From 72b1bf252dd21ace9ff23a006c4a48e0f03e5b57 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Wed, 27 Sep 2023 15:29:08 +0200 Subject: [PATCH] Fixups to render properly mathjax Signed-off-by: Marcello Seri --- hm.tex | 43 ++++++++++++++++++++++--------------------- 1 file changed, 22 insertions(+), 21 deletions(-) diff --git a/hm.tex b/hm.tex index 9883332..d4f7401 100644 --- a/hm.tex +++ b/hm.tex @@ -20,7 +20,6 @@ \setcounter{FileDepth}{0} \boolfalse{FileSectionNames} % Number the file names \booltrue{CombineHigherDepths} % Combine parts/chapters/sections -\HTMLAuthor{Marcello Seri} \HTMLLanguage{en-UK} \HTMLDescription{Lecture notes for Hamiltonian Mechanics} @@ -42,6 +41,7 @@ \usepackage{units} \usepackage{tikz} +\tikzset{every picture/.style={ampersand replacement=\&}} \usepackage{pgfplots} \usepackage{tikz-3dplot} \pgfplotsset{compat=1.16} @@ -2640,7 +2640,7 @@ \chapter*{Preface} g_{ij}(q) = \sum_{k=1}^N m_k\left\langle\frac{\partial \vb*{x}_k}{\partial q^i},\frac{\partial \vb*{x}_k}{\partial q^j}\right\rangle \end{equation} is the riemannian metric induced by the euclidean metric on $Q$ given by - $ds^2 = \sum_{j=1}^N m_j\langle\dd \vb*{x}_j, \dd{\vb*{x}_j}\rangle$. + $ds^2 = \sum_{j=1}^N m_j\langle\dd{\vb*{x}_j}, \dd{\vb*{x}_j}\rangle$. \end{theorem} This can be further generalized to mechanical systems which are constrained on the tangent bundle $TM$ of a manifold $M$ of dimension $m$. @@ -3005,15 +3005,15 @@ \chapter*{Preface} \end{equation} can be explicitly computed using its definition: \begin{align} - \dd H(q,p) & = \dd \left( p_i\dot q^i(q,p) - L(q, \dot q(q,p)) \right) \\ - & = \dot q^i \dd p_i + p_i \dd \dot q^i(q,p) - \frac{\partial L}{\partial q^i} \dd q^i - \frac{\partial L}{\partial \dot q^i} \dd \dot q^i \\ - & = \dot q^i \dd p_i - \dot p_i \dd q^i, \label{eq:dHlc2} + \dd H(q,p) & = \dd{}\left( p_i\dot{q}^i(q,p) - L(q, \dot{q}(q,p)) \right) \\ + & = \dot{q}^i \dd{p_i} + p_i \dd{\dot{q}^i}(q,p) - \frac{\partial L}{\partial q^i} \dd{q^i} - \frac{\partial L}{\partial \dot{q}^i} \dd{\dot{q}^i} \\ + & = \dot{q}^i \dd{p_i} - \dot{p}_i \dd{q^i}, \label{eq:dHlc2} \end{align} where we used the Euler-Lagrange equations to get \begin{equation} - p_i = \frac{\partial L}{\partial \dot q^i} + p_i = \frac{\partial L}{\partial \dot{q}^i} \qquad\mbox{and}\qquad - \dot p_i = \frac{\dd }{\dd t}\frac{\partial L}{\partial \dot q^i} = \frac{\partial L}{\partial q^i}. + \dot p_i = \frac{\dd }{\dd t}\frac{\partial L}{\partial \dot{q}^i} = \frac{\partial L}{\partial q^i}. \end{equation} The theorem follows comparing coefficients in \eqref{eq:dHlc} and \eqref{eq:dHlc2}. \end{proof} @@ -3627,12 +3627,12 @@ \chapter*{Preface} \end{equation} which imply \begin{equation} - \langle\vb*{p}, \dd\vb*{x}\rangle = \|\vb*{p}\|\|\dd \vb*{x}\| = \frac{E}{c(\vb*{x})}\|\dd\vb*{x}\|. + \langle\vb*{p}, \dd{\vb*{x}}\rangle = \|\vb*{p}\|\|\dd{\vb*{x}}\| = \frac{E}{c(\vb*{x})}\|\dd{\vb*{x}}\|. \end{equation} For the action \eqref{eq:variationalMaupertuis} we end up with \begin{equation} - S_0 = E \int_{\vb*{x}_1}^{\vb*{x}_2} \frac{\|\dd\vb*{x}\|}{c(\vb*{x})}, + S_0 = E \int_{\vb*{x}_1}^{\vb*{x}_2} \frac{\|\dd{\vb*{x}}\|}{c(\vb*{x})}, \end{equation} which is $E$ multiplied by the time of propagation of light between the points $\vb*{x}_1$ and $\vb*{x}_2$. @@ -3645,7 +3645,7 @@ \chapter*{Preface} \end{equation} where \begin{equation} - \dd s^2 = \dd \vb*{x}^2 = \dd x^2 + \dd y^2 + \dd z^2. + \dd s^2 = \dd{\vb*{x}}^2 = \dd x^2 + \dd y^2 + \dd z^2. \end{equation} \end{theorem} @@ -4145,10 +4145,10 @@ \chapter*{Preface} \end{equation} Using the definition, we have \begin{equation} - \widetilde p_i \dd \widetilde q^i = - \frac{\partial q^k}{\partial \widetilde q^i} p_k \frac{\partial \widetilde q^i}{\partial q^l} \dd q^l - = \delta^k_l p_k \dd q^l - = p_k \dd q^k. + \widetilde{p}_i \dd{\widetilde{q}^i} = + \frac{\partial q^k}{\partial \widetilde{q}^i} p_k \frac{\partial \widetilde{q}^i}{\partial q^l} \dd{q^l} + = \delta^k_l p_k \dd{q^l} + = p_k \dd{q^k}. \end{equation} \end{proof} @@ -4976,11 +4976,12 @@ \chapter*{Preface} \subsection{Time-dependent hamiltonian systems} + \textcolor{red}{TODO: correct and rewrite} To discuss canonical transformations for time-dependent hamiltonian systems, we consider again the extended phase space $T^*M\times \mathbb{R}^2$ with the coordinates $(q^1,\ldots,q^n,p_1,\ldots,p_n, q^{n+1}=t, p_{n+1}=E)$ introduced in Section~\ref{sec:timedepH} and symplectic form $\widetilde\omega = \dd p_i\wedge \dd q^i - \dd E\wedge \dd t$ discussed in Example~\ref{ex:timedepH}. In terms of the tautological one--form, we have \begin{equation} - \widetilde\omega = \dd \widetilde\eta, \quad \widetilde\eta = p_i \dd q^i - E \dd t. + \widetilde{\omega} = \dd{\widetilde{\eta}}, \quad \widetilde{\eta} = p_i \dd{q^i} - E \dd{t}. \end{equation} A transformation $\widetilde\Phi : T^*M\times \mathbb{R}^2 \to T^*M\times \mathbb{R}^2$ that associates $(q,p,t,E)$ to $(Q,P,T,\widetilde E)$ is canonical if it preserves the symplectic form, that is, if ${\widetilde\Phi}^* \widetilde \omega = \widetilde \omega$. Exactly as in the time-independent case, one can show that there exists a function $S(q,p,t,E)$ such that \begin{equation}\label{eq:timedepgen} @@ -5806,9 +5807,9 @@ \chapter*{Preface} What we have seen so far seems very abstract, however there is a practical and direct approach to make computations in the action--angle variables that also helps clarifying the first half of the name. - Since $\dd p \wedge \dd q = \dd I \wedge \dd \phi$, the difference $p\dd q - I\dd \phi$ is a closed one--form on the neighborhood $U(M_{E^0})$. Thus, locally, + Since $\dd{p} \wedge \dd {q} = \dd{I} \wedge \dd {phi}$, the difference $p\dd q - I\dd \phi$ is a closed one--form on the neighborhood $U(M_{E^0})$. Thus, locally, \begin{equation} - p\dd q - I\dd \phi = \dd S + p\dd{q} - I\dd{\phi} = \dd{S} \end{equation} for some generating function $S = S(q,\phi)$. Consider the fundamental cycles $\gamma_1, \ldots, \gamma_n$ on $\mathbb{T}^n$, @@ -5820,16 +5821,16 @@ \chapter*{Preface} where $k=1,\ldots,n$. Integrating $\dd S$ on the cycle $\gamma_i$ one obtains \begin{equation} - I_k = \frac1{2\pi} \oint_{\gamma_k} p \dd q, \quad k=1,\ldots,n. + I_k = \frac1{2\pi} \oint_{\gamma_k} p \dd{q}, \quad k=1,\ldots,n. \end{equation} This formula, that looks like Maupertuis action, can be used to compute the canonical actions. To compute the conjugate angles, we use the generating function $\widetilde S = \widetilde S(q,I)$ of the canonical transformation $(q,p) \mapsto (\phi, I)$. That is, the function such that \begin{equation} - \dd\widetilde S = p\dd q - \phi \dd I. + \dd{\widetilde{S}} = p\dd{q} - \phi \dd{I}. \end{equation} - Since $I = I(E)$, a restriction of the closed one--form $\dd\widetilde S$ on the torus $M_E$ can be written as $\dd\widetilde S\big|_{M_E} = p\dd q$ and, therefore, + Since $I = I(E)$, a restriction of the closed one--form $\dd{\widetilde S}$ on the torus $M_E$ can be written as $\dd{\widetilde S}\big|_{M_E} = p\dd{q}$ and, therefore, \begin{equation} - \widetilde S(q,I) = \int_{x_0(q,E)}^{(q, p(q,E))} p \dd q, \quad E = E(I), + \widetilde S(q,I) = \int_{x_0(q,E)}^{(q, p(q,E))} p \dd{q}, \quad E = E(I), \end{equation} where the integral along a path on $M_E$ \emph{locally} does not depend on the choice of the path itself (globally this is generally false, a clarification would require a discussion of holonomy). Thus, the canonical angles can be determined as