a^2$
$ a_2$
$a^{2+2}$
$a_{i,j}$
$x_2^3$
${x_2}^3$
$10^{10^{ \,\!{8} }$
$10^{10^{ \overset{8}{} }}$
$10^{10^8}$
$\sideset{_1^2}{_3^4}\prod_a^b$
${}_1^2\!\Omega_3^4$
$\overset{\alpha}{\omega}$
$\underset{\alpha}{\omega}$
$\overset{\alpha}{\underset{\gamma}{\omega}}$
$\stackrel{\alpha}{\omega}$
$ x', y'', f', f''\!$
$x', y'', f', f''$
$x^\prime, y^{\prime\prime}$
$x\prime, y\prime\prime$
$\dot{x}, \ddot{x}$
$ \hat a \ \bar b \ \vec c$
$\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}$
$\overline{g h i} \ \underline{j k l}$
$A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C$
$\overbrace{ 1+2+\cdots+100 }^{5050}$
$\underbrace{ a+b+\cdots+z }_{26}$
$\sum_{k=1}^N k^2$
$\textstyle \sum_{k=1}^N k^2$
$\prod_{i=1}^N x_i$
$\textstyle \prod_{i=1}^N x_i$
$\coprod_{i=1}^N x_i$
$\textstyle \coprod_{i=1}^N x_i$
$\lim_{n \to \infty}x_n$
$\textstyle \lim_{n \to \infty}x_n$
$\int\limits_{1}^{3}\frac{e^3/x}{x^2}\, dx$
$\int_{1}^{3}\frac{e^3/x}{x^2}\, dx$
$\textstyle \int\limits_{-N}^{N} e^x\, dx$
$\textstyle \int_{-N}^{N} e^x\, dx$
$\iint\limits_D \, dx\,dy$
$\iiint\limits_E \, dx\,dy\,dz$
$ \iiiint\limits_F \, dx\,dy\,dz\,dt$
$\int_C x^3\, dx + 4y^2\, dy$
$\oint_C x^3\, dx + 4y^2\, dy$
$ \bigcap_1^n p$
$\bigcup_1^k p$
$\frac{2}{4}=0.5$
$\tfrac{2}{4} = 0.5$
$\dfrac{2}{4} = 0.5 \qquad \dfrac{2}{c + \dfrac{2}{d + \dfrac{2}{4}}} = a$
$\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a$
$\binom{n}{k}$
$\tbinom{n}{k}$
$\dbinom{n}{k}$
$\begin{matrix} x & y \\ z & v \end{matrix}$
$\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{bmatrix}$
$f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}$
$\begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \\ \end{align}$
$\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}$
$\begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0\\ \end{array}$

UNISCIEL / Franck

Échantillon de formules, extrais d'un projet UNISCIEL fournies par Franck (environ 10% des formules les plus distinctes dans les 20 premiers % des modules)

$\frac{dEm}{dt}$
$(\overrightarrow{e_{\rho}},\overrightarrow{e_{\varphi}})$
$\overrightarrow{a}=(\ddot{\rho}-\rho\dot{\varphi}^{2})\overrightarrow{e_{\rho}}+(2\dot{\rho}\dot{\varphi}+\rho\ddot{\varphi})\overrightarrow{e_{\varphi}}$
$\frac{d\varepsilon}{dt}=0$
$\varepsilon_p=mgz+Cste$
$\overrightarrow{L}_{/O}=\overrightarrow{OA}\wedge m\overrightarrow{v}$
$\overrightarrow{v}=\dot{\rho}\overrightarrow{e_{\rho}}+\rho\dot{\varphi}\overrightarrow{e_{\varphi}}$
$\ddot{\varphi}+\frac{g}{l}\sin\varphi=0$
$\overrightarrow{a}=-l\dot{\varphi}^{2}\overrightarrow{e_{\rho}}+l\ddot{\varphi}\overrightarrow{e_{\varphi}}$
$\varphi(t)=A \cos(\omega_{0}t+\alpha)$
$\varepsilon_{k}=\frac{1}{2}ml^{2}\dot{\varphi}^{2}$
$\sin(\overrightarrow{OA},\overrightarrow{P})\overrightarrow{e_{z}}=l mg sin(-\varphi)\overrightarrow{e_{z}}\overrightarrow{M_{P/O}}=-mgl\sin\varphi\overrightarrow{e_{z}}$
$\ddot{\varphi}+\omega_{0}^{2}\varphi=0$
$\frac{d\varepsilon}{dt}=ml^{2}\dot{\varphi}\ddot{\varphi}+mgl \sin\varphi \dot{\varphi}=0$
$\overrightarrow{a}=-R\dot{\theta}^{2}\overrightarrow{e_{\rho}}+R\ddot{\theta}\overrightarrow{e_{\theta}}$
$-g\sin\theta=\frac{1}{2} R ~\dot{\theta}^{2}-g$
$\overrightarrow{v}=\overrightarrow{\Omega}\wedge \overrightarrow{OA}=l~\dot{\theta}\overrightarrow{e_\theta}+l~\dot{\varphi} \sin \theta ~\overrightarrow{e_\varphi}$
$\overrightarrow{L_{/O}}=\overrightarrow{OA}\wedge m\frac{d\overrightarrow{OA}}{dt}$
$m\ddot{\rho}-C^{2}/m\rho^{2}=-k(\rho-l_0)+\mu_c m g$
$\frac{d\varepsilon}{dt}=\dot{\rho}(m \ddot{\rho}- \frac{C^{2}}{m\rho^{3}} +k(\rho-l_0))=m \dot{\rho}(\ddot{\rho}- m\rho \dot{\varphi}^{2} +k(\rho-l_0))$
$\sum \overrightarrow{F_{ext}}=m\overrightarrow{a}$
$\frac{1}{2}m v_B^{2}-0=mg(z_A-z_B)=mg(z_A-z_C+z_C-z_{O'}+z_{O'}-z_B)=mg(h-R+R\cos \alpha_1)$
$\frac{1}{2}m v_S^{2}-0=mg(z_A-z_S)=mg(h-R)$
$v_S=\sqrt{2g(h-R)}$
$a_1=g \sin \alpha_1$ et $N_1=mg \cos \alpha_1$
$m\frac{v^{2}}{R}=N_2-mg\cos\alpha$
$\overrightarrow{a(C)}=2g\frac{h}{R}\overrightarrow{n}=2g\frac{h}{R}\overrightarrow{u_z}$
$\overrightarrow{a_{a/R}}=\overrightarrow{a_{eR'/R}}+\overrightarrow{a_{cR'/R}}+\overrightarrow{a_{r/R'}}$
$\overrightarrow{f_{ie}}=-m'g\sin \alpha_1 \overrightarrow{u_1}$
$\mathcal{R}_0=\{O,\overrightarrow{e}_x,\overrightarrow{e}_y,\overrightarrow{e}_z \}$
$\begin{eqnarray} \overrightarrow{a}_{M/\mathcal{R}_0}&=&\left( \frac{d^2 \overrightarrow{OM} }{dt^2} \right)_{\mathcal{R}_0}, \\ &=&\ddot{z}(t)\overrightarrow{e}_z \end{eqnarray}$
$\begin{eqnarray} \overrightarrow{a}_{M/\mathcal{R}_0}&=&\left( \frac{d^2 \overrightarrow{OM} }{dt^2} \right)_{\mathcal{R}_0},\\ &=&\underbrace{ \left( \frac{d^2 \overrightarrow{OA} }{dt^2} \right)_{\mathcal{R}_0}}_{\overrightarrow{a}_{A/\mathcal{R}_0}} + \underbrace{\left( \frac{d^2 \overrightarrow{AM} }{dt^2} \right)_{\mathcal{R}_0}}_{\overrightarrow{0}}. \end{eqnarray}$
$\underbrace{\overrightarrow{a}_{M/\mathcal{R}_0}}_{\textrm{acc. absolue}}=\underbrace{\overrightarrow{a}_{M/\mathcal{R}_A}}_{\textrm{acc. relative}}+\underbrace{\overrightarrow{a}_{A/\mathcal{R}_0}}_{\textrm{acc. d'entrainement}}$
$\begin{eqnarray} m\overrightarrow{a}_{M/\mathcal{R}_A}&=&\overrightarrow{P}+\overrightarrow{R}+\overrightarrow{f}_e\\ 0\overrightarrow{e}_z&=&(-mg+R-m\ddot{z}(t) )\overrightarrow{e}_z \end{eqnarray}$
$(\ddot{z}(t)=0)$
$\mathcalR_0=\{0,\vec{e_x},\vec{e_y},\vec{e_z}\}$
$\begin{eqnarray} R_Y&=&2m r \Omega^2 \sinh\left(\Omega t \right),\\ R_Z&=&mg. \end{eqnarray}$
$ \begin{eqnarray} m \ddot{X}(t)&=& m X(t)\dot{\theta}^2, \\ 0 &=&R_Y - m2\dot{X}(t)\dot{\theta}(t), \\ 0 &=&R_Z-mg. \end{eqnarray}$
$X(t)=r\cosh\left(\Omega t \right)$
$\displaystyle{ \begin{array}{c l} \overrightarrow {OP}(t) & = x(t) \vec e_x + y(t) \vec e_y \\ & = X(t) \vec e_X \end{array}$
$\theta = (\overrightarrow e_x, \overrightarrow e_X) = \Omega t$
$\begin{array}{c l c} x(t) & = X(t) \cos(\Omegat) & (1) \\ y(t) & =X(t) \sin(\Omegat) & (2) \end{array}$
$\overrightarrow{a}_e=- X(t)\dot{\theta}^2 \overrightarrow{e}_X +X(t) \ddot{\theta}(t) \overrightarrow{e}_Y$
$\textit{i.e.}$
$m \overrightarrow{a}_{P/\mathcal{R}_1} + m\overrightarrow{a}_e + m\overrightarrow{a}_c= \sum\overrightarrow{F}_{ext}.$
$P(\lambda)=m\lambda^2-m\Omega^2$
$\dot{X}(t=0)=\Omega \left(A_1- A_2 \right)=0$
$\Omega \neq 0$
$\displaystyle{\lim_{t\rightarrow \infty} \varepsilon(t)}$
$\theta(t)=\theta^*+\varepsilon(t)$
$|\varepsilon(t)|\ll 1$
$\left(\overrightarrow{e}_i \cdot \overrightarrow{e}_j \right)=\cos\left(\overrightarrow{e}_i, \overrightarrow{e}_j \right)$
$\overrightarrow{OM} =-R \sin\left( \theta \right) \overrightarrow{e}_X - R \cos \left( \theta \right) \overrightarrow{e}_Z.$
$\displaystyle{ \left( \frac{d \overrightarrow{OM} }{dt } \right)_{\mathcal{R}_2}=\left( \frac{d \overrightarrow{OM} }{dt } \right)_{\mathcal{R}_0} +\overrightarrow{\Omega}_{\mathcal{R}_2/\mathcal{R}_0} \wedge \overrightarrow{OM}} ~~~~ (2)$
$\left( \frac{d \overrightarrow{OM} }{dt } \right)_{\mathcal{R}_2}= \left( \frac{d R\overrightarrow{e}_r }{dt } \right)_{\mathcal{R}_2} =\overrightarrow{0}$
$\overrightarrow{a}_M/\mathcal{R}_0 = R \ddot{\theta} \overrightarrow{e}_\theta -R\dot{\theta}^2\overrightarrow{e}_r +2R \dot{\theta}\omega\cos(\theta)\overrightarrow{e}_Y - R \omega \sin(\theta) \omega \overrightarrow{e}_X.$
$\overrightarrow{a}_{M/\mathcal{R}_0} = \overrightarrow{a}_{M/\mathcal{R}_1} +2R \dot{\theta}\omega\cos(\theta)\overrightarrow{e}_Y - R \omega \sin(\theta) \omega \overrightarrow{e}_X.$
$\overrightarrow{f}_\textrm{i}=\underbrace{m R \omega \sin(\theta) \omega \overrightarrow{e}_X}_{\textrm{force centrifuge}} + \underbrace{2R \dot{\theta}\omega\cos(\theta)\overrightarrow{e}_Y}_\textrm{force de Coriolis}$
$\theta^\star=0 [\pi]$
$\omega>\omega_c$
$\ddot{\varepsilon}(t)=-\frac{g}{R}\cos\theta^\star \varepsilon(t)+ \omega^2 \left( \cos(\theta^\star)^2 -\sin(\theta^\star)^2 \right) \varepsilon(t),$
$\cos(\theta^\star)^2 -\sin(\theta^\star)^2=2 \cos(\theta^\star)^2 -1$
$\overrightarrow{\Omega}_{\mathcal{R}_{T2}/\mathcal{R}_T}$
$\Omega_Q\simeq 0,27$
$\overrightarrow{\Omega}_{\mathcal{R}_{T_2}/\mathcal{R}_S}= \left( \Omega_A+\Omega_Q\cos(\alpha) \right)\overrightarrow{e}_z + \Omega_Q\sin(\alpha)\overrightarrow{e}_x.$
$R_T \Omega_Q^2 \simeq 0,034~m.s^{-2}$
$\overrightarrow{V}_{M/\mathcal{R}_T}=\left( \frac{d \overrightarrow{TM} }{dt } \right)_{\mathcal{R}_{T_2}} + \overrightarrow{\Omega}_{\mathcal{R}_{T_2}/\mathcal{R}_{T}}\wedge \overrightarrow{TM}$
$\begin{eqnarray} \left( \frac{d \overrightarrow{e}_Y }{dt } \right)_{\mathcal{R}_S}&=&\left( \frac{d \overrightarrow{e}_Y }{dt } \right)_{\mathcal{R}_{T_2}} +\overrightarrow{\Omega}_{\mathcal{R}_{T_2}/\mathcal{R}_S} \wedge \overrightarrow{e}_Y\\ &=&\overrightarrow{0}+\Omega_A \overrightarrow{e}_z\wedge \overrightarrow{e}_Y +\Omega_Q \overrightarrow{e}_Z \wedge \overrightarrow{e}_Y\\ &=& \Omega_A \left( - \cos(\alpha) \overrightarrow{e}_X + \sin(\alpha) \overrightarrow{e}_Z \right) -\Omega_Q \overrightarrow{e}_X \end{eqnarray}$
$|~l~|\neq \pi/2$
$\frac{d^{2}x}{dt^{2}}=-\frac{g}{l}x+2 \Omega_T ~\sin \lambda \frac{dy}{dt}$
$\overrightarrow{T}=-T \overrightarrow{HA}/l$
$m\ddot{x}=-Tx /l + 2m~\Omega_T~ \sin~ \lambda \dot{y}$
$T \approx mg$
$\ddot{x}+ \omega_0^{2}~x = 2\Omega_T~ \sin~ \lambda \dot{y}$
$\varrho(t)=x_m ~exp(-i~ \Omega_T ~ \sin ~ \lambda t)~cos~ (\omega_0 t)$
$T=\frac{2\pi}{|\Omega_T \sin \lambda|}=31 \mathrm{ h} 52 \mathrm{ min}$
$\Omega=|\Omega_T \sin \lambda|=5.5\times 10^{-5} \mathrm{ rad.s}^{-1}$
$\frac{1}{(d-R)^2}\simeq \frac{1}{d^2}\left(1+2\frac{R}{d} \right)$
$\overrightarrow{C}=\rho^{2}\dot{\varphi}\overrightarrow{e_z} et \frac{d\overrightarrow{C}}{dt}=\overrightarrow{0}$
$a_A=-C^{2}u^{2}(\frac{d^{2}u}{d\varphi^{2}}+u)$
$\overrightarrow{a_A}=(\ddot{\rho}-\rho \dot{\varphi}^{2})\overrightarrow{e_\rho}+(2\dot{\rho}\dot{\varphi}+\rho \ddot{\varphi})\overrightarrow{e_\varphi}$
$a_A=(\ddot{\rho}-\rho \dot{\varphi}^{2})=(-C^{2}u^{2}\frac{d^{2}u}{d\varphi^{2}}-\frac{C^{2}u^{4}}{u}) =(-C^{2}u^{2}\frac{d^{2}u}{d\varphi^{2}}-C^{2}u^{3})=-C^{2}u^{2}(\frac{d^{2}u}{d\varphi^{2}}+u)$
$u=\frac{1+e \cos\varphi}{p}$
$(\frac{d^{2}u}{d\varphi^{2}}+u)=-\frac{e cos \varphi}{p}+\frac{1+e \cos\varphi}{p}=\frac{1}{p}$
$g = 10 \mathrm{ m.s}^{-2}$
$\overrightarrow{L_{/T}}=\overrightarrow{TM} \wedge m \overrightarrow{v_0}$
$b=\sqrt{d^2 + 2dGM/v_0^2}$
$E=\frac{1}{2}\mu v^2\varepsilon(q,m_1,m_2) \quad \textrm{avec} \quad \varepsilon(q,m_1,m_2)=q\frac{m_1-m_2-2qm_1}{qm_1+m_2}$
$\overrightarrow{p}=m_1 \frac{d \overrightarrow{r}_1}{dt}+ m_2 \frac{d \overrightarrow{r}_2}{dt}=\frac{d m_1\overrightarrow{r}_1 +m_2\overrightarrow{r}_2}{dt}=\overrightarrow{0}$
$E_\textrm{m}=\frac{1}{2}m_1 \left[\frac{d\overrightarrow{r}_1}{dt} \right]^2 + \frac{1}{2}m_2 \left[\frac{d\overrightarrow{r}_1}{dt} \right]^2 - G\frac{m_1 m_2}{|| \overrightarrow{r} || }$
$E_\textrm{m}=\frac{1}{2} \mu ||\overrightarrow{v}||^2 - G\frac{\mu M}{|| \overrightarrow{r} || }$
$\overrightarrow{L}_G=\overrightarrow{r}_1 \wedge \left(m_1 \frac{d \overrightarrow{r}_1}{dt} \right)+\overrightarrow{r}_2 \wedge \left(m_2 \frac{d \overrightarrow{r}_2}{dt} \right)$
$\left[\mu\ddot{r} - \frac{L_g^2}{ r^3} \right]=-G\frac{\mu M}{r^2}~~~~~(2)$
$\frac{d^2\overrightarrow{r}}{dt^2}=-\frac{L_G^2}{\mu^2 r^2}\left[ \frac{d^2}{d\theta^2}\left(\frac{1}{r} \right) + \frac{1}{ r} \right] \overrightarrow{e}_r$
$\frac{d^2}{d\theta^2}\left(\frac{1}{r} \right) + \frac{1}{ r}=\frac{G\mu^2 M}{ L_G^2}$
$u=A\cos(\theta + \alpha) + \frac{G\mu^2 M}{ L_G^2}$
$E_\textrm{m}=\frac{1}{2} \mu ||\overrightarrow{v}||^2 - G\frac{\mu M}{|| \overrightarrow{r} || }$
$m_1(1-2q)>m_2 \quad \textrm{ou} \quad q<\frac{1}{2}\left( 1-\frac{m_1}{m_2} \right)$
$\epsilon p = \frac{K}{r}$
$\overrightarrow{F_{12}}=-\overrightarrow{F_{21}}=\frac{-e*e}{4\pi \varepsilon_0 {A_1A_2}^{3}}\overrightarrow{A_1A_2}$
$dW=\overrightarrow{F_{12}}.\overrightarrow{dr}=\frac{-e^{2}}{4\pi \varepsilon_0 {r}^{2}}\overrightarrow{u_r}.dr\overrightarrow{u_r}$