Notebooks/Clase5b_Mecanica-SymPy.ipynb

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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<img src=\"../static/aeropython_name_mini.png\" alt=\"AeroPython\" style=\"width: 300px;\"/>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Clase 5b: Mec\u00e1nica con SymPy"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "_Si SymPy te ha parecido hasta ahora un CAS decente e incluso interesante (nada como tener los resultados en $\\LaTeX$ incrustados en el notebook y la sintaxis de Python para hacer c\u00e1lculo simb\u00f3lico) entonces espera a ver el paquete `mechanics`. Con \u00e9l, podremos manipular velocidades y aceleraciones de s\u00f3lidos expresadas en distintos sistemas de referencia con una facilidad impresionante._\n",
      "\n",
      "_Tienes disponible la documentaci\u00f3n de `mechanics` en http://docs.sympy.org/0.7.5/modules/physics/mechanics/index.html._"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Sistemas de referencia"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El objeto primordial que vamos a manejar van a ser los sistemas de referencia. Podremos definir relaciones geom\u00e9tricas entre ellos y de esta forma las transformaciones de vectores entre un sistema y otro ser\u00e1n triviales."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import init_session\n",
      "init_session(use_latex=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "IPython console for SymPy 0.7.5 (Python 3.4.2-64-bit) (ground types: python)\n",
        "\n",
        "These commands were executed:\n",
        ">>> from __future__ import division\n",
        ">>> from sympy import *\n",
        ">>> x, y, z, t = symbols('x y z t')\n",
        ">>> k, m, n = symbols('k m n', integer=True)\n",
        ">>> f, g, h = symbols('f g h', cls=Function)\n",
        "\n",
        "Documentation can be found at http://www.sympy.org\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Todo lo que necesitamos est\u00e1 en `sympy.physics.mechanics`, incluyendo la clase `ReferenceFrame`. Nada m\u00e1s crear un sistema de referencia podemos acceder a sus versores unitarios: `x`, `y` y `z`.\n",
      "\n",
      "http://docs.sympy.org/0.7.5/modules/physics/vector/vectors.html"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import ReferenceFrame"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = ReferenceFrame(\"A\")\n",
      "A.x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{\\hat{a}_x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAUBAMAAAB7UtADAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAECLdzUTvdomZZjJU\nu6tkz/T2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiklEQVQIHWNgYGBkQABlRzibNcVMAcaRL+BI\ngLGBygVgbGS62cVSAcrn+sawPwnK5vnGwPsTpm4jA+cXGFum9zGMzfO/ES6u/wuk5vB/gXxDhvhP\nDLxANfeDUhkY5L8zbPmswMD87QADA9Ozew33vzMwfUM4hPe9Icxghnesf2HsZd8Z/ufAOCAaAJAn\nJoxLDW8EAAAAAElFTkSuQmCC\n",
       "prompt_number": 3,
       "text": [
        "\u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y para definir vectores solo tenemos que **multiplicar cada componente por su versor**:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 * A.x - 1 * A.y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2\\mathbf{\\hat{a}_x} - \\mathbf{\\hat{a}_y}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAWBAMAAABzm9yKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOElEQVQoFW3RMUvDQBQH8P+RS1tDjXURdWnU\nD2BB6Jp8A29ycemigovgIm5dnaxIBwWprmZIB6cuyeQsLo71G4igNKnCGZNQ8857y9378X93RwLk\nJYpVXVRnvir5RMnZ+pYHnI0u1LOyvuQ7qMYwv3GqC5Z9D7gCBExdsOzXwJGnDSkYijRY9Udrimet\n4oF46/GkoUlSr00w7sF1NEHqVgtcYHOoCVLfAMyXdvAX3L3/rYd0kHi9A4wnQnsi8WewRfmaXj0f\nfM5NyZ9kmduDPk+64A7qy7IBdwgjXumSh9Zyb96iD6yeHB+aMuLhHRDckBwKr8S2A4RSTrF0eWDI\nCM10kFTu/MtSPnKbPnE24+7Pttmm0jlvUSm67XfC9iBaSJ4IFY2hn1ejLLI81bR99eNR6/+Q+U5q\nP3bPZGZrDwqmAAAAAElFTkSuQmCC\n",
       "prompt_number": 4,
       "text": [
        "2 \u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m - \u001b[94m\u001b[1ma_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "De ahora en adelante, para trabajar como si nos enfrent\u00e1ramos a un problema de la escuela, vamos a hacer dos cosas:\n",
      "\n",
      "* Definir un sistema inercial $1$ del que partir, para as\u00ed poder referir todos los dem\u00e1s sistemas a \u00e9l.\n",
      "* Que los versores de ese sistema sean $i, j, k$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = ReferenceFrame(\"1\", latexs=['\\mathbf{i}', '\\mathbf{j}', '\\mathbf{k}'])\n",
      "A.x + A.y + A.z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{i} + \\mathbf{j} + \\mathbf{k}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAATBAMAAADMlCYHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMpkQdond\nRGZc8tcOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVQoFWNgEDJgQAFFJ1G47P9/gfnM/x1Q\nxBnyIeJwwSIIn/V/AlwIzFj/F5XPAtVXDBWeCqWZL0AYnAEQGqYOKs1gBWNAaeYFEAZUHT/MXqg6\n3v+/IfJI6uyBalj9EyDiMPNiMNVNygQq6UdTx4ihjlUbZBJhdfcfEKXuC8QHYPO2uLgccnHxAOkD\n28vt4uKj4uLSAOSz/Hn/D0gRtvcz139QUBJ032cG+QPEqYv/wcDA//8/UDEQwMIPLVzY//932P//\nK0QNQh1nAZo6hAIo6x6Y5v/O9BkiwPsAKoGV4v9bfwCrBJogr8+JCWhCYC4ASv1WiKruNuQAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "\u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y para no tener que hacerlo siempre, un peque\u00f1o truco de magia:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Definimos nuestra propia clase para que los versores sean IJK\n",
      "class IJKReferenceFrame(ReferenceFrame):\n",
      "    def __init__(self, name):\n",
      "        super().__init__(name, latexs=['\\mathbf{%s}_{%s}' % (idx, name) for idx in (\"i\", \"j\", \"k\")])\n",
      "        self.i = self.x\n",
      "        self.j = self.y\n",
      "        self.k = self.z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = IJKReferenceFrame(\"1\")\n",
      "A.i + A.j + A.k"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{i}_{1} + \\mathbf{j}_{1} + \\mathbf{k}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAATBAMAAAB/3A4dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMpnddhCJ\nRGZaefImAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLklEQVQoFWNgEDJgwADsmhswxf7/gosx/3eA\ns+EMtv8L4Gw4owihifV/AlwYzuD6fwHOhjNYEJoYiuGiQMZWKMcWiyCyJmRpBisUHpQDFUTSxI/i\nPKj8/f8HkHQjNNlDFbP6J2DKs87HrmlTNlRtP1BTIUwfzHnyBxjYBdAEWX6x6gCFwOJATRWZaPIM\n8gfY2yagCbL8uv8AqAcsDrLJEk0eqImBGUPTF7CPwOIITUdcXCa5uHiADEBoQgiy/Hz/DyiFpgko\nguQnTJs+c/0PIF0TgzzQyaTaxBD/A6KJ//9/4gKC/f9/h/P/vyICCC30tiMFBJJHgUwggPvVCMJn\nYLgHZvwv0DdgYFZAFYTxYOIR87NgQmD6//WfDKyd3xagCEI5uMQZrszsxqYeLgYAKb1mXNaAdd8A\nAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "\u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "\u00c1lgebra vectorial"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Nuestros vectores funcionan tambi\u00e9n con s\u00edmbolos, y podemos realizar las operaciones de producto escalar y producto vectorial con ellos."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "R, V = symbols('R, V', positive=True)\n",
      "r1 = R * (A.x + A.y + A.z)\n",
      "v1 = V * (A.x - 2 * A.z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$R\\mathbf{i}_{1} + R\\mathbf{j}_{1} + R\\mathbf{k}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAATBAMAAABvkIKEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 9,
       "text": [
        "R \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + R \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + R \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$V\\mathbf{i}_{1} -  2 V\\mathbf{k}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAASBAMAAABFi4Y1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny2wqMF1p8AAAAABJRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": [
        "V \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m - 2\u22c5V \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import dot, cross"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1.dot(v1)\n",
      "dot(r1, v1)\n",
      "r1 & v1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- R V$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAOBAMAAABN+VgMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVQYGVWPMQ4BURCGP7FrV5AVF9DoFCQO\nQOIAtlCzDUpqDUfYhEZFKSqJC2w0WjfgBkKxrAgGkbz3F/Mm35/3zwz8ZfbDbs9l8ipC/R78sbzO\nAPtK7CJtylM4DR/6NVbCEipnXoPqmJLAqWYMJeIhn8bYe80IYSNphyNpjRvhbOQKafs0NSN5xlwK\nOQW/pFzlowJYC5AR5Af6TjgetozBKbekKmq436WI3wKFSitnZCKyWNesbshhZmSsST91vn11YDcT\n1/sabx/pL43+CN+kAAAAAElFTkSuQmCC\n",
       "prompt_number": 12,
       "text": [
        "-R\u22c5V"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1.cross(v1)\n",
      "cross(r1, v1)\n",
      "r1 ^ v1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  2 R V\\mathbf{i}_{1} + 3 R V\\mathbf{j}_{1} -  R V\\mathbf{k}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAAATBAMAAACO+5fVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 13,
       "text": [
        "- 2\u22c5R\u22c5V \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + 3\u22c5R\u22c5V \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m - R\u22c5V \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Podemos hallar tambi\u00e9n la norma de los vectores con su m\u00e9todo `magnitude` e incluso normalizarlos con `normalize`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(r1 ^ v1).magnitude()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{14} R V$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVBAMAAADiLT7tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABdUlEQVQoFW2RvUsDQRDF3yXkcpczH41gIxzp\nBDGBlCpJYS9W2hkQsRLS2si1FmJKO9NZWHiCNhZ+IOks8gcISWsVktOgYIhvJjFF1oGdfffmt7uz\ne8A4rMJoGhPLmOIVwzKMecMxjQPTmnWc+qxjfts105t1TsTYkZSpAl45Kq2HWBptA+8/NDSumBe/\nRLotpmwL1hDOJ2WyKC5D2jndU2ZZmK4PlDt4pYxzaMRCTjFhrDthCh3gIoddyhUObHC4IpRJJYR5\n4xkDbpWD1ZBKkynPMWZulImAIx+4rsCWykKhAtRFyT6OL0wmyrdDOo8+HqSCsxa8hghhUhCGynum\ns1nVowC7Bzf4Y46VST8BbIfLJ7eyBrgXRFY7DWWyRVhsCdn6rVbYfXCuikyyVCq/+OiGei24veqE\n6V6OlfQDzLEfPk+qjwDpYSAWI9EMdB4z/A3ydl4/U4P9oQUmeS2Gvfa9you1o3B/RHHIN/OKWpG0\nNVX/i18UPWk6HD1fSgAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "  ____    \n",
        "\u2572\u2571 14 \u22c5R\u22c5V"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(r1 ^ v1).normalize()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  \\frac{\\sqrt{14}}{7}\\mathbf{i}_{1} + \\frac{3 \\sqrt{14}}{14}\\mathbf{j}_{1} -  \\frac{\\sqrt{14}}{14}\\mathbf{k}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQIAAAAwBAMAAAAIpGlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 15,
       "text": [
        "   ____ \n",
        "-\u2572\u2571 14  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   7     \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m +     ____\n",
        "3\u22c5\u2572\u2571 14 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   14    \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m +    ____ \n",
        "-\u2572\u2571 14  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   14    \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##### Ejercicio\n",
      "\n",
      "Usando directamente la f\u00f3rmula para la derivada en ejes m\u00f3viles:\n",
      "\n",
      "$$\\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_1 = \\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_0 + \\mathbf{\\omega}_{01}\\! \\times \\mathbf{a}$$\n",
      "\n",
      "Calcula la derivada del vector de posici\u00f3n $R \\mathbf{i}_0$, siendo $A_0$ un sistema de referencia que gira respecto al inercial con velocidad angular $\\mathbf{\\omega}_{01}=\\Omega \\mathbf{k}_0$. **\u00bfCu\u00e1l es el m\u00f3dulo de la derivada?**"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "R, Omega = symbols('R, Omega', positive=True)\n",
      "A0 = IJKReferenceFrame('0')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = R * A0.i\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$R\\mathbf{i}_{0}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAASBAMAAABRHhjUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRCJ\nzUQBtxRlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxUlEQVQYGU3OPw4BQRTH8W+w/i0ZUSkkJgo1\njSjdQK0yiU6johRHEBfY7Uk4gkrLDTiCf6tRjNndjPWq9z55b+YHtnor21V1s9mhoM8WeMKgndHS\ngvOGVMDQzmQDyL1+I6SX4PkiORGSwpzMRdqlUWt8N/1Okj0eTMN0T60fQZ11CD0QfgQzxN7Aw8SQ\nIbgv8iauYx7wog3nQ65CFOOkXLMRgoLyDRpqGII5URQvjwrFrhJauwHiyn8t4l8SmsQ5EihtN3wB\noj40JUWOI5EAAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": [
        "R \u001b[94m\u001b[1m0_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "omega01 = Omega * A0.k\n",
      "omega01"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega\\mathbf{k}_{0}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAASBAMAAADBDWEbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\nEO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6ElEQVQYGVXOP04CQRTH8e+wbpbd1WQqWzEe\nQCsLGr2BVNByBE7AEi6wXoDEGENlosbGW8gJxGinFSCb9Q8ZfmOhyyvmvXwy8+YHmIN9iNyCv9pq\n290O9CuUD+CkQfhP0VL3w1mV4i9RUlQpnYvqP54O3blm2C51BJ+enka/QlCox6XIvGlqNmHnXUP6\nLDobaMNFTY9zC5lCLP2mtMdKAY7gWu0719ap5QVjWnBnCOeBs2QTHkgYYbqJiMdbT/fEwytql8ee\n9goyyyuJK6m7ReRc68bNpj0+lKBa6an/caOCrs+1WePxZA2ClklZiRwIRAAAAABJRU5ErkJggg==\n",
       "prompt_number": 18,
       "text": [
        "\u03a9 \u001b[94m\u001b[1m0_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "da = omega01 ^ a\n",
      "da"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega R\\mathbf{j}_{0}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\nEO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABM0lEQVQYGVWPvUoDQRSFv0mU/VOylfWKdjaB\ngEWapBFLrbQ0jZ3CvkEWXyApFZQsIqlsxEawiE/gbpfSvzKFaDQkmjDe2RSyF+Zw5ps7d84Aam2V\n/7IG6XyzsOev7Iu1O5Oz8whHl+e8FUEtEF96QE0p6rp4sL5FFj9EmtLYCVk3FJwfEXckUguh5xtm\nyvsUsacij1AYsaNj8bA0FilORGRtlFG9WLwgM8KRQ2tSSSLxSUy1CssD8d6zNMywXzJux4U2tOSp\nZiAD5czcTWIvZCYZy3At+1IbZZ5J4iefV5TahRsldyITx/Bmyh0uF6iGm8V3h6RzfotzckXhchMO\n5BtDKzDc5w1Xj7H1kEN9CkeVbH7IlwzMVV/6vbrJky8dvgfFhsmfL739C91umqewdX+coT/7oGEp\n9ULDIAAAAABJRU5ErkJggg==\n",
       "prompt_number": 19,
       "text": [
        "\u03a9\u22c5R \u001b[94m\u001b[1m0_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "da.magnitude()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\Omega R$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAPBAMAAADuVgmRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\nEO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVQIHU2OsU7CUBSGv0tp2t5K2om5hpHF\npAkrr+AkIy5uDLyBjS9QR0kgTYiji08AT6A+gs5M2GCowdS/nbjJ/e+Xc/7/ngOYwSXnp3sT9ycq\n+EW1WGaCXDJOBNEWcwLvIHb3kvsrKOYEv2L7IxnPYRMTfot9WXmDjhoXR7FTSXSHSjmNO1DVq9L3\nTNzbScJPdf7wv8TksQYkyqrYuHGVe9EbPWKar4y5hlcjV9YOwbLC3Np2DVvyQfDwTGc9gqnWKb0E\nWx/x65K7+glmKf+/RDSDL80FuQAAAABJRU5ErkJggg==\n",
       "prompt_number": 20,
       "text": [
        "\u03a9\u22c5R"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=\"alert alert-warning\">Si no especificaste `positive=True` vas a ver algo como $\\sqrt{\\Omega^2 R^2}$. Deber\u00eda haber una forma de simplificar esta expresi\u00f3n _a posteriori_, pero de momento no funciona del todo bien. Preparando este notebook nos hemos dado cuenta y ya les hemos avisado :) https://github.com/sympy/sympy/issues/8326\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Movimiento relativo"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\u00bfA qui\u00e9n no le gusta multiplicar matrices de rotaci\u00f3n? Para esa minor\u00eda que lo detesta, existe SymPy. Para ello debemos especificar la orientaci\u00f3n de nuestros sistemas de referencia usando el m\u00e9todo `orient`, y recuperaremos la matriz de cosenos directores usando el m\u00e9todo `dcm`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1 = IJKReferenceFrame(\"1\")\n",
      "A0 = IJKReferenceFrame(\"0\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phi = symbols('phi')\n",
      "A0.orient(A1, 'Axis', [phi, A1.z])  # Rotaci\u00f3n phi alrededor del eje A1.z\n",
      "A0.dcm(A1)  # \"Direct Cosine Matrix\""
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} & \\sin{\\left (\\phi \\right )} & 0\\\\- \\sin{\\left (\\phi \\right )} & \\cos{\\left (\\phi \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAABLCAMAAAALSZigAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n/3SqlPgLJN1NThRRO9AAAAAASUVORK5CYII=\n",
       "prompt_number": 22,
       "text": [
        "\u23a1cos(\u03c6)   sin(\u03c6)  0\u23a4\n",
        "\u23a2                  \u23a5\n",
        "\u23a2-sin(\u03c6)  cos(\u03c6)  0\u23a5\n",
        "\u23a2                  \u23a5\n",
        "\u23a3   0       0     1\u23a6"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Usando el argumento `Axis` hemos especificado que rotamos el sistema un \u00e1ngulo especificado alrededor de un eje. Otros m\u00e9todos son:\n",
      "\n",
      "* `Body`: se especifican los tres \u00e1ngulos de Euler.\n",
      "* `Space`: igual que `Body`, pero las rotaciones se aplican en orden inverso.\n",
      "* `Quaternion`: utilizando cuaternios, rotaci\u00f3n alrededor de un vector unitario $\\lambda$ una cantidad $\\theta$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=\"alert alert-success\">\u00bfQu\u00e9 es lo bueno de usar uno de estos m\u00e9todos? \u00a1Que **siempre** tenemos la transformaci\u00f3n bien definida! Es imposible meter \"a cap\u00f3n\" una matriz de rotaci\u00f3n que sea incorrecta o absurda.</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Diferente sistema de referencia\n",
      "\n",
      "Para expresar un vector en otro sistema de referencia, no hay m\u00e1s que usar los m\u00e9todos `express` o `to_matrix`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A0.x.express(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{cos}\\left(\\phi\\right)\\mathbf{i}_{1} + \\operatorname{sin}\\left(\\phi\\right)\\mathbf{j}_{1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAVBAMAAAD/f8IzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nufZmdTOs152YaNbQWA9F/gLES9pSGqLHmQAAAABJRU5ErkJggg==\n",
       "prompt_number": 23,
       "text": [
        "cos(\u03c6) \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + sin(\u03c6) \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A0.x.to_matrix(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )}\\\\\\sin{\\left (\\phi \\right )}\\\\0\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAABLCAMAAADtTJGhAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 24,
       "text": [
        "\u23a1cos(\u03c6)\u23a4\n",
        "\u23a2      \u23a5\n",
        "\u23a2sin(\u03c6)\u23a5\n",
        "\u23a2      \u23a5\n",
        "\u23a3  0   \u23a6"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### S\u00edmbolos din\u00e1micos\n",
      "\n",
      "Si queremos especificar que un s\u00edmbolo puede variar con el tiempo, hay que usar la funci\u00f3n `dynamicsymbols`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import dynamicsymbols"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha = dynamicsymbols('alpha')\n",
      "alpha"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\alpha{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAVBAMAAAA+1EraAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiZl2RCLdEO9Uu81m\nqzKufkATAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVQYGX2QsU7CUBSGvxZKqUIFHXQhdncQ\neIJObiZMhDDhwMDWxMXNmOjOG0A0bhq7uXZhI4RH4BE6kLj6t1cpEyc59/znO+fce3LhgPlJVlzt\ndwzzpL7ZY9/gjiEuUKkBXgqdAlV7YKvloUA3khV1Hk8lTrthdsLt12cbK4B+6L44Uz6EZ/Kahp/g\ntRQxUXol90bYP5IWPCu9k/sp1TlcnOfI3Rr0qFcu9YoGyykR5REniZCqut6KvU12vb2mdu1HtLTp\n+l5HAM1lYp+BVnXe2lo11IixSs/EwT/QioHRGtjZe67ceAfAfOHR33xecJIsLOAXOGcpup/x+cEA\nAAAASUVORK5CYII=\n",
       "prompt_number": 26,
       "text": [
        "\u03b1(t)"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y pedir su derivada con el m\u00e9todo `diff`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha.diff()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{d}{d t} \\alpha{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAArBAMAAAAu4Q2VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njeqPKvYHoMNlfCAqdJQAAAAASUVORK5CYII=\n",
       "prompt_number": 27,
       "text": [
        "d       \n",
        "\u2500\u2500(\u03b1(t))\n",
        "dt      "
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##### Ejercicio\n",
      "\n",
      "![Sistemas de ejes asociados a la pala](../static/ejes_pala.jpg)\n",
      "\n",
      "(Sacado de Cuerva et al. \"Teor\u00eda de los Helic\u00f3pteros\")\n",
      "\n",
      "**Obtener la matriz de rotaci\u00f3n de la pala $B$ respecto a los ejes $A1$.**"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = IJKReferenceFrame(\"A\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1 = IJKReferenceFrame(\"A1\")\n",
      "psi = dynamicsymbols('psi')\n",
      "A1.orient(A, 'Axis', [psi, A.z])\n",
      "A1.dcm(A)  # T_{A1A}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQAAAABLCAMAAACoRKlPAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 29,
       "text": [
        "\u23a1cos(\u03c8(t))   sin(\u03c8(t))  0\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2-sin(\u03c8(t))  cos(\u03c8(t))  0\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3    0           0      1\u23a6"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2 = IJKReferenceFrame(\"A2\")\n",
      "beta = dynamicsymbols('beta')\n",
      "A2.orient(A1, 'Axis', [beta, -A1.y])\n",
      "A2.dcm(A1)  # T_{A2A1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\sin{\\left (\\beta{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAP0AAABLCAMAAACMeHGmAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 30,
       "text": [
        "\u23a1cos(\u03b2(t))   0  sin(\u03b2(t))\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2    0       1      0    \u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3-sin(\u03b2(t))  0  cos(\u03b2(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A3 = IJKReferenceFrame(\"A3\")\n",
      "zeta = dynamicsymbols('zeta')\n",
      "A3.orient(A2, 'Axis', [zeta, A2.z])\n",
      "A3.dcm(A1)  # T_{A3A1}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAggAAABLCAMAAAAicY+6AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 31,
       "text": [
        "\u23a1cos(\u03b2(t))\u22c5cos(\u03b6(t))   sin(\u03b6(t))  sin(\u03b2(t))\u22c5cos(\u03b6(t)) \u23a4\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a2-sin(\u03b6(t))\u22c5cos(\u03b2(t))  cos(\u03b6(t))  -sin(\u03b2(t))\u22c5sin(\u03b6(t))\u23a5\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a3     -sin(\u03b2(t))           0           cos(\u03b2(t))      \u23a6"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B = IJKReferenceFrame(\"B\")\n",
      "theta = dynamicsymbols('theta')\n",
      "B.orient(A3, 'Axis', [theta, A3.x])\n",
      "B.dcm(A3)  # T_{BA3}"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\0 & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 32,
       "text": [
        "\u23a11      0           0    \u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a20  cos(\u03b8(t))   sin(\u03b8(t))\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a30  -sin(\u03b8(t))  cos(\u03b8(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.dcm(A2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & 0\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\\\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAABLCAMAAADeZKRwAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n7Lib3+obevuSS3P0Vr49/94M3N3Ad/8DsLHUm1U5qQIAAAAASUVORK5CYII=\n",
       "prompt_number": 33,
       "text": [
        "\u23a1     cos(\u03b6(t))             sin(\u03b6(t))            0    \u23a4\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a2-sin(\u03b6(t))\u22c5cos(\u03b8(t))  cos(\u03b8(t))\u22c5cos(\u03b6(t))   sin(\u03b8(t))\u23a5\n",
        "\u23a2                                                     \u23a5\n",
        "\u23a3sin(\u03b8(t))\u22c5sin(\u03b6(t))   -sin(\u03b8(t))\u22c5cos(\u03b6(t))  cos(\u03b8(t))\u23a6"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.dcm(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 34,
       "text": [
        "\u23a1                cos(\u03b2(t))\u22c5cos(\u03b6(t))                        sin(\u03b6(t))         \n",
        "\u23a2                                                                             \n",
        "\u23a2-sin(\u03b2(t))\u22c5sin(\u03b8(t)) - sin(\u03b6(t))\u22c5cos(\u03b2(t))\u22c5cos(\u03b8(t))  cos(\u03b8(t))\u22c5cos(\u03b6(t))   -\n",
        "\u23a2                                                                             \n",
        "\u23a3-sin(\u03b2(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b2(t))  -sin(\u03b8(t))\u22c5cos(\u03b6(t))  s\n",
        "\n",
        "               sin(\u03b2(t))\u22c5cos(\u03b6(t))                 \u23a4\n",
        "                                                   \u23a5\n",
        "sin(\u03b2(t))\u22c5sin(\u03b6(t))\u22c5cos(\u03b8(t)) + sin(\u03b8(t))\u22c5cos(\u03b2(t))\u23a5\n",
        "                                                   \u23a5\n",
        "in(\u03b2(t))\u22c5sin(\u03b8(t))\u22c5sin(\u03b6(t)) + cos(\u03b2(t))\u22c5cos(\u03b8(t)) \u23a6"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Velocidad angular\n",
      "\n",
      "Tambi\u00e9n podemos hallar la velocidad angular de un sistema respecto a otro usando el m\u00e9todo `ang_vel_in`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.ang_vel_in(A2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIYAAAAaBAMAAACA8zmaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfCTAvp8TzrrQUVuYf1S6mNPZPtm/AUV7sBI/Lf1EAAAAAElFTkSuQmCC\n",
       "prompt_number": 35,
       "text": [
        "\u03b8\u0307 \u001b[94m\u001b[1ma3_x\u001b[0;0m\u001b[0;0m + \u03b6\u0307 \u001b[94m\u001b[1ma2_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.ang_vel_in(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2} -  \\dot{\\beta}\\mathbf{j}_{A1} + \\dot{\\psi}\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAAAaBAMAAACwWn+SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 36,
       "text": [
        "\u03b8\u0307 \u001b[94m\u001b[1ma3_x\u001b[0;0m\u001b[0;0m + \u03b6\u0307 \u001b[94m\u001b[1ma2_z\u001b[0;0m\u001b[0;0m - \u03b2\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m + \u03c8\u0307 \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B.ang_vel_in(A).express(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(\\left(- \\operatorname{sin}\\left(\\psi\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{i}_{A} + (\\left(\\operatorname{sin}\\left(\\psi\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\psi\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{j}_{A} + (\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} + \\operatorname{cos}\\left(\\beta\\right) \\dot{\\zeta\\left(t\\right)} + \\dot{\\psi})\\mathbf{k}_{A}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABzYAAAAZBAMAAAChqtWOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\nVKvu110NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAR70lEQVR4Ae1bf4xc11U+49mZXY93vUsRikyU\netqotEWgDMH/RLTqkgJR/6kXxCJogA5Q01QJeNQqOBINGVElsUooLhGhqgpMUP6gQpClUVGhARal\nxa0a6IIqlR+VvEWoFVWVOG1aBzfO8n3n3Pveeffd92ZlGbCEr/Leu++ec77zne/e+2berCNyre1b\ngUO7+3a95nglFPjrKwFylWFcW0SYkCs/sSc3LmOe///MxRWvtP+Ny9D7f2LiG2lcRsn5RXQZQI2k\nrn7DZU7sFS8sPxdXPM1VAHiVVPq/OfFXrOQrBnQVrINrFK4pcE2BawpcU+CaAv83CnSGIjyaWmdI\nSwfHK9iptZdhpGuj7MqBNZzyruZl584Q1+4rR02uiiXAUjyL8efOEHdpXvpG//4rvXu1r+A331j4\nVq0iRfJYR5ooDSjuO0N2oVakITHNm3ZoqbTOMCs9h6sYvHdtP/RTxgUfh1PtEnW/UYVAXvGfH1YB\n/V2kHAX1tiiCV22kDp0hL/kYWtgKKoVbVPzIn5uHP5u3H+kMcTdnIXpZlE7M1VZyDdij1KYDKxeg\nnb0LYFO0Y+jx8O3IpLwz2xIGDiNeW+d3QweX7jpOtIauyBfRLVzVkDsp7L39H2xwVViRKUKJV28a\nX8tL3+D/2C9t+6ga58Wdt61FX++Ifpk81lFL5CI8clCS7oFGTLOyt+FirMsaTF7JKx4wkkClF3ET\nG2/VXmOcw/LU9x/lBdIyDXrwI1+okPElKTgpR0FlTsVThTJxypgKvt0osi6U6FYo82xlnat78PbU\nNMechVgRk3lCrrTkCr8acAXFNCsDpgZ6pML5J2Dn4dvRaXlnNsWNq6vzQmnujdE/fGqIM7t3jqSH\nrkRX9rONsHh4PNbgqrC2NxWvhqG0annpa/6dR+SbPqjG+RUyWA++3o99lzzUUUvkQjxyUJJqBdox\nTXdv18VYlzVoHSJ5xfOlK72IWwMN9GuMc1ieuqLuK6oiEGEN+sh4dezZ+JJKysXC8GYTwakmX34n\noTDef6JhhYRU9bkqlLnvoqej/eCd5p63EEtZIh0rIy25ko9FVYBLlLhLnP8UfYD2/N5cxCCPphZs\nN9N+T8aJY/2vntoJ5nMBLbr2R5kYDCksQP8tjxpSdYfRNUXR+HpeDhvjQ0O5lAbFe5ID9MpWQ+VG\nXpNbt54IRecqs9xCtaxbprkhZsfVYkuyzhQDHUbFipv90K8zDtREfiHFs3uitkRVgpxAZREfkVWu\ngmwrKVto1SlQK1WTXz57L1w4vnK+YYUECEel6IaJXZkFH16s6np2zT1nITpZIh0DCiUbtkvGbg3Y\nocRdEmnpctQ6K3vzIJYYj6YWbDfS/o6M08MYOzFTA7ufxASfwTW6rkzUVDspLGzPlK4VH2LhsUNi\niqe37qTxJ2Y64vLS1/xfK/I151/pMmARn/a7eWxxya2OEzONd4mwaHKVObWMRj6NxdI5BCh8ONUw\nvJH9/dA/MaOnuoYpKWT8vFpqJ6KemOmwq7OIUkM80cHNjjodOCOrQx3PnErKcWF4p1rF/beomePL\nGw0rJAA4KgadV1ysavOu5cZUti3EEzONYHCko7liyVlFtSgP7FDiLpFAK2gJ0Mre/GMk5FFp37dW\n3AbblAO9HRvufptdcR5s4fRjeqvdj6H7Mziia3YFw66wExmg3uiqIOGkWPE9lnhp0/hMXvqq/5PS\nueSDUs6Ywg+Pgq/3Q98nN3KZROXedMhRySkRlUaRZnVvl4PWTBXWEOSVBsUVI0aFq9IrcBMjbtWe\nYRyx3Epy1OdF+TxeIC2T0Ie25WhYH+brSnKUi9l25iDClHHKsjdkV8U5NM2vEHXIzVWhzGDvheDE\ni1YdiNfUnrMQnZiRjpYRS3aKlhm1KA/sUOIuKfcmv89zJ2BvnitWykMY41G06z5/d/f4buen/vbU\nlzkGW/+hiUwXsQmXtvFL0r2nNo69JN/xoet+bQIzh3ig4TL4j+cRRLgwVqzgu07/pvQfvGcmgKcz\nfQbffvb2teCa2Bl+x1n8xvSrwbf/qdM7csep92r+MJbLS1zFviiLxetGhrM8+rq7/yz4SktyzZFL\nZJUlyBW1SKNM033rLu+t2d6MXFsUV4wMvRIX05NIQ7Y5xoqF/GElJdRbozIMKrND6Mdnct8MV21J\nSUrHFNduYkZ4scaUZaCKy58ef/MkVqPILVSs6HJif6C2N+lRzz1vIRqusijo6FgsWRVNp4E1+BVe\nopS7JExG1BIu2Jvv+JIWitOt4Yj3/XU5JAu70rtN3jAy29LK+e7O4jfxwbsh8jaRLXmzyI9Pul+H\neWEocnKMjnX5dVxehYOubOFzs/t7cmR0/UhuU3iMM+1i//Sn18w1tRN2ek5eL/eNDe+Wnc7DnSdk\nYaL5LT6Xl7l5HH7pmf8kF20ZzjK546H3w0jftuRaRy6RVZYgs6xCLUK7NJ/d5b01UyVIr5I0KN5E\nz+FKKs3CMD8lygf5w95MqLdFtQmkChL62Wee+QOrDjstWUQED5QpaGpOVetjrbFRILyaFYsJ3TYq\ntubKiV2o7U3wyOSetxDd9Bd0NFcsWRVNp4Hk3QrP7pIwGXGlA7R3ofsBBFr7K1x4xDZ4BJ9WWCkL\n4fUBttcvbx0YyxlstC18ERzLlHvzTfY+d2jHPqcQzu7yBjr4fUddCRn2Zm9Dzq19DD9Hfifh0ZgS\nnUX4EzW1A6u78xn5LjkIP+Jtyspz+D7ReU7zW/xDGEer5KUvj0Mbskou2jKcUYTwA5y+bclJTj+I\ncakkssoSZJZVqEVol8b25sKIo6YKNcDRpngDPY9bk4Y0m6Rh8rA3E+ptUW0CqYKk+VGR4p+/piUR\nPEhBQVNzqlr4iNFF8klE6iTgitZGxdymxcTa3jTFrWrwyOSesxC9mPw9RelorliyKppboW6Fe5Ri\nl9hkFCsdoL0Ljw+RAu2NIh/EBUf/9o+jfXRN5G9uG+nexOKemG18cIok2/g+ex4L9Pl3CffmD4tc\ngPngrNja7PLruBzFQVe2sDdPbmPoksjqlPBoTIs3gwHc6JragdWXD2A6l+EHvP6LvO6IvKD5LZ5r\nG62Sl7l5rE7hHovKcJYJvoUF37bkWkcukVWWILOsQi3ScGlsb94046ipEqRvU5wYGXoetyYN9cgx\nVj7AC3szod4WlWFQmR1V/AVO8Amuoo//aFoSwYMUKmjLGiNL3cq4UiA80LlC4lS2UVFop7jtTVPc\nqiaPeu45C9GLGenY+taSwY+K1qYhWeEepdglITSudBTQ+8beLtDwXeGi1U+Yot3x1hd1b64Xe1Me\nHy1JH4uZxffv39vm3vyhcm8+Z7Es/CSOcm8ubW7e/jubm9vYBoheQbbVdcKjMeWjYHDGUFM7sQZf\nx958O/wwXQM+B85h8HnNb/G5vJxaHid35NO4WMtwHqyJvBZW+rYl1/muJ4qVJchaVlSL0C6N7s3+\nZ4cSY9WZOrQo3kivpF+ThtrVGVupcv/m5l9sbv40sibU26LaBFIFQbN/SZYxmaElJel0GmUVNDFT\nBK/aKlzZMN6/iKvG6Mj8uXKK696k4kXV5JHJPWchOjELOqRkJQdFa9PAohxwbkriZMSVDtDet276\nmlZ6Fy58woanrI51R3LTjN9pw96k7Tfw4rc0xkrbEvl+7uhyb/IRx/dONHYfx8ed/rwWfxFLPzef\nIjycCTvVP2MoavE4DHZiLW11d+QB+OEXr+Kp9KLmt/hcXv46xgMPCW5naxnO+CLAn83NdxvTfwkP\njkxyrSOXyD77EmQtK6pFGi6N7s2lO5Erfm5SAxxtimspCKnRQ2igX5OmaUqIxRY+NxPqbVHzZkdn\n6KI9yzVFWpJOJyygTEFTM4Xwqq2OFIXiDM7L2H55tqE2KjpXTnHdm0Fxqxo8crmn7QvRTX9Bh7mw\nUVcnpFX53PQr1AFnd0mYjLjSAdq7tLin1T/wZP23oJWpHB66vclX2k/gxe9fceUL8D+L/Lvbm3i7\n1ld2WNm9X/4x+1sQXhNk50PYu39CeDgTFmO3gAdRUzuxltcPjFdm8HsVjk0s6IPbMrio+Su0K3np\ny+PgbGkLF2sZzj2skIdhpW9bcpLLFmh7M0FWz6iWQpdpdG9eh0cemj2xqAGONsWb6Hn6qTQLwzxj\nYrGFvZlQb4tqE6j4LeiD2F6xpSURvGdSUNDU7NcYWdKdDeNLGytr/regNio6VyENaeneDIpb1QDO\n5QaltoVIemhkVdDRXLFkVTSdBkY54PwiMlpxpQO0d0mObyGyM8EX0/ehwyO2lUdkYYQVhP/0oUDb\nh2VyYIbr8rbIv4zxloo4/HcBQ0u7qGuIjnX/gV99n8YNXdnC52b3t2VpdGRNflLhMU7YW6Xzl7jQ\nNbUTdnBmUb4bdsVDjvd03i+9qea3+Fxe5tb8E6odWoYzhL5+BjN925JrHblEVlmCrGVFtQjt0uje\n3F2m8EGVIH2b4k30HC7lr0jTNCXEYgt7M6HeFtUmkCpI6LP6rNMMkpZE8ECZgqZmv8ZCxYqD8eVd\nvNXoJBhyGxV1c8ro3gyKW9Xgkcs9ZyG66S/oaK5YsiqaTkOywrO7JExGXOkAxd58A1/6Dstn7Etd\n/LbD8ge//s73dp596eeefemG47+Pe9pW/v7Wd9HWm4n80atfM3v53pdevvfFz/3XLtyx1vp3f0XQ\n2D12Cp234OjNcEILexP/COs90v3U+2aE5zhgBy978PQY3d4Mp8ROLPnC5149wkXx+l+9ZySvO/0V\nzY8x0srlZW4e/bN/h3NoGc6PvuaeB2ilb1vy3gz2XCKrLEGuqEXoMs3q3h6+IuwsnsdoUIU1UIcW\nxZvolbhgl0jTNCWKhZRhbybUW6PmzQ6h73r3GGdraUkED5R7s31U/PafhZeK071tUi4mjrVQ6c1g\nd8pwb0bFrWrwSKlxAuYsRDf9BR3NFUtWRdNpSIEdSrFL4mSEld6bdfb2Nn5rD1+iv1fuw1silj+P\npqa2zo6ab8k4PcGx67d51i4+hs6gH13j3qTdN8DyzYAtutpdOCvWDXqjeBUjboxyLS99s/6VeIBP\nbSDv65IHcrVEcX9VcAOtjqql0EmaO5/+hP7pzlRhDVZHBUWHHEbFyJt90a8xLkoNezOF1aKboyru\nTiDCFtAVJ39TUs7Ndr3iDt62SnFyMRHdUVG3qRm6Z/pr3JtR8fBEUu8Yqlfknr8QK7Logq1Qyipa\nB66gBM1C6A3KpQTtP/n08bEc2MbDBUdTU9viWM0PZpy+h2MdTJCIdvHddoJ+dB0Maak3wEZJomvF\nSbF+RYcUr2LEjVGu5aVv1r8SD/CpDeR9XfJArpYIT9thBdNulNbimDcKPWWvpPQU/kDLe4ulswZw\nqGwpRmkJvX3RrzEuSv3DGqAOaNHNUZUgJxBhC+iKk78pKedmO1PxEUZHcXIxEd1RUbepGZYmq88d\nvCQSFRerWr1jqF6RY/5CrMiia7tCKatoHbiCEjQLobbSS9DlkRxd079x8u+cjY02Y9/fyDgtjER+\ncbBOC7toJ3BkXdUaT2/UP480uirWVJ2JV2+kVc9L37y/R1gY9Sd2n/ctk4c66ok8nO+XahE6SbMy\nrP4DfDrzSFoVIzHidj/064zzpZboLHq/UaVAqvY8aEc5vzDSivvjx5SYiZOPCcxLKupWKr568eiW\npIqrd1kze/tYiFVZmKeVUsBPgasoVc2mjHGg/4S/Ggztf+4N/4MvHWqNtkUdzRSG7zPrML9bdyW7\naBQ066rWeDrGP/6w5V0VyzaQTZA5l2elVctL37x/GUnOgzW7z/uWyQO5eoEezvdLtQidpDm+Lse+\nNSzd6cwjaVWMxIjb/dCvM86XWqKz6P1GlQKp2vOgHeX8bNcqPvWUEjNx8jGBeUlF3UrFB7c/OZZU\ncfUua2ZvHwuxKgvztFIK+ClwFaWqma70EnTheeke/whkG9oREGsX2q3dGDuV683lnXYPgLrkXUtP\nl7LBtYBVPB9p/ZKW6P/qKPSLR929OhLBG7ANkCEJubLAKl55V9Dy0L5fujZKn8Xwcfumn0jjMer9\niApLUWcT88KXDk1OPkUMSAQNLk0V23g+JqJH5MpcNVIqvGO4rn+9aUjjIqxLvwbfApSdoqjEWVFy\nBBM/+W/nJOof99CwSgAAAABJRU5ErkJggg==\n",
       "prompt_number": 37,
       "text": [
        "((-sin(\u03c8)\u22c5sin(\u03b6) + cos(\u03b2)\u22c5cos(\u03c8)\u22c5cos(\u03b6))\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5cos(\u03c8)\u22c5\u03b6\u0307 + sin(\u03c8)\u22c5\u03b2\u0307) \u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m + ((sin(\u03c8)\u22c5cos(\u03b2)\u22c5cos(\u03b6) + sin(\u03b6)\u22c5cos(\u03c8))\u22c5\u03b8\u0307 - sin(\u03b2)\u22c5sin(\u03c8)\u22c5\u03b6\u0307 - cos(\u03c8)\u22c5\u03b2\u0307) \u001b[94m\u001b[1ma_y\u001b[0;0m\u001b[0;0m + (sin(\u03b2)\u22c5cos(\u03b6)\u22c5\u03b8\u0307 + cos(\u03b2)\u22c5\u03b6\u0307 + \u03c8\u0307) \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Derivada en ejes m\u00f3viles\n",
      "\n",
      "Hacer una derivada con la f\u00f3rmula lo hace cualquiera, pero SymPy puede encargarse autom\u00e1ticamente."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v1 = A1.x\n",
      "v1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{i}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAASBAMAAAC3N9OQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMplmRBCJ\n3XZModtBAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNgEDJgQADm/w4IDgPr/wQkHkMx\nMgeVzY+iktUfRV8/mMcYwMBQCNQF4c1fwFCRC+Nx909gYLCF8Xj2NiDx6hgFkHgJHBcQPE4Dvg9g\nHv///wwMezsaf0PkgAYxtDKwf4fzmIHG/2RgMAJJMDDICzAE/p0QcT8PAHetIcptQTo1AAAAAElF\nTkSuQmCC\n",
       "prompt_number": 38,
       "text": [
        "\u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#v1.diff(dynamicsymbols._t, A2)\n",
      "dv1 = v1.diff(symbols('t'), A)\n",
      "dv1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(\\operatorname{sin}^{2}\\left(\\psi\\right) \\dot{\\psi} + \\operatorname{cos}^{2}\\left(\\psi\\right) \\dot{\\psi})\\mathbf{j}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAAZBAMAAAD9ML7rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 39,
       "text": [
        "\u239b   2            2      \u239e\n",
        "\u239dsin (\u03c8)\u22c5\u03c8\u0307 + cos (\u03c8)\u22c5\u03c8\u0307\u23a0 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dv1.to_matrix(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\\\sin^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )} + \\cos^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )}\\\\0\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVgAAABLCAMAAAACh2/zAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 40,
       "text": [
        "\u23a1                    0                    \u23a4\n",
        "\u23a2                                         \u23a5\n",
        "\u23a2   2       d             2       d       \u23a5\n",
        "\u23a2sin (\u03c8(t))\u22c5\u2500\u2500(\u03c8(t)) + cos (\u03c8(t))\u22c5\u2500\u2500(\u03c8(t))\u23a5\n",
        "\u23a2           dt                    dt      \u23a5\n",
        "\u23a2                                         \u23a5\n",
        "\u23a3                    0                    \u23a6"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(dv1 & A1.j).simplify()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{d}{d t} \\psi{\\left (t \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAArBAMAAADf6lYYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 41,
       "text": [
        "d       \n",
        "\u2500\u2500(\u03c8(t))\n",
        "dt      "
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Puntos, velocidades y la rueda que no desliza"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El \u00faltimo paso que nos queda para completar la cinem\u00e1tica es la posibilidad de definir puntos en s\u00f3lidos y aplicar su campo de velocidades. SymPy tambi\u00e9n permite esto, y para ello no tenemos m\u00e1s que importar la clase `Point`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.mechanics import Point"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "O = Point(\"O\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para trabajar como lo har\u00edamos en la escuela, vamos a especificar que $O$ es el origen de $A$, y para eso vamos a imponer que su velocidad es cero con el m\u00e9todo `set_vel`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "O.set_vel(A, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para definir nuevos puntos, podemos utilizar el m\u00e9todo `locate_new`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e_b = symbols('e_b')\n",
      "E_b = O.locatenew('E_b', e_b * A1.x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y para obtener vectores de un punto a otro, el m\u00e9todo `pos_from`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E_b.pos_from(O)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e_{b}\\mathbf{i}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAASBAMAAADS2iHvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVQYGWNggIMXE+BMJAbn/wtIPDiT9f8G\nOBuZsRKZQ4jNj2oIo7JJAEgL630Us9MDWBXY7gLF/cHCfEA164CKOhlEFjCowIXPJzAs2c7AwN1j\nfJCB4SZMmNH/AAPDdAYGfrDWrzBhJnEHiLACUITts7UAxOzFfA/AwhxAYQPmQmYHiPAGlgKwMPMF\nBpEAPgOeC2BhngnMH8DCDCY2Bxg4Elg28P//z8Ag7uL3ESIMNJqBX4BjAohmcGXg+oUQZhc4Bxbl\nPMDA8IeBYQaYw8DA5aQAZr1/wBD3/UBI/WYAfoo7IFK/ix4AAAAASUVORK5CYII=\n",
       "prompt_number": 46,
       "text": [
        "e_b \u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=\"alert alert-info\">La notaci\u00f3n de este paquete est\u00e1 influenciada por el libro Kane, T. R. & Levinson, D. A. \"Dynamics, Theory and Applications\". Es ligeramente distinto a como estudiamos nosotros en la escuela, pero \u00a1est\u00e1n abiertos a que les hagamos cualquier tipo de sugerencia! https://github.com/sympy/sympy/issues/2584#issuecomment-31552654</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por \u00faltimo, el **campo de velocidades de un s\u00f3lido r\u00edgido** se formula usando el m\u00e9todo `v2pt_theory`.\n",
      "\n",
      "$$v^P_A = v^O_A + \\omega_{A_1 A} \\times \\mathbf{OP}$$\n",
      "\n",
      "Este m\u00e9todo pertenece *al punto del cual queremos conocer la velocidad* y recibe tres par\u00e1metros:\n",
      "\n",
      "* `O`, punto de velocidad conocida respecto a A\n",
      "* `A`, sistema de referencia donde queremos calcular la velocidad\n",
      "* `A1`, sistema de referencia donde est\u00e1n fijos ambos puntos (_sistema de arrastre_)\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por tanto, para hallar la velocidad del punto que acabamos de crear:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E_b.v2pt_theory(O, A, A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e_{b} \\dot{\\psi}\\mathbf{j}_{A1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAYBAMAAABZ8fMmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 47,
       "text": [
        "e_b\u22c5\u03c8\u0307 \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##### Ejercicio\n",
      "\n",
      "![Disco que rueda](../static/disco.png)\n",
      "\n",
      "(Apuntes de \u00d3scar L\u00f3pez Rebollal)\n",
      "\n",
      "**\u00a1Halla la velocidad y la aceleraci\u00f3n de $P$!**"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos nuestros sistemas de referencia\n",
      "A1 = IJKReferenceFrame('1')\n",
      "A0 = IJKReferenceFrame('0')\n",
      "A2 = IJKReferenceFrame('2')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos los s\u00edmbolos din\u00e1micos necesarios\n",
      "xi, theta = dynamicsymbols('xi, theta')\n",
      "xi, theta"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\xi{\\left (t \\right )}, & \\theta{\\left (t \\right )}\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAaBAMAAACgM6QiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niG5Bkn2IAAAAAElFTkSuQmCC\n",
       "prompt_number": 49,
       "text": [
        "(\u03be(t), \u03b8(t))"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Orientamos los sistemas de referencia\n",
      "A0.orient(A1, 'Axis', [0, A1.k])  # A0 no gira respecto a A1\n",
      "A2.orient(A0, 'Axis', [theta, A0.k])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A2.dcm(A1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 51,
       "text": [
        "\u23a1cos(\u03b8(t))   sin(\u03b8(t))  0\u23a4\n",
        "\u23a2                        \u23a5\n",
        "\u23a2-sin(\u03b8(t))  cos(\u03b8(t))  0\u23a5\n",
        "\u23a2                        \u23a5\n",
        "\u23a3    0           0      1\u23a6"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos el punto C, centro del disco, y especificamos su velocidad\n",
      "# respecto a A1\n",
      "C = Point('C')\n",
      "C.set_vel(A1, xi.diff() * A1.x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Localizamos el punto P, punto fijo del disco, respecto a C, en\n",
      "# el sistema A2 (que gira solidariamente con el disco)\n",
      "R = symbols('R')\n",
      "P = C.locatenew('P', -R * A2.j)\n",
      "P.pos_from(C)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-  R\\mathbf{j}_{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQYGWXOMWoCQRTG8T84q6siLuIBUthZ\nWHgA9wbuEdJES2MrgnuEba0SsEkXwVIIamPrASxEbEUlsCpIJrODMDi+avi9b957YJUYeY/idOJW\nOyAto0en+I57Jis3ljdVsOPzYzEfPjRCW6EH+RsHubJaMUwi3IblIh4OAhUdrFj3PfMne8L50l4I\n0y+U6klVIPMJN+2pUFxMvviKq1aoObmN+2e8GSTn6PnkT8bV+YUrXrKXVGS8Bs5VjLWvDS/kGyyH\neo4zM35/lZP8np3dkP73WFSn3Sff/ZKT8mj7tj7X9A8NW0cG87ppKwAAAABJRU5ErkJggg==\n",
       "prompt_number": 53,
       "text": [
        "- R \u001b[94m\u001b[1m2_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Hallamos la velocidad de P en A1, expresada en A0\n",
      "# \u00a1Con esta llamada ya estamos diciendo que C y P son fijos en A2!\n",
      "P.v2pt_theory(C, A1, A2).express(A0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} + \\dot{\\xi})\\mathbf{i}_{0} + R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{j}_{0}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAZBAMAAAAVlVzpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 54,
       "text": [
        "(R\u22c5cos(\u03b8)\u22c5\u03b8\u0307 + \u03be\u0307) \u001b[94m\u001b[1m0_x\u001b[0;0m\u001b[0;0m + R\u22c5sin(\u03b8)\u22c5\u03b8\u0307 \u001b[94m\u001b[1m0_y\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Misi\u00f3n cumplida :)**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "---"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "_Hemos hecho un repaso bastante profundo de las posibilidades del paquete `mechanics` de SymPy. Nos hemos dejado algunas cosas en el tintero pero no demasiadas: esta funcionalidad a\u00fan se est\u00e1 expandiendo y necesita pulir algunos detalles._\n",
      "\n",
      "**Referencias**\n",
      "\n",
      "* Cap\u00edtulo de **aeromec\u00e1nica** del libro de Cuerva y otros http://nbviewer.ipython.org/gist/Juanlu001/7711865\n",
      "* Estabilidad longitudinal de un Boeing 747 http://nbviewer.ipython.org/github/AlexS12/Mecanica_Vuelo/blob/master/MVII_MatrizSistema.ipynb\n",
      "\n",
      "_\u00bfSer\u00e1s t\u00fa el siguiente que publique un notebook usando SymPy? ;)_"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si te ha gustado esta clase:\n",
      "\n",
      "<a href=\"https://twitter.com/share\" class=\"twitter-share-button\" data-url=\"https://github.com/AeroPython/Curso_AeroPython\" data-text=\"Aprendiendo Python con\" data-via=\"pybonacci\" data-size=\"large\" data-hashtags=\"AeroPython\">Tweet</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>\n",
      "\n",
      "---"
     ]
    },
    {
     "cell_type": "heading",
     "level": 6,
     "metadata": {},
     "source": [
      "<img src=\"../static/linkedin.png\" alt=\"AeroPython\" style=\"width: 25px\" align=\"right\";/> Curso impartido por: [Juan Luis Cano](http://es.linkedin.com/in/juanluiscanor)  & [\u00c1lex S\u00e1ez](http://es.linkedin.com/pub/alejandro-saez-mollejo/55/22/473)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 6,
     "metadata": {},
     "source": [
      "En colaboraci\u00f3n: Alberto Lorenzo"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "<h4 align=\"right\">\u00a1S\u00edguenos en Twitter!"
     ]
    },
    {
     "cell_type": "heading",
     "level": 6,
     "metadata": {},
     "source": [
      "<a href=\"https://twitter.com/Pybonacci\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @Pybonacci</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>\n",
      "\n",
      "<a href=\"https://twitter.com/Alex__S12\" class=\"twitter-follow-button\" data-show-count=\"false\" align=\"right\";>Follow @Alex__S12</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>\n",
      "\n",
      "<a href=\"https://twitter.com/newlawrence\" class=\"twitter-follow-button\" data-show-count=\"false\" align=\"right\";>Follow @newlawrence</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": [
      "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/deed.es\"><img alt=\"Licencia Creative Commons\" style=\"border-width:0\" src=\"http://i.creativecommons.org/l/by/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Curso AeroPython</span> por <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Juan Luis Cano Rodriguez y Alejandro S\u00e1ez Mollejo</span> se distribuye bajo una <a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/deed.es\">Licencia Creative Commons Atribuci\u00f3n 4.0 Internacional</a>."
     ]
    },
    {
     "cell_type": "heading",
     "level": 5,
     "metadata": {},
     "source": [
      "<script src=\"//platform.linkedin.com/in.js\" type=\"text/javascript\"></script>\n",
      "<script type=\"IN/MemberProfile\" data-id=\"http://es.linkedin.com/in/juanluiscanor\" data-format=\"inline\" data-related=\"false\"></script>\n",
      "<script src=\"//platform.linkedin.com/in.js\" type=\"text/javascript\"></script>\n",
      "<script type=\"IN/MemberProfile\" data-id=\"http://es.linkedin.com/in/alejandrosaezm\" data-format=\"inline\" data-related=\"false\"></script>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "---\n",
      "_Las siguientes celdas contienen configuraci\u00f3n del Notebook_\n",
      "\n",
      "_Para visualizar y utlizar los enlaces a Twitter el notebook debe ejecutarse como [seguro](http://ipython.org/ipython-doc/dev/notebook/security.html)_\n",
      "\n",
      "    File > Trusted Notebook"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%html\n",
      "<a href=\"https://twitter.com/Pybonacci\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @Pybonacci</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<a href=\"https://twitter.com/Pybonacci\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @Pybonacci</a>\n",
        "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.HTML at 0x430e208>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Esta celda da el estilo al notebook\n",
      "from IPython.core.display import HTML\n",
      "css_file = '../static/styles/style.css'\n",
      "HTML(open(css_file, \"r\").read())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<link href='http://fonts.googleapis.com/css?family=Source+Sans+Pro|Josefin+Sans:400,700,400italic|Ubuntu+Condensed' rel='stylesheet' type='text/css'>\n",
        "\n",
        "El estilo se ha aplicado =)\n",
        "\n",
        "<style>\n",
        "\n",
        "\n",
        "\n",
        "#notebook_panel { /* main background */\n",
        "    background: #f7f7f7;\n",
        "}\n",
        "\n",
        "div.cell { /* set cell width */\n",
        "    width: 900px;\n",
        "}\n",
        "\n",
        "div #notebook { /* centre the content */\n",
        "    background: #fff; /* white background for content */\n",
        "    width: 950px;\n",
        "    margin: auto;\n",
        "    padding-left: 0em;\n",
        "}\n",
        "\n",
        "#notebook li { /* More space between bullet points */\n",
        "    margin-top:0.7em;\n",
        "}\n",
        "\n",
        "/* draw border around running cells */\n",
        "div.cell.border-box-sizing.code_cell.running { \n",
        "    border: 1px solid #111;\n",
        "}\n",
        "\n",
        "/* Put a solid color box around each cell and its output, visually linking them*/\n",
        "div.cell.code_cell {\n",
        "    font-family: 'Source Sans Pro', sans-serif;\n",
        "    background-color: rgb(256,256,256);\n",
        "    font-size: 110%;\n",
        "    border-radius: 0px; \n",
        "    padding: 0.5em;\n",
        "    margin-left:1em;\n",
        "    margin-top: 1em;\n",
        "}\n",
        "\n",
        "div.text_cell_render{\n",
        "    font-family: 'Josefin Sans', serif;\n",
        "    line-height: 145%;\n",
        "    font-size: 125%;\n",
        "    font-weight: 500;\n",
        "    width:750px;\n",
        "    margin-left:auto;\n",
        "    margin-right:auto;\n",
        "}\n",
        "\n",
        "\n",
        "/* Formatting for header cells */\n",
        ".text_cell_render h1, .text_cell_render h2, .text_cell_render h3,\n",
        ".text_cell_render h4, .text_cell_render h5 {\n",
        "    font-family: 'Ubuntu Condensed', sans-serif;\n",
        "}\n",
        "/*\n",
        ".text_cell_render h1 {\n",
        "    font-family: Flux, 'Ubuntu Condensed', serif;\n",
        "    font-style:regular;\n",
        "    font-weight: 400;    \n",
        "    font-size: 30pt;\n",
        "    text-align: center;\n",
        "    line-height: 100%;\n",
        "    color: #335082;\n",
        "    margin-bottom: 0.5em;\n",
        "    margin-top: 0.5em;\n",
        "    display: block;\n",
        "}\n",
        "*/\n",
        ".text_cell_render h1 {\n",
        "    font-weight: 600;\n",
        "    font-size: 35pt;\n",
        "    line-height: 100%;\n",
        "    color: #234765;\n",
        "    margin-bottom: 0.1em;\n",
        "    margin-top: 0.3em;\n",
        "    display: block;\n",
        "}\n",
        "\n",
        ".text_cell_render h2 {\n",
        "    margin-top:16px;\n",
        "    font-size: 27pt;\n",
        "    font-weight: 550;\n",
        "    margin-bottom: 0.1em;\n",
        "    margin-top: 0.3em;\n",
        "    font-style: regular;\n",
        "    color: #234765;\n",
        "}\t\n",
        "\n",
        ".text_cell_render h3 {\n",
        "    font-size: 20pt;\n",
        "    font-weight: 550\n",
        "    text-align: left;\n",
        "    margin-bottom: 0.1em;\n",
        "    margin-top: 0.3em;\n",
        "    font-style: regular;\n",
        "    color:  #376f9e;\n",
        "}\n",
        "\n",
        ".text_cell_render h4 {    /*Use this for captions*/\n",
        "    font-size: 18pt;\n",
        "    font-weight: 450\n",
        "    text-align: left;\n",
        "    margin-bottom: 0.1em;\n",
        "    margin-top: 0.3em;\n",
        "    font-style: regular;\n",
        "    color:  #376f9e;\n",
        "}\n",
        "\n",
        ".text_cell_render h5 {  /*Use this for small titles*/\n",
        "    font-size: 18pt;\n",
        "    font-weight: 550;\n",
        "    color: rgb(163,0,0);\n",
        "    font-style: italic;\n",
        "    margin-bottom: .1em;\n",
        "    margin-top: 0.8em;\n",
        "    display: block;\n",
        "    color:  #b21c0d;\n",
        "}\n",
        "\n",
        ".text_cell_render h6 { /*use this for copyright note*/\n",
        "    font-family: 'Ubuntu Condensed', sans-serif;\n",
        "    font-weight: 300;\n",
        "    font-size: 14pt;\n",
        "    line-height: 100%;\n",
        "    color: #252525;\n",
        "    text-align: right;\n",
        "    margin-bottom: 1px;\n",
        "    margin-top: 1px;\n",
        "}\n",
        "\n",
        ".CodeMirror{\n",
        "        font-family: 'Duru Sans', sans-serif;\n",
        "        font-size: 100%;\n",
        "}\n",
        "\n",
        "</style>\n",
        "<script>\n",
        "    MathJax.Hub.Config({\n",
        "                        TeX: {\n",
        "                           extensions: [\"AMSmath.js\"],\n",
        "                           equationNumbers: { autoNumber: \"AMS\", useLabelIds: true}\n",
        "                           },\n",
        "                tex2jax: {\n",
        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
        "</script>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "<IPython.core.display.HTML at 0x430e358>"
       ]
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     ],
     "prompt_number": 1
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