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{"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"}}
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<!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script data-outdated-warner src="assets/warner.js"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 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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 &gt; 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 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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&quot;&quot;&quot; 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 &gt; 0. $ &quot;&quot;&quot;</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>
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