\documentclass[11pt,a4paper,twocolumn,catalan]{article} \usepackage{babel} \usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage[dvips]{epsfig} \usepackage{indentfirst} \usepackage[a4paper,right=2cm,top=2.5cm]{geometry} \setlength\arraycolsep{2pt} \newcommand{\RR}{\mathbb{R}} \newcommand{\MM}{\mathbb{M}} \newcommand{\NN}{\mathbb{N}} \newcommand{\J}{\mathrm{J}} \newcommand{\K}{\mathcal{K}} \newcommand{\QQ}{\mathcal{Q}} \newcommand{\dd}{\mathrm{d}} \newcommand{\tg}{\mathrm{tg}} \newcommand{\tgh}{\mathrm{tgh}} \newcommand{\cotg}{\mathrm{cotg}} \newcommand{\dB}{\mathrm{dB}} \newcommand{\Np}{\mathrm{Np}} \newcommand{\eV}{\mathrm{eV}} \newcommand{\hatn}{\hat{n}} \newcommand{\expon}[1]{\mathrm{e}^{#1}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\dx}{\dd x} \newcommand{\s}{\mathrm{s}} \newcommand{\MeV}{\mathrm{MeV}} \newcommand{\dr}{\dd r} \newcommand{\dtr}{\dd^3 r} \newcommand{\dt}{\dd t} \newcommand{\cte}{\mathrm{cte}} \newcommand{\ds}{\dd s} % \newcommand{\parcial}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\deriv}[2][]{\frac{\dd #1}{\dd #2}} \newcommand{\E}[1]{\times 10^{#1}} \newcommand{\AAA}{\ensuremath{\,\mathrm{\AA}}} \newcommand{\cm}{\mathrm{cm}} % \newcommand{\vD}{\vec{D}} \newcommand{\vH}{\vec{H}} \newcommand{\vj}{\vec{\jmath}} \newcommand{\vE}{\vec{E}} \newcommand{\vB}{\vec{B}} \newcommand{\vP}{\vec{P}} \newcommand{\vM}{\vec{M}} \newcommand{\vN}{\vec{N}} \newcommand{\vS}{\vec{S}} \newcommand{\va}{\vec{a}} \newcommand{\vnabla}{\vec{\nabla}} \newcommand{\vL}{\vec{L}} \newcommand{\vr}{\vec{r}} \newcommand{\vp}{\vec{p}} \newcommand{\vsigma}{\vec{\sigma}} \newcommand{\vR}{\vec{R}} \newcommand{\vA}{\vec{A}} \newcommand{\vepsilon}{\vec{\varepsilon}} \newcommand{\vJ}{\vec{J}} \newcommand{\vk}{\vec{k}} \newcommand{\vv}{\vec{v}} \newcommand{\vX}{\vec{X}} \newcommand{\vY}{\vec{Y}} \newcommand{\ve}{\vec{e}} \newcommand{\vx}{\vec{x}} \newcommand{\vu}{\vec{u}} \newcommand{\vy}{\vec{y}} \newcommand{\vxi}{\vec{\xi}} \newcommand{\vbeta}{\vec{\beta}} \newcommand{\vn}{\vec{n}} \newcommand{\vomega}{\vec{\omega}}\newcommand{\vK}{\vec{K}} \newcommand{\vF}{\vec{F}} \newcommand{\vz}{\vec{z}} \newcommand{\vm}{\vec{m}} \newcommand{\vq}{\vec{q}} \newcommand{\vmu}{\vec{\mu}} \newcommand{\vl}{\vec{\ell}} \newcommand{\vs}{\vec{s}} \newcommand{\vQ}{\vec{Q}} \newcommand{\valpha}{\vec{\alpha}} \newcommand{\vnu}{\vec{\nu}} \newcommand{\vV}{\vec{V}} \newcommand{\parcte}[3]{ \left( \parcial{#1}{#2} \right)_{\!#3} } \newcommand{\sep}{\vspace{1mm} \hrule \vspace{1mm} \noindent} \newcommand{\inv}[1]{{#1}^{-1}} % \newcommand{\la}{\left<} \newcommand{\ra}{\right>} \newcommand{\bok}[3]{ \la #1 \right| #2 \left| #3 \ra } \newcommand{\ketbra}[2]{ \left| #1 \ra\la #2 \right| } \newcommand{\commut}[2]{\left[ #1 , #2 \right]} \author{pod} \title{Mecánica teórica} \date{Otoño, 2001} \begin{document} \maketitle \noindent Forza generalizada $$Q_j = \sum \vF_i \cdot \parcial[\vr_i]{q_j} \ , \quad L(q,\dot{q},t) = T-V$$ Ec. Lagrange $$ \deriv{t} \left[ \parcial[T]{\dot{q}_j} \right] - \parcial[T]{q_j} = Q_j $$ $$ \quad \deriv{t} \left[ \parcial[L]{\dot{q}_j} \right] - \parcial[L]{q_j} = 0$$ Disipativos $$\deriv{t}\left[\parcial[L]{\dot{q}_j}\right] - \parcial[L]{q_j} + \parcial[\mathcal{F}]{\dot{q}_j} = 0$$ $$\mathcal{F} = \frac{1}{2} \sum ( k_x v_{x,i}^2 + k_x v_{x,i}^2 + k_x v_{x,i}^2 ) = \frac{1}{2} \deriv[W_f]{t} $$ Potencial general $$ Q_j = - \parcial[U]{q_j} + \deriv{t} \left[ \parcial[U]{\dot{q}_j} \right] $$ \sep Acción $ S = \int_{t_1}^{t_2} L(q,\dot{q},y)\dt\ $, ppi. Hamilton $\delta S = 0$ \\ Momento $$p_i = \parcial[L]{\dot{q}} \ , \qquad p_i = \textrm{cte} \leftrightarrow L\ne L(q_i)$$ f. energía $$ h(q,\dot{q},t)= \sum_j \dot{q}_j \parcial[L]{\dot{q}_j} \ , \qquad \deriv[h]{t} = - \parcial[L]{t} $$ Si $L=L_1 + L_2 + L_3 \ , \ \to V\ne V(\dot{q}) \Rightarrow h = E$ \sep Formulismo de Hamilton $$ H(p,q,t) = \dot{q}_i p_i - L(q,\dot{q},t) $$ $$\dot{q}_i = \frac{\partial H}{\partial p_i} \: ,\qquad \dot{p}_i = -\frac{\partial H}{\partial q_i} $$ $$\deriv[H]{t} = \frac{\partial H}{\partial t} = -\frac{\partial L}{\partial t}\quad ,\qquad p_i = \frac{\partial L}{ \partial\dot{q}_i}$$ si $ L = L_0 + L_1 + L_2 \ ,\quad L_2 = T\ (q_i \ne q_i(t)) $ \\ $\to\quad L_0 = -V \ne -V(\dot{q}) \ ,\ \to H = L_2-L_0 = E $ \\ \noindent $L = L_0(q,t) + \dot{q}^t \mathbf{a} + \frac{1}{2} \dot{q}^t T \dot{q} $ \\ $H(q,p,t) = \frac{1}{2} (p^t-\mathbf{a}^t) T^{-1} (p-\mathbf{a}) - L_0(q,t) $ \\ \noindent $$\delta I = \delta\int^{t_2}_{t_1} L \dd t = \delta\int^{t_2}_{t_1}(\dot{q}_i p_i - H)\dd t = 0 $$ $$\Delta\int^{t_2}_{t_1}\!\!\!L\dt = \int^{t_2+\Delta t_2}_{t_1+\Delta t_1}\!\!\!L(\alpha)\dt - \int^{t_2}_{t_1} \!\!\!L(0)\dt $$ Mínima acción $$\Delta\int^{t_2}_{t_1}\!\!\!p_i\dot{q}_i\dt = 0 $$ \sep Transformaciones canónicas. $\quad K = H + \parcial[F]{t}$ $$ p_i \dot{q}_i - H(q,p,t) = P_i Q_i - K(Q,P,t) + \deriv{t}F(p,P,q,Q,t) $$ \begin{enumerate} \item $F = F_1(q,Q,t)$: $p_i = \parcial[F_1]{q_i} \ , \qquad P = - \parcial[F_1]{Q_i}$ $$\parcial[p_i]{Q_k} = \frac{\partial^2 F_1}{\partial Q_k \partial Q_i} = - \parcial[P_k]{q_i}$$ \item $F = F_2(q,P,t) - Q_i P_i$. $p_i = \parcial[F_2]{q_i} \ , \qquad Q_i = \parcial[F_2]{P_i}$ $$\parcial[p_i]{P_k} = \frac{\partial^2 F_2}{\partial P_k \partial q_i} = \parcial[Q_k]{q_i}$$ \item $F = q_i p_i + F_3(q, Q, t)$. $q_i = - \parcial[F_3]{q_i} \ , \qquad P_i = - \parcial[F_3]{Q_i}$ $$\parcial[q_i]{Q_k} =-\frac{\partial^2 F_3}{\partial Q_k \partial p_i} = \parcial[P_k]{q_i}$$ \item $F = q_i p_i - Q_i P_i + F_4(p,P,t)$. $q_i = -\parcial[F_4]{p_i} \ , \quad Q_i = \parcial[F_4]{P_i}$ $$\parcial[q_i]{P_k} = \frac{\partial^2 F_4}{\partial P_k \partial p_i} = - \parcial[Q_k]{q_i}$$ \end{enumerate} Condiciones $$ \{Q_i, Q_j\} = \{q_i, q_j\} = \{P_i, P_j\} = \{p_i, p_j\} = 0 $$ $$ \{P_i, Q_j\} = \{p_i, q_j\} = \delta_{ij} $$ Liouville $$ M_{\alpha,\beta} =\parcial[x_\alpha]{y_\beta} \ ,\ M^t J M = J $$ $$\phi_{t,s}(x) = ( \varphi^1_{t,s}(x),\ldots,\varphi^{2n}_{t,s}(x)) \ ,\ V_s = V_t $$ \noindent Paréntesis de Poisson $$ \{ f , g \} = \sum_i^n \left(\parcial[f]{p_i} \parcial[g]{q_i} - \parcial[f]{q_i} \parcial[g]{p_i} \right) $$ $$ \deriv[g]{t} = \parcial[g]{t} + \{ H , g \} \ , \ \{ f \circ \psi , g \circ \psi\} = \{ f , g \} \circ \psi$$ Jacobi $\{u , \{ v, w\} \} + \{v , \{ w, u\} \} + \{w , \{ u, v\} \} = 0$ \\ Eq. Hamilton $ \dot{q}_k = \{H , q_k\} \ , \quad \dot{p}_k = \{H , p_k\}$ \sep Hamilton-Jacobi $K = H+\parcial[F]{t} = 0 \ ,\ S = F_2 $ \\ $$ H \left( q_1 , \ldots , q_n ; \parcial[S]{q_1} , \ldots , \parcial[S]{q_n} ; t \right ) + \parcial[S]{t} = 0 $$ $$ p_i = \parcial[S]{q_i} = \alpha_i \ ,\quad Q_i = \parcial[S]{P_i} = \parcial[S]{\alpha_i} = \beta_i $$ \noindent $$ S(q,\alpha,t) = W(q,\alpha) - \alpha_1 t \ ,\ W(q, \alpha) = \sum_i W_i( q_i , \alpha ) $$ si $H \ne H(q_j) \to p_j = P_j = \gamma \ , \ W_j = \gamma q_1 $ \noindent Variables acción-angulo $H(q,p) = \alpha_1 = E$ \\ Momento $ J = \oint p\dd q$ \\ $$ Q = \parcial[S(q, \alpha, t)]{\alpha_1} \ , \qquad \bar{\omega} = \parcial[W(q,J)]{J}$$ $$\dot{\bar{\omega}} = \parcial[H(J)]{J} = \nu(J) \ , \qquad \bar{\omega} = \int \nu(J) \dd t = \nu(J) t + \beta$$ \sep S. no inercial $$ \left. \deriv{t} \right)_f = \left. \deriv{t} \right)_g + \vomega \times \ ,\ \, \vv_f = \vV + \vv_r + \omega \times \vr$$ $$ \vF= m \va_f = m \ddot{\vR} + m \va_r + m \vomega \times ( \vomega \times \vr ) + 2 m \vomega \times \vv_r $$ si $\ddot{\vR} = 0 \to \vF_\textrm{ef} = m \va_r $ \\ $$ \vF_\textrm{ef} = m \va_f - m \vomega \times ( \vomega \times \vr ) - 2 m\vomega \times \vv_r $$ \sep Sólido rígido $$ I_{i,j}=\sum_\alpha m_\alpha\left[ \delta_{i,j} \sum_k x^2_{\alpha,k} - x_{\alpha,i} x_{\alpha,i}\right]$$ $$ I_{i,j} = \int_v \left[ \delta_{i,j} \sum_k x_k^2 - x_i x_j \right] \rho(\vr) \dd V \ ,\ T_\textrm{rot} = \frac{1}{2} \sum_{i,j} I_{i,j} \omega_i\omega_j $$ $$ T_\textrm{rot} = \frac{1}{2} \vL\vomega \ ,\ L_i = \sum_j I_{i,j}\ \omega_j \ , \ \vL = \{ I \}\ \vomega $$ $$ I_{i,j}^\mathrm{cm} = J_{i,j} - M \left[ a^2 \delta_{i,j} - a_i a_j \right] \ ,\ \va : x_i \to {x'_i}^\textrm{cm} $$ \noindent Angulos de Euler $ \lambda = \lambda_\psi\ \lambda_\theta\ \lambda_\varphi $ $$ ( I_i - I_j ) \omega_i \omega_j - \sum_k I_k \dot{\omega}_k \epsilon_{ijk} = 0 $$ $$ ( I_i - I_j ) \omega_i \omega_j - \sum_k (I_k \dot{\omega}_k - N_k) \epsilon_{ijk} = 0 $$ \end{document}