-
Notifications
You must be signed in to change notification settings - Fork 0
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Documenter.jl
committed
Jul 9, 2024
1 parent
2b55c49
commit 91a96b3
Showing
6 changed files
with
7 additions
and
15 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -1 +1 @@ | ||
{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-09T16:24:00","documenter_version":"1.5.0"}} | ||
{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-09T16:34:14","documenter_version":"1.5.0"}} |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -1,11 +1,3 @@ | ||
<!DOCTYPE html> | ||
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Problem definition · AveragedKepler</title><meta name="title" content="Problem definition · AveragedKepler"/><meta property="og:title" content="Problem definition · AveragedKepler"/><meta property="twitter:title" content="Problem definition · AveragedKepler"/><meta name="description" content="Documentation for AveragedKepler."/><meta property="og:description" content="Documentation for AveragedKepler."/><meta property="twitter:description" content="Documentation for AveragedKepler."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="index.html">AveragedKepler</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href="index.html">Problem definition</a></li><li><span class="tocitem">3D Visualization</span><ul><li><a class="tocitem" href="synthesis.html">Optimal Synthesis</a></li><li><a class="tocitem" href="spheres.html">Spheres</a></li><li><a class="tocitem" href="geodesics.html">Geodesics</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href="index.html">Problem definition</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="index.html">Problem definition</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/control-toolbox/averaged_kepler" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/control-toolbox/averaged_kepler/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-averaged-Kepler-problem"><a class="docs-heading-anchor" href="#The-averaged-Kepler-problem">The averaged Kepler problem</a><a id="The-averaged-Kepler-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-averaged-Kepler-problem" title="Permalink"></a></h1><p>We consider the Hamiltonian</p><p class="math-container">\[ | ||
H(r, \theta, p_r, p_\theta) = v p_\theta + \Vert p \Vert_{g} | ||
$ | ||
|
||
where $v\]</p><p>is a constant, <span>$p = (p_r, p_\theta)$</span>, and <span>$\Vert \cdot \Vert_{g}$</span> is the norm induced by the metric</p><p class="math-container">\[ | ||
g = \mathrm{d}r^2 + m_\lambda^2(r)\, \mathrm{d}\theta^2, \quad | ||
m_\lambda^2(r) = \frac{\sin^2 r}{1 - \lambda \sin^2 r} | ||
$ | ||
|
||
with $\lambda = 4/5\]</p><p>.</p><p>Along the geodesics, we have <span>$H+p^0 = 0$</span>. The parameter <span>$p^0$</span> is constant equal to <span>$-1$</span> (hyperbolic), <span>$0$</span> (abnormal) or <span>$1$</span> (elliptic).</p><p><strong>Remark.</strong> We can parameterize the geodesics by the norm of the initial convector, setting <span>$\Vert{p_0}\Vert_g = 1$</span>. This amounts to parameterize by the initial angle <span>$\alpha_0$</span>: $ p<em>r = \sin \alpha</em>0, \quad p<em>\theta = m</em>\lambda(r) \cos \alpha<em>0. $ In that case, the hyperbolic geodeics are given by $ p</em>\theta\, v + 1 = v\, m<em>\lambda(r) \cos \alpha</em>0 + 1 > 0. $</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="synthesis.html">Optimal Synthesis »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Tuesday 9 July 2024 16:24">Tuesday 9 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> | ||
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Problem definition · AveragedKepler</title><meta name="title" content="Problem definition · AveragedKepler"/><meta property="og:title" content="Problem definition · AveragedKepler"/><meta property="twitter:title" content="Problem definition · AveragedKepler"/><meta name="description" content="Documentation for AveragedKepler."/><meta property="og:description" content="Documentation for AveragedKepler."/><meta property="twitter:description" content="Documentation for AveragedKepler."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="index.html">AveragedKepler</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href="index.html">Problem definition</a></li><li><span class="tocitem">3D Visualization</span><ul><li><a class="tocitem" href="synthesis.html">Optimal Synthesis</a></li><li><a class="tocitem" href="spheres.html">Spheres</a></li><li><a class="tocitem" href="geodesics.html">Geodesics</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href="index.html">Problem definition</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="index.html">Problem definition</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/control-toolbox/averaged_kepler" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/control-toolbox/averaged_kepler/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-averaged-Kepler-problem"><a class="docs-heading-anchor" href="#The-averaged-Kepler-problem">The averaged Kepler problem</a><a id="The-averaged-Kepler-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-averaged-Kepler-problem" title="Permalink"></a></h1><p>@doc raw""" We consider the Hamiltonian</p><p>$</p><pre><code class="nohighlight hljs">H(r, \theta, p_r, p_\theta) = v p_\theta + \Vert p \Vert_{g}</code></pre><p>$</p><p>where <span>$v$</span> is a constant, <span>$p = (p_r, p_\theta)$</span>, and <span>$\Vert \cdot \Vert_{g}$</span> is the norm induced by the metric</p><p>$</p><pre><code class="nohighlight hljs">g = \mathrm{d}r^2 + m_\lambda^2(r)\, \mathrm{d}\theta^2, \quad | ||
m_\lambda^2(r) = \frac{\sin^2 r}{1 - \lambda \sin^2 r}</code></pre><p>$</p><p>with <span>$\lambda = 4/5$</span>.</p><p>Along the geodesics, we have <span>$H+p^0 = 0$</span>. The parameter <span>$p^0$</span> is constant equal to <span>$-1$</span> (hyperbolic), <span>$0$</span> (abnormal) or <span>$1$</span> (elliptic).</p><p><strong>Remark.</strong> We can parameterize the geodesics by the norm of the initial convector, setting <span>$\Vert{p_0}\Vert_g = 1$</span>. This amounts to parameterize by the initial angle <span>$\alpha_0$</span>: $ p<em>r = \sin \alpha</em>0, \quad p<em>\theta = m</em>\lambda(r) \cos \alpha<em>0. $ In that case, the hyperbolic geodeics are given by $ p</em>\theta\, v + 1 = v\, m<em>\lambda(r) \cos \alpha</em>0 + 1 > 0. $ """</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="synthesis.html">Optimal Synthesis »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Tuesday 9 July 2024 16:34">Tuesday 9 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
Oops, something went wrong.