\documentclass[11pt,a4paper,twocolumn,catalan]{article}
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\author{pod}
\title{\bf Ampliación de Mecánica Teórica}
\date{Primavera, 2002}


\begin{document}
\maketitle

\noindent
$$ \deriv t \parcial[L]{\dot \vq} - \parcial[L] \vq = 0 \ ,\quad L = T-V $$
$$ \vp = \parcial[L]{\dot\vq} \ ,\quad H = \vp \cdot \dot\vq - L $$
$$ \dot\vq = \parcial[H]{\vp} \ , \quad \dot\vp = - \parcial[H]{\vp} \ ,\quad \deriv[H]{t} = \parcial[H]{t} = -\parcial[L]{t}$$
$$ \deriv[f]{t} = \parcial[f]{t} + [f,H] \, $$
$$ \ [f,g] = \parcial[f]{\vq} \parcial[g]{\vp} - \parcial[f]{\vp} \parcial[g]{\vq} =  \left( \parcial[f]{\vx} \right)^t J \left( \parcial[g]{\vx} \right)$$
$$ \dot p_i = [p_i, H] \ , \quad \dot q_i = [q_i, H] \ , \quad [p_i, q_j] = -\delta_{ij} $$
$$ \vx=(\vq,\vp) \ , \quad J =\begin{pmatrix}
  0 & 1 \\
  -1 & 0 \\
\end{pmatrix} \ , \quad \dot \vx = J \parcial[H]{\vx} = [ \vx,H] $$

\sep
 Transformaciones canónicas.  $\quad H' = H + \parcial[F]{t}$
 $$ \left( \vp\ \dd\vq - H \dt\right) - \left( \vP\ \dd\vQ - H' \dt \right)=  \dd F $$
 \begin{enumerate}
  \item $F = F_1(q,Q,t)$: $$\vp = \parcial[F_1]{\vq} \ , \qquad \vP = -  \parcial[F_1]{\vQ}$$
  \item $F = F_2(q,P,t) - \vQ \vP$: $$\vp = \parcial[F_2]{\vq} \ , \qquad \vQ = \parcial[F_2]{\vP}$$
  \item $F = \vq \vp + F_3(q, Q, t)$: $$\vq = - \parcial[F_3]{\vq} \ , \qquad \vP = -  \parcial[F_3]{\vQ}$$
  \item $F = \vq \vp - \vQ \vP + F_4(p,P,t)$: $$\vq = -\parcial[F_4]{\vp} \ , \quad \vQ  = \parcial[F_4]{\vP}$$
\end{enumerate}
$$ \dot \vX = M J M^t \parcial[H]{\vX} \to MJM^t = J $$
Invariante integral $ \deriv{t} I = 0 $
 $$ I(t) = \mathop{\int \cdots \int}_{\Omega_t} \dd\vp\,\dd\vq\ \rho(\vq,\vp,t) \leftrightarrow \parcial[\rho]{t} + [\rho,H] = 0 $$

\sep
 $$ \parcial[S]{t} + H\left( \vq , \parcial[S]{\vq} , t \right) = 0 \quad,\quad \vp = \parcial[S]{\vq} \ , \quad \vbeta = \parcial[S]{\valpha} $$
\sep
 $$ J_k(\valpha) = \oint p_k(q_k,\valpha) \dd q_k \quad , \quad \dot{\vomega} = \parcial[H']{\vJ} = \vnu(\vJ) $$

\sep Variació nde las constantes
 $H = H_0 + \lambda H_1$, t.c $S_0$
 \begin{eqnarray*}
  \valpha_1(t) &=& - \int\dt \parcial[H_1]{\vbeta} \\
  \valpha_2(t) &=& - \int\dt \left\{ \parcial[^2 H_1]{\valpha\ \partial\vbeta} \cdot \valpha_1 + \parcial[^2 H_1]{\vbeta\ \partial\vbeta}
  \cdot \vbeta_1 \right\} \\
  \vbeta_1 (t) &=&\ \int\dt \parcial[H_1]{\valpha} \\
  \vbeta_2 (t) &=&\ \int\dt \left\{ \parcial[^2 H_1]{\vbeta\ \partial\valpha} \cdot \vbeta_1 + \parcial[^2 H_1]{\valpha\ \partial\valpha}
  \cdot \valpha_1 \right\}
 \end{eqnarray*}
 Met.media $$ H'(\alpha, \beta, t) \approx \la H_1(\alpha, \beta, t) \ra = \frac1{T_0} \int_{T_0} \dt'\ H_1(\alpha, \beta, t') $$

\sep Poincaré-Von Ziepel
 $$ E_1 = H^{(1)}_{\vec0} = \int_{[0,1]^n} \dd\vomega_0 H_1(\vJ) \equiv \la H_1(\vJ) \ra $$
 \begin{align*}
 E_2 &= \la H_2( \vJ, \vomega_0 ) \ra + \frac12 \la
 \parcial[S_1]{\vomega_0} \cdot\parcial[^2 H_0]{\vJ\ \partial \vJ} \cdot \parcial[S_1] {\vomega_0} \ra\\&
 +\la\parcial[H_1]{\vJ} \cdot \parcial[S_1]{\vomega_0}\ra 
 \end{align*}
 $$ S_1(\vomega_0, \vJ) = \frac{-1}{2\pi i} \sum_{\vm\ne0} \frac{H_{\vm}^{(1)} (\vJ)}{\vm \cdot \vnu_0 (\vJ)}\ \expon{2\pi i \vm \cdot \vomega_0}$$
 $$ \vnu = \vnu_0 + \epsilon\, \partial[E_1]{\vJ} + \epsilon^2\, \partial[E_2]{\vJ} + \cdots $$
 $$ J_0 = J + \epsilon\, \parcial[S_1]{\vomega_0} + \cdots $$
 $$ \vomega = \omega_0 + \epsilon\, \parcial[S_1]{\vJ} + \cdots $$
 $$ (n=1) \ E_1(J) = H_1(\J, \omega_0) + \nu_0(\vJ) \parcial[S_1]{\omega_0} $$

\noindent
\begin{itemize}
  \item Taylor $E= H_0( \parcial[S]{\vomega_0} ) + \epsilon H_1 ( \parcial[S]{\vomega_0} , \vomega_0 ) + \cdots$
  \item Desenvol i $E = H'_0+\epsilon H'_1+\cdots$
  \item Fourier $S_{\vm}$
  \item Integrar $\vomega_0$.
\end{itemize}


\sep Twist Map
 $$ J_{n+1} = J_n \quad , \quad  \omega_{n+1} = \alpha(J_{n+1})+ \omega_n $$
Pertorb. $\epsilon H_1$
 $$ J_{n+1} = J_n + \epsilon f(J_{n+1}, \omega_n) $$ $$ \omega_{n+1} = \alpha(J_{n+1})+ \omega_n + \epsilon g(J_{n+1}, \omega_n) $$

\sep Teor. adiabática
 $$ \lambda = \lambda(\epsilon t) \Longrightarrow \Delta J = o(\epsilon^2) $$

\sep
 Sist. dinámicos $ \dot\vx = F(\vx,t) $
 $$ \vk = \cte \ , \ \vF = \vF(\vx) \Longleftrightarrow \vF \cdot \overrightarrow{\textrm{grad}} = 0 $$
 $$ I(t) = \mathop{\int \cdots \int}_{\Omega(t)} \dd^N \vx \ \rho(\vx,t) =\cte \Longleftrightarrow $$
 $$ \Longleftrightarrow \parcial[\rho]{t} + \mathrm{div} \left[ \rho(\vx,t) \vF(\vx,t) \right] = 0 $$
 Rectificat $\{x_i\} \to \{y_i\}$ tq $y_i = \delta_{1i} $
 $$ F_k(\vx) = \sum_l \left. \parcial[F_k]{x_l} \right|_{\vx=\vec0} x_l + o(\vx^2) = \sum_l a_{kl}\ x_l  \ , \quad \dot\vx = A \vx $$
 $$ \lambda_k = \alpha_k + i\omega_k \longrightarrow x_k = \sum_l c_{kl} \expon{\alpha_l t} \cos( \omega_l t + \varphi_l )$$

\begin{itemize}
  \item $\lambda_1\ne\lambda_2 \in\RR-\{0\}$
\begin{itemize}
  \item $\lambda_1/\lambda_2>0$ nodo, parábola ($\lambda_i > 0$ inestable)
  \item $\lambda_1/\lambda_2<0$ p. ensilladura, hiperbola
\end{itemize}
  \item $\lambda_1 = \lambda_2^*$ foco ($\alpha>0$ inestable, $\alpha=0$ centro)
  \item $\lambda_1 = \lambda_2$ nodo.
\end{itemize}


\sep\sep
 Oscil. amónic (frec. $\varpi$)
$$ q = \sqrt{ \frac{J_0}{\pi \varpi m} } \sin 2\pi\omega_0 \ , \quad p = \sqrt{ \frac{J_0 \varpi m }{\pi} }\cos 2 \pi \omega_0 $$
$$ F = \sum_{\vl} a_{\vl}\ \expon{2\pi i \vl \vomega t} \ \textrm{on} \quad a_{\vl} = \expon{2\pi i \vbeta \vl} \int_{[0,1]^N}\!\!\!\!\!\!\!\!\!\! \dd^n \vomega\ F\ \expon{-2\pi i \vl \vomega} $$
\end{document}
