\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}
