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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Angular Momentum and Energy\n",
+    "Prepared by: [Ceyda Alan](https://github.com/calan04), [Emmanuel Airiofolo](https://github.com/Emma-airi), [Joost Hubbard](https://github.com/Joosty) and [Angadh Nanjangud](https://www.angadhn.com/)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In this lecture we cover the following topics:\n",
+    "1. [](content:Two-Body-Relative-Dynamics)\n",
+    "2. [](content:The-Specific-Angular-Momentum)\n",
+    "3. [](content:Consequences-of-Specific-Angular-Momentum-=-Constant)\n",
+    "4. [](content:Kepler's-Second-Law)\n",
+    "5. [](content:Polar-Coordinates)\n",
+    "6. [](content:Conservation-of-Specific-Orbital-Energy:-$E$)\n",
+    "7. [](content:Admissible-Orbital-Radius)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Two-Body-Relative-Dynamics)=\n",
+    "## Two-Body Relative Dynamics"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```{math}\n",
+    "{\\bf{r}} (t_0) = \\bf{r_0}\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "```{math}\n",
+    ":label: L2_1\n",
+    "\\bf{\\ddot r} = -\\frac{\\mu}{r^3}\\bf{r}\n",
+    "```\n",
+    "\n",
+    "- System of 3 scalar second order differential equations\n",
+    "- It can be reduced to a system of 6 first order equations\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_2 \n",
+    "\\bf{\\dot r} = \\bf{v}\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_3 \n",
+    "\\bf{\\dot v} = -\\frac{\\mu}{r^3}\\bf{r}\n",
+    "```\n",
+    "\n",
+    "- To solve this system we need 6 initial conditions with $\\bf{r_0}$ and $\\bf{v_0}$ as known intial values\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_4 \n",
+    "{\\bf{r}} (t_0) = \\bf{r_0}\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_5 \n",
+    "{\\bf{v}} (t_0) = \\bf{v_0}\n",
+    "```\n",
+    "\n",
+    "- The system admits at maximum 6 independent integrals of motion\n",
+    "\n",
+    "> An Integral of Motion is a function $f({\\bf{r}} , {\\bf{v}} , t )$ that is constant for all times, t\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_6 \n",
+    "f({\\bf{r}} , {\\bf{v}} , t) = constant\n",
+    "```\n",
+    "\n",
+    "- The value of the constant is determined by the intial conditions\n",
+    "- An integral of motion reduces the degree of freedom of our problem\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:The-Specific-Angular-Momentum)=\n",
+    "## The Specific Angular Momentum"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "- The relative angular momentum of $m_2$ with respect $m_1$ is \n",
+    "```{math}\n",
+    ":label: L2_7\n",
+    "{\\bf H}_{2/1} = {\\bf{r}}\\times( m_2 {\\bf{v}}) = {\\bf{r}}\\times( m_2 {\\bf{\\dot r}})\n",
+    "```\n",
+    "\n",
+    "- per unit mass gives the specific angular momentum\n",
+    "```{math}\n",
+    ":label: L2_8 \n",
+    "{\\bf{h}}=\\frac{{\\bf H}_{2/1}} {m_2} = \\bf{r}\\times\\bf{\\dot r}\n",
+    "```\n",
+    "\n",
+    "- the time derivative of $\\bf{h}$ is\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_9 \n",
+    "\\frac{d\\bf{h}}{dt} =\\frac{d}{dt}(\\bf{r}\\times\\bf{\\dot r}) = \\bf{\\dot r}\\times\\bf{\\dot r} + \\bf{r}\\times\\bf{\\ddot r} =  \\bf{r}\\times\\bf{\\ddot r}\n",
+    "```\n",
+    "\n",
+    "- From the dynamics $\\bf{\\ddot r}=-\\frac{\\mu}{r^3}\\bf{r}$, thus;\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_10\n",
+    "\\frac{d\\bf{h}}{dt}=\\bf{r}\\times\\bf{\\ddot r}=\\bf{r}\\times-\\frac{\\mu}{r^3}\\bf{r}=0\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_11\n",
+    "\\frac{d\\bf{h}}{dt}=0\n",
+    "```\n",
+    "\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_12\n",
+    "\\bf{h} = constant\n",
+    "```\n",
+    "\n",
+    "This result has the following implications:\n",
+    "\n",
+    "- ${\\bf{h}}({\\bf{r}} , {\\bf{\\dot r}} , t)$ = constant.\n",
+    "- $\\bf{h}$ is an integral of motion.\n",
+    "- All 3 scalar quantities are constants: $h_x({\\bf{r}} , {\\bf{\\dot r}} , t)$ = constant, $h_y({\\bf{r}} , {\\bf{\\dot r}} , t)$ = constant, $h_z({\\bf{r}} , {\\bf{\\dot r}} , t)$ = constant."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Consequences-of-Specific-Angular-Momentum-=-constant)=\n",
+    "## Consequences of Specific Angular Momentum = constant"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_13\n",
+    "\\bf{h} = \\bf{r}\\times\\bf{v}\n",
+    "```\n",
+    "\n",
+    "hence:\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_14\n",
+    "\\bf{h}\\perp\\bf{r}\\times\\bf{v} \n",
+    "```\n",
+    "\n",
+    "**Direction** \n",
+    "\n",
+    "![Figure 1](images/L2_1.png)\n",
+    "\n",
+    "- $\\bf{r}$ and $\\bf{v}$ must at all times lie in a plane perpendicular to $\\bf{h}$ \n",
+    "- The motion is planar\n",
+    "\n",
+    "**Magnitude**\n",
+    "\n",
+    "![Figure 2](images/L2_2.png)\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_15 \n",
+    "\\bf{h} = \\bf{r}\\times\\bf{v} = \\bf{r}\\times(v_r\\bf{e_r} + v_\\theta\\bf{e_\\theta})\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_16\n",
+    "=\\bf{r}\\times(v_r\\bf{e_r}) + \\bf{r}\\times(v_\\theta\\bf{e_\\theta})\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_17\n",
+    "=rv_\\theta{\\bf{\\hat{h}}} ⇒ rv_\\theta = constant\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_18\n",
+    "dA = \\frac{1}{2}rvdt\\sin(\\phi)\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_19\n",
+    "\\frac{dA}{dt} = \\frac{1}{2}rv\\sin(\\phi) = \\frac{1}{2}rv_\\theta \n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_20\n",
+    "⇒ \\frac{dA}{dt}=\\frac{1}{2}h = constant\n",
+    "```\n",
+    "\n",
+    "> *Kepler's Second Law: Areal velocity is constant*"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Kepler's-Second-Law)=\n",
+    "## Kepler's Second Law\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_20\n",
+    "⇒ \\frac{dA}{dt}=\\frac{1}{2}h = constant\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Polar-Coordinates)=\n",
+    "## Polar Coordinates"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "If the motion is planar we can use polar coordinates to fully describe it. \n",
+    "\n",
+    "![Figure 3](images/L2_3.png)\n",
+    "\n",
+    "- In polar coordinates $\\bf{r} = r\\bf{e_r}$\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_21\n",
+    "\\frac{d\\bf{r}}{dt} = \\bf{v} = {\\dot r}\\bf{e_r} + {r}\\bf{e_\\theta}\\dot \\theta \n",
+    "```\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_22\n",
+    "\\bf{h}=\\bf{r}\\times\\bf{v}=r\\bf{e_r}\\times({\\dot r}\\bf{e_r} + {r}\\bf{e_\\theta}\\dot \\theta)\n",
+    "```\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_23\n",
+    "⇒r^2\\dot \\theta{\\bf{k}} = rv_\\theta {\\bf{k}} = constant\n",
+    "```\n",
+    "\n",
+    ">\n",
+    "```{math} \n",
+    ":label: L2_24\n",
+    "\\frac{dA}{dt}=\\frac{1}{2}h=\\frac{1}{2}r^2\\dot \\theta\n",
+    "```\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_25\n",
+    "{\\frac{d^2\\bf{r}}{dt}} = {\\bf{a}} = ({\\ddot r} - r\\dot\\theta^2) {\\bf{e_r}} + (r\\ddot \\theta+2\\dot r\\dot \\theta) {\\bf{e_\\theta}}\n",
+    "```\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_26\n",
+    "\\frac{-\\mu}{r^2}\\bf{e_r}\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_27\n",
+    "{\\ddot r} - r\\dot \\theta^2 = \\frac{-\\mu}{r^2}\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_28\n",
+    "r\\ddot \\theta + 2{\\dot r}\\dot \\theta = 0 ↔ \\frac{d}{dt}(r^2\\dot \\theta)=2r{\\dot r}\\dot \\theta+r^2\\ddot \\theta=r(r\\ddot \\theta+2{\\dot r}\\dot \\theta) =0\n",
+    "```\n",
+    "\n",
+    "- {eq}`L2_28` is equivalent to $\\bf{h}$ = constant which is a scalar"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Conservation-of-Specific-Orbital-Energy:-$E$)=\n",
+    "## Conservation of Specific Orbital Energy: $E$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "\n",
+    "Take the two-body dynamics and the dot product with $\\bf{\\dot r}$;\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_29\n",
+    "\\bf{\\dot r}\\cdot\\bf{\\ddot r}=\\frac{-\\mu}{r^2}\\bf{\\hat{r}}\\cdot\\bf{\\dot r}\n",
+    "```\n",
+    "\n",
+    "1. Note that:\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_30\n",
+    "\\frac{d}{dt}(\\frac{1}{2}v^2)=\\frac{1}{2}\\frac{d}{dt}(\\bf{\\dot r\\cdot\\dot r})\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_31\n",
+    "⇒\\frac{1}{2}{\\bf{\\ddot r}} {\\cdot\\bf{\\dot r}} + {\\frac{1}{2}\\bf{\\dot r}}{\\cdot\\bf{\\ddot r}} = {\\bf{\\dot r}}{\\cdot\\bf{\\ddot r}}\n",
+    "```\n",
+    "\n",
+    "thus; \n",
+    "\n",
+    "- \n",
+    "```{math}\n",
+    ":label: L2_32\n",
+    "⇒({\\bf{\\dot r}}\\cdot{\\bf{\\ddot r}}) = \\frac{d}{dt}(\\frac{1}{2}v^2)\n",
+    "```\n",
+    "\n",
+    "> Notice that this is the derivative of the kinetic energy\n",
+    "\n",
+    "- In polar coordinates $v^2={v_r}^2+{v_\\theta}^2={\\dot r}^2+r^2\\dot \\theta^2$\n",
+    "\n",
+    "2. Note that:\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_33\n",
+    "\\frac{d}{dt}(\\frac{\\mu}{r})=\\frac{\\partial}{\\partial r}\\frac{\\mu}{r}\\frac{dr}{dt} = -\\frac{\\mu}{r^2}{\\dot r}\n",
+    "```\n",
+    "\n",
+    "thus \n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_34\n",
+    "-\\frac{\\mu}{r^2}{\\bf{\\hat{r}}}{\\cdot\\bf{\\dot r}} = \\frac{\\mu}{r^2}\\dot r = \\frac{d}{dt}(\\frac{\\mu}{r})\n",
+    "```\n",
+    "\n",
+    ">Notice that this is the derivative of the potential of the gravotational force\n",
+    "\n",
+    "Combining the Derivatives of the Kinetic Energy {eq}`L2_32` and Potential of Gravitationl Force {eq}`L2_34`, we get the Conservation of Specific Orbital Energy {eq}`L2_36`.\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_35\n",
+    "\\frac{d}{dt}(\\frac{1}{2}v^2-\\frac{\\mu}{r})=0\n",
+    "```\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_36\n",
+    "\\frac{1}{2}v^2-\\frac{\\mu}{r} = constant\n",
+    "```\n",
+    "\n",
+    ">Notice that this is the conservation of specific orbital energy"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "(content:Admissible-Orbital-Radius)=\n",
+    "## Admissible Orbital Radius"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_37\n",
+    "E=\\frac{1}{2}v^2-\\frac{\\mu}{r}=\\frac{1}{2}{\\dot r}^2+\\frac{1}{2}r^2\\dot \\theta-\\frac{\\mu}{r}=\\frac{1}{2}{\\dot r}^2+\\frac{1}{2}\\frac{h^2}{r^2}-\\frac{\\mu}{r}\n",
+    "```\n",
+    "\n",
+    "- $\\frac{1}{2}\\frac{h^2}{r^2}-\\frac{\\mu}{r}$ is the effective potential $\\phi(r)$\n",
+    "- $\\frac{1}{2}\\frac{h^2}{r^2}$ is the potential of the contributed force\n",
+    "- $\\frac{\\mu}{r}$ is the potential gravitational force\n",
+    "\n",
+    "![Figure 4](images/L2_4.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The motion is only possible for those values of $r$ such that $E\\geq\\phi_{eff}$\n",
+    "\n",
+    "![Figure 5](images/L2_5.png)\n",
+    "\n",
+    "- For $E<0$ the motion occurs between $r_{min}-r_{max}$\n",
+    "- For $E\\geq0$ the motion is unbounded\n",
+    "- For $E=min\\phi(r), r_{min}=r_{max}$ hence we have a circular orbit\n",
+    "\n",
+    "![Figure 6](images/L2_6.png)\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Given initial position $\\bf{r_0}$ and velocity $\\bf{v_0}$ we can \n",
+    "\n",
+    "- Compute \n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_38\n",
+    "\\bf{h}=\\bf{h_0}=constant\n",
+    "```\n",
+    "     \n",
+    ">orbital plane\n",
+    "\n",
+    ">$r_0^2\\dot \\theta_0^2\\rightarrow$ Areal velocity\n",
+    "\n",
+    "- Compute\n",
+    "\n",
+    "```{math} \n",
+    ":label: L2_39\n",
+    "E_0=\\frac{1}{2}v_0^2 - \\frac{\\mu}{r_0}\n",
+    "```\n",
+    "\n",
+    "if $E<0$ bounded motion and $E\\geq0$ unbounded motion\n",
+    "\n",
+    "- Compute\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_40\n",
+    "r_{min/max}\n",
+    "```\n",
+    "\n",
+    "when\n",
+    "\n",
+    "```{math}\n",
+    ":label: L2_41\n",
+    "\\phi_{eff}=E=E_0\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
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