physics-vector.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "from IPython.utils.text import columnize\n",
    "import sympy as sp\n",
    "import sympy.physics.vector as spv\n",
    "\n",
    "import IPython.display as ipd\n",
    "\n",
    "spv.init_vprinting()\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def directory(obj):\n",
    "    return [\n",
    "        str for str in dir(obj) if callable(getattr(obj, str) ) # & ~str.startswith(\"_\")\n",
    "    ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Audio          JSON               Video               display_markdown      \n",
      "Code           Javascript         VimeoVideo          display_pdf           \n",
      "DisplayHandle  Latex              YouTubeVideo        display_png           \n",
      "DisplayObject  Markdown           clear_output        display_pretty        \n",
      "FileLink       Math               display             display_svg           \n",
      "FileLinks      Pretty             display_html        publish_display_data  \n",
      "GeoJSON        ProgressBar        display_javascript  set_matplotlib_close  \n",
      "HTML           SVG                display_jpeg        set_matplotlib_formats\n",
      "IFrame         ScribdDocument     display_json        update_display        \n",
      "Image          TextDisplayObject  display_latex     \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(columnize(directory(ipd)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Catalan           Naturals            __dir__            __ne__             \n",
      "ComplexInfinity   Naturals0           __doc__            __new__            \n",
      "Complexes         NegativeInfinity    __eq__             __reduce__         \n",
      "EmptySequence     NegativeOne         __format__         __reduce_ex__      \n",
      "EmptySet          One                 __ge__             __repr__           \n",
      "EulerGamma        Pi                  __getattr__        __setattr__        \n",
      "Exp1              Rationals           __getattribute__   __sizeof__         \n",
      "GoldenRatio       Reals               __gt__             __slots__          \n",
      "Half              TribonacciConstant  __hash__           __str__            \n",
      "IdentityFunction  UniversalSet        __init__           __subclasshook__   \n",
      "ImaginaryUnit     Zero                __init_subclass__  _classes_to_install\n",
      "Infinity          __call__            __le__             false              \n",
      "Integers          __class__           __lt__             register           \n",
      "NaN               __delattr__         __module__         true               \n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(columnize(dir(sp.S)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "HALF = sp.S.Half\n",
    "PI = sp.pi\n",
    "E = sp.exp\n",
    "POSITIVEINFINITY = sp.S.Infinity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Support Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def reference_frame(\n",
    "    frame: str, x=r\"\\imath\", y=r\"\\jmath\", z=r\"\\mathbf k\"\n",
    ") -> spv.ReferenceFrame:\n",
    "    return spv.ReferenceFrame(\n",
    "        frame,\n",
    "        latexs=(\n",
    "            rf\"\\; {{}}^\\mathcal {frame} \\hat {x}\",\n",
    "            rf\"\\;{{}}^\\mathcal {frame} \\hat {y}\",\n",
    "            rf\"\\: {{}}^\\mathcal {frame} \\hat {{z}}\",\n",
    "        ),\n",
    "    )\n",
    "\n",
    "\n",
    "def RAOreference_frame(frame: str, x=r\"e_1\", y=\"e_2\", z=r\"e_3\") -> spv.ReferenceFrame:\n",
    "    return spv.ReferenceFrame(\n",
    "        frame,\n",
    "        latexs=(\n",
    "            rf\"\\; {{}}^{frame} \\hat {x}\",\n",
    "            rf\"\\;{{}}^{frame} \\hat {y}\",\n",
    "            rf\"\\: {{}}^{frame} \\hat {z}\",\n",
    "        ),\n",
    "    )\n",
    "\n",
    "\n",
    "def vector(F: spv.ReferenceFrame, rx, ry, rz=0) -> spv.Vector:\n",
    "    return rx * F.x + ry * F.y + rz * F.z\n",
    "\n",
    "\n",
    "def vector_cos(\n",
    "    F: spv.ReferenceFrame, magnitude, angle_x, angle_y, angle_z=sp.pi / 2\n",
    ") -> spv.Vector:\n",
    "    return magnitude * (\n",
    "        sp.cos(angle_x) * F.x + sp.cos(angle_y) * F.y + sp.cos(angle_z) * F.z\n",
    "    )\n",
    "\n",
    "\n",
    "def vector_line(start: spv.Vector, finish: spv.Vector) -> spv.Vector:\n",
    "    return finish - start\n",
    "\n",
    "\n",
    "def vector_line_eqn(\n",
    "    F: spv.ReferenceFrame, start: spv.Vector, finish: spv.Vector, kappa\n",
    ") -> spv.Vector:\n",
    "    return start + vector_line(start, finish).normalize() * kappa\n",
    "\n",
    "\n",
    "def angle_between_vectors(a: spv.Vector, b: spv.Vector):\n",
    "    return sp.acos(a.dot(b) / a.magnitude() / b.magnitude())\n",
    "\n",
    "\n",
    "def make_2D_symbols(symbol_list):\n",
    "    return [sp.symbols(f\"{ch}_x:y\", real=True) for ch in symbol_list]\n",
    "\n",
    "\n",
    "F, E = make_2D_symbols([\"F\", \"E\"])\n",
    "assert str(F) == \"(F_x, F_y)\"\n",
    "assert str(E) == \"(E_x, E_y)\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\; {}^E \\hat E_1, \\  \\;{}^E \\hat E_2, \\  \\: {}^E \\hat E_3\\right)$"
      ],
      "text/plain": [
       "(e_x, e_y, e_z)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = RAOreference_frame(\"E\", \"E_1\", \"E_2\", \"E_3\")\n",
    "E.x, E.y, E.z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\; {}^A \\hat e_1, \\  \\;{}^A \\hat e_2, \\  \\: {}^A \\hat e_3\\right)$"
      ],
      "text/plain": [
       "(a_x, a_y, a_z)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = RAOreference_frame(\"A\")\n",
    "\n",
    "A.x, A.y, A.z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( c_{{1}{1}}, \\  c_{{1}{2}}, \\  c_{{1}{3}}, \\  c_{{2}{1}}, \\  c_{{2}{2}}, \\  c_{{2}{3}}, \\  c_{{3}{1}}, \\  c_{{3}{2}}, \\  c_{{3}{3}}\\right)$"
      ],
      "text/plain": [
       "(c_{{1}{1}}, c_{{1}{2}}, c_{{1}{3}}, c_{{2}{1}}, c_{{2}{2}}, c_{{2}{3}}, c_{{3\n",
       "}{1}}, c_{{3}{2}}, c_{{3}{3}})"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = sp.symbols(\"c_{{1:4}{1:4}}\")\n",
    "c"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# DCM\n",
    "The orientation of a specific frame relative to its parent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{{1}{1}} & c_{{1}{2}} & c_{{1}{3}}\\\\c_{{2}{1}} & c_{{2}{2}} & c_{{2}{3}}\\\\c_{{3}{1}} & c_{{3}{2}} & c_{{3}{3}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡c_{{1}{1}}  c_{{1}{2}}  c_{{1}{3}}⎤\n",
       "⎢                                  ⎥\n",
       "⎢c_{{2}{1}}  c_{{2}{2}}  c_{{2}{3}}⎥\n",
       "⎢                                  ⎥\n",
       "⎣c_{{3}{1}}  c_{{3}{2}}  c_{{3}{3}}⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A_DCM_E = sp.Matrix(3, 3, c)\n",
    "A_DCM_E"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Orient Explicit__\n",
    "\n",
    "Sets the orientation of this reference frame relative to a \n",
    "parent reference frame by explicitly setting the direction \n",
    "cosine matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "A.orient_explicit(E, A_DCM_E)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Returns the direction cosine matrix__\n",
    "\n",
    "The returned A matrix can be used A to express the orthogonal unit vectors \n",
    "of this frame in terms of the orthogonal unit vectors of otherframe.\n",
    "\n",
    "Notice that for ${}^{\\mathcal E} \\mathbf {e}_2$ is represents in the second row of E.dcm(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}c_{{1}{1}} & c_{{1}{2}} & c_{{1}{3}}\\\\c_{{2}{1}} & c_{{2}{2}} & c_{{2}{3}}\\\\c_{{3}{1}} & c_{{3}{2}} & c_{{3}{3}}\\end{matrix}\\right], \\  c_{{2}{1}}\\; {}^A \\hat e_1 + c_{{2}{2}}\\;{}^A \\hat e_2 + c_{{2}{3}}\\: {}^A \\hat e_3\\right)$"
      ],
      "text/plain": [
       "⎛⎡c_{{1}{1}}  c_{{1}{2}}  c_{{1}{3}}⎤, c_{{2}{1}} a_x + c_{{2}{2}} a_y + c_{{2\n",
       "⎜⎢                                  ⎥                                         \n",
       "⎜⎢c_{{2}{1}}  c_{{2}{2}}  c_{{2}{3}}⎥                                         \n",
       "⎜⎢                                  ⎥                                         \n",
       "⎝⎣c_{{3}{1}}  c_{{3}{2}}  c_{{3}{3}}⎦                                         \n",
       "\n",
       "}{3}} a_z⎞\n",
       "         ⎟\n",
       "         ⎟\n",
       "         ⎟\n",
       "         ⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.dcm(A), E.y.express(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\left( a_{1}, \\  a_{2}, \\  a_{3}\\right), \\  \\left( b_{1}, \\  b_{2}, \\  b_{3}\\right)\\right)$"
      ],
      "text/plain": [
       "((a₁, a₂, a₃), (b₁, b₂, b₃))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, b = [sp.symbols(f\"{ch}_1:4\") for ch in [\"a\", \"b\"]]\n",
    "a, b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{{1}{1}} & c_{{1}{2}} & c_{{1}{3}}\\\\c_{{2}{1}} & c_{{2}{2}} & c_{{2}{3}}\\\\c_{{3}{1}} & c_{{3}{2}} & c_{{3}{3}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡c_{{1}{1}}  c_{{1}{2}}  c_{{1}{3}}⎤\n",
       "⎢                                  ⎥\n",
       "⎢c_{{2}{1}}  c_{{2}{2}}  c_{{2}{3}}⎥\n",
       "⎢                                  ⎥\n",
       "⎣c_{{3}{1}}  c_{{3}{2}}  c_{{3}{3}}⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.dcm(A)  # represent E unit in term of A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a_{1} c_{{1}{1}} + a_{2} c_{{1}{2}} + a_{3} c_{{1}{3}}\\\\a_{1} c_{{2}{1}} + a_{2} c_{{2}{2}} + a_{3} c_{{2}{3}}\\\\a_{1} c_{{3}{1}} + a_{2} c_{{3}{2}} + a_{3} c_{{3}{3}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a₁⋅c_{{1}{1}} + a₂⋅c_{{1}{2}} + a₃⋅c_{{1}{3}}⎤\n",
       "⎢                                             ⎥\n",
       "⎢a₁⋅c_{{2}{1}} + a₂⋅c_{{2}{2}} + a₃⋅c_{{2}{3}}⎥\n",
       "⎢                                             ⎥\n",
       "⎣a₁⋅c_{{3}{1}} + a₂⋅c_{{3}{2}} + a₃⋅c_{{3}{3}}⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.dcm(A) * sp.Matrix(3, 1, a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle (a_{1} c_{{1}{1}} + a_{2} c_{{1}{2}} + a_{3} c_{{1}{3}})\\; {}^E \\hat E_1 + (a_{1} c_{{2}{1}} + a_{2} c_{{2}{2}} + a_{3} c_{{2}{3}})\\;{}^E \\hat E_2 + (a_{1} c_{{3}{1}} + a_{2} c_{{3}{2}} + a_{3} c_{{3}{3}})\\: {}^E \\hat E_3$"
      ],
      "text/plain": [
       "(a₁⋅c_{{1}{1}} + a₂⋅c_{{1}{2}} + a₃⋅c_{{1}{3}}) e_x + (a₁⋅c_{{2}{1}} + a₂⋅c_{{\n",
       "2}{2}} + a₃⋅c_{{2}{3}}) e_y + (a₁⋅c_{{3}{1}} + a₂⋅c_{{3}{2}} + a₃⋅c_{{3}{3}})\n",
       "e_z"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a[0] * A.x + a[1] * A.y + a[2] * A.z).express(E)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Frame differentiation DCM\n",
    "\n",
    "![](./images/Example02-01.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "R, omega, t = sp.symbols(\"R, omega, t\", real=True, positive=True)\n",
    "F = RAOreference_frame(\"F\")\n",
    "A = RAOreference_frame(\"A\")\n",
    "A.orient_axis(F, F.z, omega * t)\n",
    "vecR_in_A = R * A.x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( -  R \\omega \\sin{\\left(\\omega t \\right)}\\; {}^F \\hat e_1 + R \\omega \\cos{\\left(\\omega t \\right)}\\;{}^F \\hat e_2, \\  0\\right)$"
      ],
      "text/plain": [
       "(-R⋅ω⋅sin(ω⋅t) f_x + R⋅ω⋅cos(ω⋅t) f_y, 0)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecR_in_A.diff(t, F).express(F).simplify(), vecR_in_A.diff(t, A)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unit vectors in another frame\n",
    "\n",
    "The following __rows__ gives the unit vectors of $\\mathcal A$ expressed in\n",
    "$\\mathcal F$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\omega t \\right)} & \\sin{\\left(\\omega t \\right)} & 0\\\\- \\sin{\\left(\\omega t \\right)} & \\cos{\\left(\\omega t \\right)} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡cos(ω⋅t)   sin(ω⋅t)  0⎤\n",
       "⎢                      ⎥\n",
       "⎢-sin(ω⋅t)  cos(ω⋅t)  0⎥\n",
       "⎢                      ⎥\n",
       "⎣    0         0      1⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dcm(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left\\{ A_{x} : F_{x} \\cos{\\left(\\omega t \\right)} + F_{y} \\sin{\\left(\\omega t \\right)}, \\  A_{y} : - F_{x} \\sin{\\left(\\omega t \\right)} + F_{y} \\cos{\\left(\\omega t \\right)}, \\  A_{z} : F_{z}\\right\\}$"
      ],
      "text/plain": [
       "{Aₓ: Fₓ⋅cos(ω⋅t) + F_y⋅sin(ω⋅t), A_y: -Fₓ⋅sin(ω⋅t) + F_y⋅cos(ω⋅t), A_z: F_z}"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.variable_map(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( R \\cos{\\left(\\omega t \\right)}\\; {}^F \\hat e_1 + R \\sin{\\left(\\omega t \\right)}\\;{}^F \\hat e_2, \\  \\left[\\begin{matrix}R \\cos{\\left(\\omega t \\right)}\\\\- R \\sin{\\left(\\omega t \\right)}\\\\0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛R⋅cos(ω⋅t) f_x + R⋅sin(ω⋅t) f_y, ⎡R⋅cos(ω⋅t) ⎤⎞\n",
       "⎜                                 ⎢           ⎥⎟\n",
       "⎜                                 ⎢-R⋅sin(ω⋅t)⎥⎟\n",
       "⎜                                 ⎢           ⎥⎟\n",
       "⎝                                 ⎣     0     ⎦⎠"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecR_in_A.express(F), (A.dcm(F)) * sp.Matrix([R, 0, 0])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dynamic variables\n",
    "\n",
    "![](./images/Example02-02.jpg/)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__With dynamic symbols__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\dot{\\theta}$"
      ],
      "text/plain": [
       "θ̇"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega, t = sp.symbols(\"Omega t\", real=True, positive=True)\n",
    "x, y, theta = spv.dynamicsymbols(\"x, y, theta\", level=0)\n",
    "theta.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = RAOreference_frame(\"F\", x=r\"E_x\", y=r\"E_y\", z=r\"E_z\")\n",
    "A = RAOreference_frame(\"A\", x=r\"e_x\", y=r\"e_y\", z=r\"e_z\")\n",
    "A.orient_axis(F, F.z, theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "vecP = x * A.x + y * A.y\n",
    "dvecP_A, dvecP_F = vecP.dt(A), vecP.dt(F)\n",
    "ddvecP_A, ddvecP_F = (dvecP_A.diff(sp.Symbol(\"t\"), A), dvecP_F.diff(sp.Symbol(\"t\"), F))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle (- \\Omega y + \\dot{x})\\; {}^A \\hat e_x + (\\Omega x + \\dot{y})\\;{}^A \\hat e_y$"
      ],
      "text/plain": [
       "(-Ω⋅y + ẋ) a_x + (Ω⋅x + ẏ) a_y"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dvecP_F.subs(theta.diff(), omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle (- \\Omega \\left(\\Omega x + \\dot{y}\\right) \\cos{\\left(\\theta \\right)} + \\Omega \\left(\\Omega y - \\dot{x}\\right) \\sin{\\left(\\theta \\right)} - \\left(\\Omega \\dot{x} + \\ddot{y}\\right) \\sin{\\left(\\theta \\right)} - \\left(\\Omega \\dot{y} - \\ddot{x}\\right) \\cos{\\left(\\theta \\right)})\\; {}^F \\hat E_x + (- \\Omega \\left(\\Omega x + \\dot{y}\\right) \\sin{\\left(\\theta \\right)} - \\Omega \\left(\\Omega y - \\dot{x}\\right) \\cos{\\left(\\theta \\right)} + \\left(\\Omega \\dot{x} + \\ddot{y}\\right) \\cos{\\left(\\theta \\right)} - \\left(\\Omega \\dot{y} - \\ddot{x}\\right) \\sin{\\left(\\theta \\right)})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "(-Ω⋅(Ω⋅x + ẏ)⋅cos(θ) + Ω⋅(Ω⋅y - ẋ)⋅sin(θ) - (Ω⋅ẋ + ÿ)⋅sin(θ) - (Ω⋅ẏ - ẍ)\n",
       "⋅cos(θ)) f_x + (-Ω⋅(Ω⋅x + ẏ)⋅sin(θ) - Ω⋅(Ω⋅y - ẋ)⋅cos(θ) + (Ω⋅ẋ + ÿ)⋅cos(θ\n",
       ") - (Ω⋅ẏ - ẍ)⋅sin(θ)) f_y"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ddvecP_F.subs(theta.diff().diff(), 0).subs(theta.diff(), omega).simplify())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAcCAYAAAC0yEstAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjcuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/bCgiHAAAACXBIWXMAABJ0AAASdAHeZh94AAAFHUlEQVR4nO2ae4hVVRTGf6bZwyFFKDLFAguV1FAjjVSQfJQS0QvLMJX6o4cQlaXW5OcH9tayEkKlNCQtk0orokiiUKPGEl89icpEM80YTcU0pz/2vnnnes6dx70378T9/lnDXnufb91ZZ6+91tqnVV1dHRWUJ2x3A5D0Q1PXtim+ORUUEauiPK+pC0/Kp7TdyfYNzbGogtLBdj/bl+Wb0yotFNs+BXgNGCdpXwnsq6AA2F4MVEv6OUmfb8c+CiysOLVs8RCwwHarJGWiY21fCFwpaUUpLaug+ZC0FdgOjE/Sp+3Yh4FXSmVUBUXDIqDadutcxXGOtd0JuB54tfR2VVAgPgZOB0bnKpLKnTFAbXNqp/8bbJ8G3A+MJZQcu4DFgCQdPtGckupsrwNuBlZm65JC8QhgfQlsblGIkasGqAY2AM8Ce4FpwNwy4lwPDMtNopIceymwqWjWtkDYbgu8TdgxQyWNkTQFGABsBW6zfXaZcG4COgLdswfrOdb2WUAH4PdiGt0CMRnoD0yRtCYzKOlP4E3C/21wmXDuifKC7MHcM7ZLlHvT2G1/AAwHrpP0RtZ4K2AhIf1+QtLUxvyafLB9HzALmCxpdoK+O7AR+EzSkEL54jMzZ9wOYH7ClMxLX7QdWyBnbZRdsgdzQ3FVlKmOjQYcBWbmpNmzCE5dUAynRqyOcmCK/nmgNTCpSHwA1xCi1pKUBOnUKP8qE86Mr9plD+bu2MwB/HeaBZI2xHbWeGAcsMj2g8C9wDLg9jw/oKn4EjhIOGfqIfawhwPPSdpYRM5M6dDZ9owE/bAofykTziNR1tukuY49EOUZDRhSTSiLZtiuAh4B3if0lY82sLbRkHTYdg0wxPY5krYD2G4HPA38BkwvFl/EoChvbGDeV2XC2T7K/dmDuY7dGWVex0raZnsOMJUQDtcC10oqZnjKYA0whBCOM2f6dMKZMlFSbdrCpiK+MF2BzZJ6J+irCMnKTkk/lQlnxle/Zg/mnrHbgEOEeN8QdmX9faukA6kzC0MmQxwAYLsHcA/wKfBykbk6R7k9RT8SOBl4t4w4O0RZr6FUz7ExjH4B9Mxnie2bCMlS5i25O9/8ArEWqONYAjWXkDDdJanYn3+0jfJQin5ilC+VEWdPwhG6OXswqUHxEdA3zQrbowg7ZQvQB/iGUDz3SFtTCCT9AXwNXGx7LHA5ME9SKbpjmRf1uLLC9kBgFPCepM/LiLMv8ImkI9mDSY5dCnSxfWYC0SBgOSFkj5C0i3AT1AZ4PInV9iLbdbYnpBjWGKwmNLvnAbsJyVsiCuGTtJvwEvW33SfrmecSLkVqgTvLgTML/Ui4sDnOsZK2AB8Saqts4y8C3olEwyXtiPOXA+uAq20ndUYyHEcSdI1F5pytAqZJ2pNnbqF8M+MzVtl+xvZ8Qt+2PTA6JYE5EZyZe/OOhM2WaFAuppJ1gWv7fEI5UweMTLj5mRblUwnP6g3so7CE48coa4AXG5hbEJ+kJcAEQil1ByEULgN6SVpbRpzENXMk7c9V5PvmaSnwZCFnme0OhHbYbEkPFPCclYQifqCkmlLzNdG2/5wz8lYRIutQSQdz9fm+eZoETLad90vGBjAYOExoJjQLMWG6Cnghn1OLxdcMnAhOABN66Mc5FfLsWPi3yd5H0uslMi6NtyvhorkbcAvwPXBJCWvlFgXbvYABklKPpby7UdK3wFtFtqsxuAJ4jPCJzgpCBl5x6jF8l8+p0MCOraDl4h+J2BYkWj5BXQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left( x, \\  y, \\  \\theta, \\  \\dot{\\theta}\\right)$"
      ],
      "text/plain": [
       "(x, y, θ, θ̇)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sp.symbols(\"Omega t\", real=True, positive=True)\n",
    "x, y, theta = spv.dynamicsymbols(\"x, y, theta\", level=0)\n",
    "Omega = theta.diff()\n",
    "x, y, theta, Omega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = RAOreference_frame(\"F\", x=r\"E_x\", y=r\"E_y\", z=r\"E_z\")\n",
    "A = RAOreference_frame(\"A\", x=r\"e_x\", y=r\"e_y\", z=r\"e_z\")\n",
    "A.orient_axis(F, F.z, theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\dot{x}\\; {}^A \\hat e_x + \\dot{y}\\;{}^A \\hat e_y, \\  (- y \\dot{\\theta} + \\dot{x})\\; {}^A \\hat e_x + (x \\dot{\\theta} + \\dot{y})\\;{}^A \\hat e_y, \\  (- \\Omega y + \\dot{x})\\; {}^A \\hat e_x + (\\Omega x + \\dot{y})\\;{}^A \\hat e_y\\right)$"
      ],
      "text/plain": [
       "(ẋ a_x + ẏ a_y, (-y⋅θ̇ + ẋ) a_x + (x⋅θ̇ + ẏ) a_y, (-Ω⋅y + ẋ) a_x + (Ω⋅x +\n",
       " ẏ) a_y)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecP = x * A.x + y * A.y\n",
    "vecP.dt(A), vecP.dt(F), vecP.dt(F).subs(theta.diff(), omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle (- \\Omega^{2} x \\cos{\\left(\\theta \\right)} + \\Omega^{2} y \\sin{\\left(\\theta \\right)} - 2 \\Omega \\sin{\\left(\\theta \\right)} \\dot{x} - 2 \\Omega \\cos{\\left(\\theta \\right)} \\dot{y} - \\sin{\\left(\\theta \\right)} \\ddot{y} + \\cos{\\left(\\theta \\right)} \\ddot{x})\\; {}^F \\hat E_x + (- \\Omega^{2} x \\sin{\\left(\\theta \\right)} - \\Omega^{2} y \\cos{\\left(\\theta \\right)} - 2 \\Omega \\sin{\\left(\\theta \\right)} \\dot{y} + 2 \\Omega \\cos{\\left(\\theta \\right)} \\dot{x} + \\sin{\\left(\\theta \\right)} \\ddot{x} + \\cos{\\left(\\theta \\right)} \\ddot{y})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "⎛   2             2\n",
       "⎝- Ω ⋅x⋅cos(θ) + Ω ⋅y⋅sin(θ) - 2⋅Ω⋅sin(θ)⋅ẋ - 2⋅Ω⋅cos(θ)⋅ẏ - sin(θ)⋅ÿ + cos\n",
       "\n",
       "  ⎞       ⎛   2             2\n",
       "(θ)⋅ẍ⎠ f_x + ⎝- Ω ⋅x⋅sin(θ) - Ω ⋅y⋅cos(θ) - 2⋅Ω⋅sin(θ)⋅ẏ + 2⋅Ω⋅cos(θ)⋅ẋ + s\n",
       "\n",
       "            ⎞\n",
       "in(θ)⋅ẍ + cos(θ)⋅ÿ⎠ f_y"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dvecP = vecP.diff(sp.symbols(\"t\"), F) \n",
    "ddvecP = dvecP.diff(sp.symbols(\"t\"), F) \n",
    "(\n",
    "    ddvecP.subs(theta.diff().diff(), 0)\n",
    "    .subs(theta.diff(), omega)\n",
    "    .simplify()\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = RAOreference_frame(\"F\", x=r\"E_x\", y=r\"E_y\", z=r\"E_z\")\n",
    "A = RAOreference_frame(\"A\", x=r\"e_x\", y=r\"e_y\", z=r\"e_z\")\n",
    "A.orient_axis(F, F.z, theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle (- \\Omega x \\sin{\\left(\\theta \\right)} - \\Omega y \\cos{\\left(\\theta \\right)} - \\sin{\\left(\\theta \\right)} \\dot{y} + \\cos{\\left(\\theta \\right)} \\dot{x})\\; {}^F \\hat E_x + (\\Omega x \\cos{\\left(\\theta \\right)} - \\Omega y \\sin{\\left(\\theta \\right)} + \\sin{\\left(\\theta \\right)} \\dot{x} + \\cos{\\left(\\theta \\right)} \\dot{y})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "(-Ω⋅x⋅sin(θ) - Ω⋅y⋅cos(θ) - sin(θ)⋅ẏ + cos(θ)⋅ẋ) f_x + (Ω⋅x⋅cos(θ) - Ω⋅y⋅sin\n",
       "(θ) + sin(θ)⋅ẋ + cos(θ)⋅ẏ) f_y"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecP = x * A.x + y * A.y\n",
    "dvecP = vecP.diff(sp.Symbol(\"t\"), F) \n",
    "ddvecP = dvecP.diff(sp.Symbol(\"t\"), F)\n",
    "dvecP.subs(theta.diff(), omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle (- \\Omega^{2} x \\cos{\\left(\\theta \\right)} + \\Omega^{2} y \\sin{\\left(\\theta \\right)} - 2 \\Omega \\sin{\\left(\\theta \\right)} \\dot{x} - 2 \\Omega \\cos{\\left(\\theta \\right)} \\dot{y} - \\sin{\\left(\\theta \\right)} \\ddot{y} + \\cos{\\left(\\theta \\right)} \\ddot{x})\\; {}^F \\hat E_x + (- \\Omega^{2} x \\sin{\\left(\\theta \\right)} - \\Omega^{2} y \\cos{\\left(\\theta \\right)} - 2 \\Omega \\sin{\\left(\\theta \\right)} \\dot{y} + 2 \\Omega \\cos{\\left(\\theta \\right)} \\dot{x} + \\sin{\\left(\\theta \\right)} \\ddot{x} + \\cos{\\left(\\theta \\right)} \\ddot{y})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "⎛   2             2\n",
       "⎝- Ω ⋅x⋅cos(θ) + Ω ⋅y⋅sin(θ) - 2⋅Ω⋅sin(θ)⋅ẋ - 2⋅Ω⋅cos(θ)⋅ẏ - sin(θ)⋅ÿ + cos\n",
       "\n",
       "  ⎞       ⎛   2             2\n",
       "(θ)⋅ẍ⎠ f_x + ⎝- Ω ⋅x⋅sin(θ) - Ω ⋅y⋅cos(θ) - 2⋅Ω⋅sin(θ)⋅ẏ + 2⋅Ω⋅cos(θ)⋅ẋ + s\n",
       "\n",
       "            ⎞\n",
       "in(θ)⋅ẍ + cos(θ)⋅ÿ⎠ f_y"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ddvecP.subs(theta.diff().diff(), 0).subs(theta.diff(), omega).simplify())"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__With Functions of time__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y, theta, x(t), y(t), theta(t))"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega, t = sp.symbols(\"Omega t\", real=True, positive=True)\n",
    "x, y, theta = sp.symbols(\"x, y, theta\", cls=sp.Function)\n",
    "x, y, theta, x(t), y(t), theta(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = RAOreference_frame(\"F\", x=r\"E_x\", y=r\"E_y\", z=r\"E_z\")\n",
    "A = RAOreference_frame(\"A\", x=r\"e_x\", y=r\"e_y\", z=r\"e_z\")\n",
    "A.orient_axis(F, F.z, theta(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x{\\left(t \\right)}\\; {}^A \\hat e_x + y{\\left(t \\right)}\\;{}^A \\hat e_y$"
      ],
      "text/plain": [
       "x(t) a_x + y(t) a_y"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecP = x(t) * A.x + y(t) * A.y\n",
    "vecP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABWwAAAAfCAYAAABzh4TgAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjcuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/bCgiHAAAACXBIWXMAABJ0AAASdAHeZh94AAAXTklEQVR4nO2de7TlRJWHv25QYHSNDPhAELmMCuI0dkMjIotuQUFscVRk6YyMYPMQRgQZBOShuNkOoCJg4/hsRgUVFiD4GLEVUFEHeTUMDYI4wixQkIcPbFER5XHnj6p056ZzkpwkldQ5Z39r3ZV7UpVK7apfVe2kksqs6elpDMMwDMMwDMMwDMMwDMMwjP6Z3XcGDMMwDMMwjMlFVY9Q1SP6zodhGIZhGIZhtEkTP3eWPWFrGIZhGIZh9IGqHgyc6X++S0SW9pkfwzAMwzAMw2iDpn6uPWFrGIZhjCWqeqiq3tZ3PgwjwTQ5E1X9e+A04DDgcOB0Vd2831wZhmEYRvyYT2HEhmlyJm34uXbD1jAMwxhX5gIr+s6EYaQwTXpUdTZwNnCRiJwlIp8Bvgqc7cMMwzAMwxiM+RRGbJgmPW35uWsHyp9hGIZh9M1c4OK+M2EYKcZek6r6PWCXnKCNReS+5IeIPAEsTEcQkX0DZ88wDMMwxoWx9ymMkWPsNdm1n2s3bA3DMIyRR1W3BD4F7AjcDhwAzAFO6DNfxuQywZrcBngv8LnUvmkReaCn/BiGYRjGSDPBPoURKROsyU79XLthaxiGYYw0qroFcB3utZODgC2BLwPrYa/lGD0wqZpU1ecB6wM/EJH7e86OYRiGYYw8k+pTGPEyqZrsw8/NvWGrql8AXg1sLiJ/6iIjoahji6rOB64HDhSRzzY49xRwJ3COiCyum07JOQbap6rrAUcDewNTwK+BLwIiIo8Om96A+IVl1VZZjhp96q4vymweVo9tl+Eola/1HUPX1yeAZSJyuP99h6p+F1gUarazbXsnnbb7jypp5sQfWU2GGHNqanY+8DhwY8X4I0lMfvKk+Rux+xpVwseVSb7m8uFBtTdOugtdX33XlT/GfApjFabJZkRgb0Lnfu6s6enpGTtUdTvc3fKjROSMtk+oqrOANwL7AtsDGwJ/AG4FvgIsFZGHWzpXoS2q+m7gdOBfROS8TNhXgR2AF4jIH2uef4qwg9FA+1T12cDlwBa4xY3vAl4LvAhXxgcPmV7tsmqjLJsySbrrgwo2D6XHUGU4KuVrfUe1cB9nU+AXwPYisjy1fymwqYgsKiiKWoSyd1Jpu/8oS3PcNBlyzBlWs6r6YeA9QNqZvlNEts6Jezmwa0mSJ4jISWXn7ZIK5W3+RiBGxdeoEt4Fk6TFPv0mH96J9kZBd1UIWV+x1JUPM5/CME02pG97M/E793Pzvk52CvAQbj2KVlHVvwMuAy7CrXXxHeA04ALgmcBHgVtVdW5LpyyzZVu/vSEn7IPARsC7Gpz/l8BWwHEN0igi1z5VfTLwDdzszC4i8k8icgzwUlzjOlBVN6qanqdJWbVRlrWZQN31wUCba+oxVBmOSvla31EtHNw6Qo8DN2X2zyfcKzmh7J1U2u4/CtNk/DQZcswZVrPzcWPtvNTf7gX5egzQgr+vVDxvlxTp1fyNsIyKr1ElPCgTqMVe/CboXHtR624IQtZXLHUF5lMYDtNkM/q2N03nfu6MG7bq1qLYFbhQRP5cMdOVUNW1ga/59M/BPc68j4gcLyKH4DrtdwDPAS5X1c0anq+KLfOBP+IWSZ6BiFwH/BQ4WFXXqpMHEXlURH4qqa/FtUWJfUfhbDtGRH6Uys8fcbM2s4EFQ6QHDcqqjbKsyyTqrmsq2DyUHkOW4aiUr/Ud1cI908BawDqpPL0cN0iuGHBMbULaW/H8i1V1WlV3rnN8bLTdf1RMc2w0GXrMqaHZbYArReSO1N+9Ofl+HrAB8GMRObHg7ycVzlmZpu2nqLzN3wjLKPkaVcJDMola7NFvgg6114XuuvAzQtVXTHXl0zWfYvg8mJ+LaTKVbgz2puncz82uYbs/MAs3+9o2BwALcTO8+4nIjLUY/O9Pq+rTgA/hZoHflISr6mXAbsBeIvKV1P5ZwOeBtwEfFpFjy2xJPcqc8LiqJv/vIyJf8v+fD5yIE8mlmTReBxyOe1R9A+C3OJFcICKf9HGmyHndI73fp/8hf46nArcAJ4rIJdl8Z8i1T1evf3IfsDTnuN/6bXamZlB6jcuqYngomuruSL/vKBE5PZu4uq8j3gxcKyILCaw7n872wJHATsDTgQeBHwP/KSIXZuK+GTgUmAs8GbgDOA84Q0T+kolbqukBFNlcR4+hy7AvLQLWd3ja7DtuAP4CnKaqpwIvBBK9rsiJn87jsO0bwtsbnMj6kLb7j4FpjoImfT6H8XeCjzkVwpPzbY6r9/8psxHYzm+XF8aKjyI/eaz8XJ/OOPcV0bSdAIyV7xur3+TT70N7seoO6L2+YqsrMJ+iU0KNWz6+jV2OLq+9YrA3OV8vfu7szO9dcY83X9M04Rze7rfvzzoOGc4EVgJ7quoGqf1HA08AJ2Xufp+G62jOSjmxUGzLDbiBAOAqZj6W/INUvGSWY7f0wap6EPB1XGP9Bm7NjGW4r+LtV2Bbls1w63FM4RaVvgCYA3xdVXcpOXaQfXvivlx3nuR/iGVdv/1rxfQaldUQ4aFoqrsr/XaHAcf9B26G6VD/O5juAFT17f7YN/jt6cA3ca+3HZKJewpOU1vhBp+P4zq8U4BLVfVJqbhNNF1kcx09Bi3DkrCgWN/Rft/hZzUPABbhFo4/2p/3YXJmWjMM274hvL1BibAPabv/KEpzFDQJw/k7ofvLKuEJ83FPXawoiQeje8O2qLzHxs+FiegrYmo7bTM2vm/kfhP0o71YdRdDfcVWV2A+RWeEGrd8fBu7PB1fe/Vub4pe/NxVT9iq6lNwazDcJi1/8VZV18E9Iv1n4NqiuCLyiKpei1sL4iX4u9wicpOqfhHXsewDnK2qxwPvBi4E/rWqLSJyoaqu79M6R0TyZjRgdQEvzOw/GNdw5orIrzK2Pr3Ivgw742YKV00FqOp5wLdxwr8i76AS+/bw201U9cScw3f127urpNdCWVUNb502dIebQfkzbh2ZbPpvwjXsj4nIzaF1p6ovws1ePQQsEJFbM+HPSf3/Mty6UHfjFgS/3+8/Dvd6xWtxGjvFH1JL0xX6jaH02EHbLQsLjfUda9K47xCRc4FzM7vfPyh+isrt2+/rwt5gxNaHtN1/lKU5Ipqs7O901F9WCU+YD9wuIn8oiQerHdkd07rL8JG2/dEmFJX3uPm5495XRNh2WmPcfF/i9pugH+1Fp7sUvdVXpHUF5lN0QuBxC2zsyp4r+LVXTPZ6evFz00/YboK7m9362j+4R4dnAStF5IkK8R9MHZfmfcAjwImqeihwMs652CeTbhVbksWJBz7SLCK/9+d7bk7wY8AaMyEi8puCc2b5OXBS5vhL8V/dKziuyL6d/PafAcn5S9JNr5dRVl5Ny6o0PBCNdednu5YDm6rqxsl+34GcAfyK1Z1TaN29AzfJ8u/ZQcgfd0/q5/5+e1IyCPk4j+FeE3kCODCTRB1Nl9k8rB6Dt92etJjG+o6Z+e6t7xiyfUMH9gYmtj6k7f6jSppRazJFFX+nC1+nsr0icpyIbFkUB0hew9zG/9yP/Lo9LKabtZ6i8h43P3fc+4qo2k7LjJvvC/H6TdCD9iLVXZq+6iu6uvL5jqG+ovEpAhJ63AIbuxoT4fgSvZ+7dur/Df32dwNOfBfu9YSqnCsib/X//95vn6mqa/vGUERyF3pGXkTkHlVdAhyLe1z6KuCNIpJ9HbLQFs+2uAb345K8PAg8K7PvXNxj8Leq6gW4R65/JCK/LkkrywoReTxn/93AywqOy7XPC/25wC0isnX2IFV9Ks6eB0TkrrL0UjQpq2HCk3zeRX2tpWlFd7jH5BfiHt1P1v55v4+/n2/kEF53yasD3yo5NjkPwPeyASLyM1W9B9hcVdcXkZXU1/RAm2vqMXQZVglL5/Eu2tHiqnCs78ijlb6jJlXbN3RnL1Cqvyt09TpNCTPWgsshtj6k7f6jMM2UXbFrsqq/01V/WSV8GLYAnob7aMOCssh1CdB+isp73Pzcce8roms75vsCI3TNBb1rrwvd1ekn+6yvWOsKzKfIpWX9hRy3wMauNollfBkmfBha9XPTT9gmX11bNy8i8H/A/w7xd29yoIg8DNyFu0M+ryhD/lWef/A/b8uJkm4UB/i0sxTaou7rqVsDP5GcBaUzrJdKDwAROQP36PUvgHfhHp1/QFWvUNXt1kxiICsH7H+MNdcXTjPIvk389l7y2R14Em4tlyrpNS6rIcMTamstTYu6S9Y1eamP+0LgCOBqVq+bAoF1h1vjBuCXJceC6yRg8IzUfel4DTRdZHMdPYYuwyphaVrRYoL1HQNpq++oQ9X2Dd3Zm7CEmesxKW79LHzesmFfK0lvfb+NpQ9pu/8oTHOENJlQ5u901V9WCR+GRA83tZTeIJbQbvsZWN7j5ucy/n1FjG3HfN/RuuaCfrXXlu6W0GI/2XN9xVpXYD7FIJbQnv7W99vWxy2wsatlohhfhgwfhlb93PQTtslaHBvmRRSRVzY813nA8TiB71sQ762413KuFpGfpwNU9S24BbLvx30t73Dc4+9ZCm3BLRa9LiVfeFPV2bjGf2c2TES+AHzBr5uxI27R6P1xi1RvJZm1TVpmkH1P9ttBgt3Pbz9XMT1ooayqhKdpQWtpGusONws5zeqZu4/jHOF3ysyPOYTW3Uq/3QT4aVEarH7CYiPcRUCWZ2fi1dV0kc119Bi87faoxSRN6ztS1Ok7VHU6L25VRGRW6mfV9g0d2JvJ55Kc4xcDrwfOFpHvl6WRYaXfxtKHtN1/lKU5Kpqs6u8E7y+rhNdgaEdW3bpmXwK2SMZoVT0TeA2wY94TLgHaT1l5j5Ofu9Jvx7WviK7tmO87ctdc0JP2WtbdkpzjF1O/n+yzvqKrKzCfoiSfS3KOX0w9/a302yDjFkz22NXTtVd0Y3VF6vi59wBn+ImBZN884Nr0Ddv7cDMvpesy1ORUnNOwt6qeLyLLcjK6GW5NlSdwC2Gnw16Du+N+K/AK4IfAgap6pohkG2WZLfP89saSPG+JWwdqxaAI/jH5ZcAyX9n7AwuAi0vSbsIg+5I1WDbKHqCqO+AubL4lItdVTA/aK6vSsgxEI90BiMjvVPU2YDtV3Rt4JfApEcmWSWjdXYPrABZRPhDdiHtFYGcyA5GqPh/3ysGdqdc8VjGkpotsrqPHLtpuX1qcgfUdqxi678g6ok0Yon1DN/aGJLY+pO3+oyzNeSnbiuhVk0P4O135Om1rts6TBxcBx+DW4Xu7qh6FW/Mt92ZtIMrKe5z83HHvK0a17VRlnHzfdJ5WEo/fBP1pL1bdzaCH+oqxrsB8iq7oZNyCyRy7err2GtWxuo6fezXu459pPgqcteqVAn83+4fA071QW8WvRfEm4GHgfFXdPR2uqi/APfL9LOAIEbkmFbYTzlm/B3iVd85PwD0h/KGcc5XZktylf6gk28ld/ysyeX21fwQ7yzP9Nu/1tdYYZJ+4Ba9vA+ar6otT+d0MOB83S3RI1fQ8jcpqiPAgNNFdhiuBvwE+A/wGd9GYPVdQ3QGfwr0KdIK6L2HOQGd+gTCZjXufqj4jFWct3AzrbOCzqf21NF1kcx09dlCGZWFBsb4jl1b7DlU91DsBw1DavqEze0MSVR/Sdv9RliYjoMlh/J2O+ssq4ZXxFzbzcDeJytYbW4W39Xhgsaoei1vrbA8RyXsyJghl5T1Ofi5j3leMYtsZhnHyfWP1m3xYX9qLUnfQb31FWldgPkVXBBu3fJiNXQMIde0Vq71F1PVzydywVdW9gLmAZEV3MbAXbs2MO5pkNo2qTgGL/c8rcTMf31bVI0XkDN+ZXIqrtNuADVT1ROD7uEWGL8GJdjcRuQ9ARC5S1euB16vqAhH57yFsucFvT1bVOcCfgFtF5MuZeK8CHmf1WioJ5wOPqOqVuHWiZuFmVV7i0/5Oeak0ZpB9J+EWxf6uqn4JeArwZtxj53vIzI+zVEmvaVlVDW+dJrrLeQXjR8BBwFNxzu2DA04bTHci8hNVPQT4NHCjqn4duB3XWW0H/AHYxce9SlVPBd4D3KKqF/nzLQLm+PL4SCr5JpousrmOHkO23bKw0FjfEb7vmEtqhlRVTwO2EpE9Co6p2r4hvL3BiLQPabv/KEozak2q6lyG93dC95dVwodhK1w7ewg4Rtf8oEjCUhGZsa6biFymqstxunitiFzfQn6GJbe8x83PnZC+YtTaTiXGzfclbr8J+tFedLpL0Xd9xVZXYD5FJwQet8DGriJCXnvFaG8Rdf3ca4DTVXUDnB2nAh8Qkd/Ozhx4MfAAxesd1WEKEP+3KLU/mU14Ps5xAGdkEvdAnFMxDeye8yTFcX6bbVBQYIuI/BA4DFcYh/lzzUvHUdWnAW8ALhGRuzNJHIu7C74tbuZjP9yi0McAu4jIozn5aZtc+0TkPJyj9ivc2jSvAS4E5ojIVTXSa1pWpeEBmaKe7nbOSStZ02Q5mRm3DCF1h4icBeyEG4B3Bo4GXoebmfpEJu4xwFtwg9W+uLXMZuNmsHaTmV8GbaLpIpvr6DFYGfaoxQTrO1IE6jtmOA04R6rsxk7V9g0B7e2CCPuQtvuPgWnGrEn/5EAdfyfomBNAs/P99m9ZPeZm/07AXVhl8/IKXFnOZvW6Zl0zqLynGC8/dxL6ilFrO1WZYrx832j9JuheexHrLqHv+oqmrsB8ior2tEbAcQts7Coi5LVXjPYWUdfPvQH4K25y4QjgUbxmZ01PT2czfRxwCrCt5K/hNzI0sUVVDwM+BizMeaohCtquq7rplZXVKJRlGar6X8AewA4isrwk7ljrLo9YtOiPHViGo1q+bRNLfTXtO1R1S9wrUDviHK4DcE+s7Ql8DzeAPyl1yC0isnVOOpXbt48fxN5JJYTfMeqaHIaQY04smvVPCv0A96GUPYF1RGRR8VHB8hKNnzxp/kYsY5c/diTaThPM911NLNqbBN01JZa68seaT2GYJhtee03KWK2q1+BuYL8N2FtELgE3i5Dlo8AvgA90l71g1LJFVdfDzTxd3HfFldB2XQ2dXllZjVBZDkTdYtj/iFsMu/RmDuOvuzx61yIUl+GIl2/b9F5fTfsOVd0CuA63PtAc3Mz3l4H1cLO8j+Nm2QFeivvi68KcdIZt3xDA3gknhN8xspqsQZAxJxbNqlvPbRnwERE5B7d+7e6quqCnLMXkJ0+av9H72AWj03aaYL7vGvSuvUnQXUv0XldgPkUL+RgnTJMz04lifIlQs1cD7wSuTm7WQs4NWxF5BNgHuF5Vn9Jd/tqngS1TwFLgqBD5aou266pmelMUl1VZeJSo6nNV9VhVPQv4PO7Lmu+pcuy46y6PSLQIxWVYFDZRRFJfUzTrOz4BLBORw0XkDhH5JvBd4H4ReUBEngA2xr1yslxE7heR30Gz9g3B7J1YQvgdo6bJJgQcc8rCg6NuLa9v415VOxlARG7GfUTlg33kKSY/edL8jUjGLhiBtlMH830HE4n2phhD3bVNJHUF5lMYHtNktONLWXjXrMB9rOzd6Z1rLIlgGAao6kG4rxauBC4H/k0yH0AxDKMfVHVT3Ezr9umZWVVdCmyavCqtqgq8QkQWZI639m20SlNNGoZh9I2NjYYRB+ZTGLFh117hUdXLgZ+JyDvT+9fuKT+GETUishQ342IYRnxsg3vt5qbM/vnAZanf84A11jqy9m0EoJEmDcMw+sbGRsOIBvMpjNiwa68AqOps4Bm4tYDnAG/Oxslbw9YwDMMwYmYaWAtYJ9mhqi/Hfbl1RSre1sDNnebMmFRMk4ZhGIZhtIH5FEZsmCbDsBC4D7fkw155y5rYE7aGYRjGqHED8BfgNFU9FXgh8EkftiIVbzbwYlXdGHhYRFZ2mUljojBNGoZhGIbRBuZTGLFhmgyAiHyfkodo7QlbwzAMY6Twax4dACwCrgeOBs4BHgZuT0V9L/BG4B7g5I6zaUwQpknDMAzDMNrAfAojNkyT/fH/XVeF7x8m0YQAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle (- \\Omega x{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} - \\Omega y{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} y{\\left(t \\right)} + \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} x{\\left(t \\right)})\\; {}^F \\hat E_x + (\\Omega x{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - \\Omega y{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} x{\\left(t \\right)} + \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} y{\\left(t \\right)})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "⎛                                                 d                    d\n",
       "⎜-Ω⋅x(t)⋅sin(θ(t)) - Ω⋅y(t)⋅cos(θ(t)) - sin(θ(t))⋅──(y(t)) + cos(θ(t))⋅──(x(t)\n",
       "⎝                                                 dt                   dt\n",
       "\n",
       " ⎞       ⎛                                                d\n",
       ")⎟ f_x + ⎜Ω⋅x(t)⋅cos(θ(t)) - Ω⋅y(t)⋅sin(θ(t)) + sin(θ(t))⋅──(x(t)) + cos(θ(t))\n",
       " ⎠       ⎝                                                dt\n",
       "\n",
       " d       ⎞\n",
       "⋅──(y(t))⎟ f_y\n",
       " dt      ⎠"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(vecP.diff(t, F).simplify().subs(theta(t).diff(t), omega))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle (- \\Omega^{2} x{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + \\Omega^{2} y{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} - 2 \\Omega \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} x{\\left(t \\right)} - 2 \\Omega \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} y{\\left(t \\right)} - \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} + \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)})\\; {}^F \\hat E_x + (- \\Omega^{2} x{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} - \\Omega^{2} y{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 \\Omega \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} y{\\left(t \\right)} + 2 \\Omega \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} x{\\left(t \\right)} + \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)} + \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)})\\;{}^F \\hat E_y$"
      ],
      "text/plain": [
       "⎛\n",
       "⎜   2                   2                                d\n",
       "⎜- Ω ⋅x(t)⋅cos(θ(t)) + Ω ⋅y(t)⋅sin(θ(t)) - 2⋅Ω⋅sin(θ(t))⋅──(x(t)) - 2⋅Ω⋅cos(θ(\n",
       "⎜                                                        dt\n",
       "⎝\n",
       "\n",
       "                           2                     2      ⎞       ⎛\n",
       "    d                     d                     d       ⎟       ⎜   2\n",
       "t))⋅──(y(t)) - sin(θ(t))⋅───(y(t)) + cos(θ(t))⋅───(x(t))⎟ f_x + ⎜- Ω ⋅x(t)⋅sin\n",
       "    dt                     2                     2      ⎟       ⎜\n",
       "                         dt                    dt       ⎠       ⎝\n",
       "\n",
       "\n",
       "          2                                d                        d\n",
       "(θ(t)) - Ω ⋅y(t)⋅cos(θ(t)) - 2⋅Ω⋅sin(θ(t))⋅──(y(t)) + 2⋅Ω⋅cos(θ(t))⋅──(x(t)) +\n",
       "                                           dt                       dt\n",
       "\n",
       "\n",
       "             2                     2      ⎞\n",
       "            d                     d       ⎟\n",
       " sin(θ(t))⋅───(x(t)) + cos(θ(t))⋅───(y(t))⎟ f_y\n",
       "             2                     2      ⎟\n",
       "           dt                    dt       ⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(\n",
    "    vecP.diff(t, F)\n",
    "    .diff(t, F)\n",
    "    .simplify()\n",
    "    .subs(theta(t).diff(t).diff(t), 0)\n",
    "    .subs(theta(t).diff(t), omega)\n",
    ")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rotation Formulae\n",
    "\n",
    "$$\n",
    "{}^{\\mathcal A} \\mathbf v = \n",
    "        {}^{\\mathcal A}\\frac{d \\mathbf r}{d t} =\n",
    "            {}^{\\mathcal B}\\frac{d \\mathbf r}{d t} +\n",
    "                {}^A{\\mathbf \\omega^B} \\times \\mathbf r \n",
    "$$\n",
    "\n",
    "$$\n",
    "{}^{\\mathcal A} \\mathbf v = \n",
    "    {}^{\\mathcal B} \\mathbf v +\n",
    "        {}^A{\\mathbf \\omega^B} \\times \\mathbf r\n",
    "$$\n",
    "\n",
    "\n",
    "$$\n",
    "{}^{\\mathcal A} \\mathbf a = \n",
    "    {}^{\\mathcal B} \\mathbf a \n",
    "        + {}^A \\alpha^\\mathcal{B} \\times \\mathbf r \n",
    "        + { 2 \\cdot {}^A{\\mathbf \\omega^B} } \\times {}^{\\mathcal B} \\mathbf v\n",
    "        + x {}^A{\\mathbf \\omega^B} \\times ({}^A{\\mathbf \\omega^B} \\times \\mathbf r²)\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y, theta, x(t), y(t), theta(t))"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega, t = sp.symbols(\"Omega t\", real=True, positive=True)\n",
    "x, y, theta = sp.symbols(\"x, y, theta\", cls=sp.Function)\n",
    "x, y, theta, x(t), y(t), theta(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x{\\left(t \\right)}\\; {}^A \\hat e_x + y{\\left(t \\right)}\\;{}^A \\hat e_y$"
      ],
      "text/plain": [
       "x(t) a_x + y(t) a_y"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecP = x(t) * A.x + y(t) * A.y\n",
    "vecP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} x{\\left(t \\right)}\\; {}^A \\hat e_x + \\frac{d}{d t} y{\\left(t \\right)}\\;{}^A \\hat e_y$"
      ],
      "text/plain": [
       "d              d\n",
       "──(x(t)) a_x + ──(y(t)) a_y\n",
       "dt             dt"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vecP.diff(t, A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAdwAAAAcCAYAAADLPA71AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjcuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/bCgiHAAAACXBIWXMAABJ0AAASdAHeZh94AAAM+UlEQVR4nO2de7Rm5RzHPzNE0aKlWii6WJYRk2YqpkVNpAtioZZbSxrKtRhRmlDfvlZDl6mpXFq00FAWaRC5pUh0YWYylcoiVFKTEqFUU44/fvud2ec9+7zv3u+793535+zPP/uc/Tx779/zfZ7fs593P7cZY2NjtEzE9mHAoZK2G7UtjxZazXpj+3AASUtHbct0oS2TxWk1680wfvzY8s2ZMuwArB61EY8yppVmtrcDfgPcKGlun7jvBj6V/H2fpC/UYGLLNCuTJTGtNKvTj2cObOXUZ1oVupKYbpqdDpwMPN/24yaLZPtZwBLg/cBC4BTb29Zj4rRnupXJMphumtXmx+0vXMD2LOBM4MXAH4CDgdnAMaO0q8lMd81s7wc8AiwFPg48j4xKyvZM4GzgfElnJedeApxt+2WS/leXzVOd6V4mB2G6a1a3H2e+cG1/BXgFsK2k+4ono1wGscf2TsBK4BBJX+wR7znArwkx3wXMAr4JbETOVp7tjYAjgQOAbYC7gK8CkrS2zvTUQRmaDfDM0jUbVNMkv08GXifpHttrgLlkpD1xxPld596W91mjpOx6oOj9iuRP68fFmQp+PIyeo/DjCZ+Ube8MvBU4oSEv25722P6Q7THbB6TPS1oFfAc43vbGPR7xWeAHkhZKuknS94FLgDWS7sxh39OBFUTr6Bri88S/gKOBz4wgPXUwlGZFqUqzITRdBFwq6brk/98Ccwpc33jKrgd63a+kMt/6cXEe9X48pJ61+3FWH+4niYJ2ZpUPLkA/e3ZMjqsywj4FPA34QNaFtp8J7Amc2hX0MDlaeMn3/u8RreGXSXqTpKOAecCtwCG2n9Z1WWXpqYNhNRuQKjUrpKntbYBDgWNTp6fcC5fy64Fe9xuqzLd+XJwp5seF9RyVH4974SafGPYEzpP03yofnIec9uwE/IfofxiHpF8DvwPebfsxGdfOJb7fX5Nxz9U5TDwiiXuUpMtTz/0P8G1C391qTE9fbC9IWpEvHeR6htesEFVrNoCmS4FNgVtsP2z7YcLR59iekSdNVTNsHpddD+S437BlvvXj4kwZPx5Qz5H4cXcf7juAGcA3yn6Q7Q8TI7yOkHRKRvgs4FrgV5I638ontcf2icBHUqcesd35+0BJ5yR/fx04jsjsH3fdZgx4DPB44KHkvrsTLa2T+qSn099zB5A1NPzvyTHdMq46PXUwsGZJ3IuAvYD9JX0rdX4G8GXgIOBESYuSoDo0y6Wp7b2B3YmK4aFU0POTe2wL/Cnjuv2BQ4AXAk8CbknSeqKkRyZ73gjpWQ+UlYcl5l/rx8UZRrMm1uW59RylH3e/cPckWj1X5bm4IL9MjrtMEv5pogAcltOeVcAywrmvAH6SCvt56u9Oi3UvJmbEKuBBYIntk4DnAp9LwlZPYmeH1wObAF/MGlABbJgc0xladXrqYBjNICq3q4k+lwtSBXUJkfazUhU11KNZX01tb0D0650i6equsL8mf84l5ahJa/sc4M3ATcSAlAeBVwKLiUEqB2U9b8T0qwfKysOy8q/14+IMo1kT6/Jceo7aj9e9cG0/kfh+fWNFg6WuBv5L9IuMw/YbCKHOkHRtHnsknWd7EyKhy3pMQF6RHOd3B0i63fbBRB/AG4nPK8uAD5PxWaOLfZPjlraPywjfMzn+pa701MGQmiHpGttfJdJ5IDGs/qPAh4DzgPd04taoWR5NFwKbA2dk2PEP23cnti5PBZ1OOOkJwDGSHgawfSRwKfA22ydKuqHHc2slTz1QVh6WlX+tHxdnSM0aV5f3CUszUj9O/8LdkmiV3NHvokGQtNb2CmC+7S0k3Q7rMuNU4G+M78DOY0+nk/3qySJIutf2A8BWk4SfC5zbdfrYrLhd7Joc39wnXicTaklPHQyhWYePA28CjktGFi4mWqUHavx8trrKQF9NJS0hfsFNFr55+n/b84D3ARdIOror7lrby4i5j/NYX0aaQN56oKw8LKXMt35cnEE1a2JdnlfPUftx+oW7aXL8R1ZE2zcDW/e7YYpzJb2169zlRAtkF6DT93Ms8Azg7ZLuzWtPwo7AWuC6HnEA7gGemsfoPCQFayvgt5K2zwjfOHnmnZJuTk7Xnp4+efazVD9Jh2WSFvS777BIus32acSw/E8Tn5H2k/RQV9Q6NSu1jBCr0cwA7p/kl9Ps5DjUAI0K8jiP5mXm4Uh8GFo/HpIm1uWllxFK9uP0C7czcmzDrIjAH4EH8tw04faMc53v7POAb9l+LnA4cCXxOSNNT3tsPxbYHrhB0oN9bNkodb8y2DI5ZqURYB9gA+D7qXOjSM9pRP9UmjnAawm9b+4KW53jnmVxV+rvgyXdnxGnTs3KLiN7J8e39Il365DPOY1y87hfPZBmqDwcsQ9D68fD0MS6vIoyUqofp1+4f0uOm2ZFlPTyPDfswxXE6LhOZ/tniE8Nh0rq3raopz3EElwb0uMTBKxbkmsT4M/J/2Vsj9RpDU9WON6eHL+UOldJenoh6bSM6xcQjnq2pEv73SO5ZmjNJK1rAdp+C/FZZw0x+nMh8N6My2rRrIimebC9IdFPdJmk3Qtc9wZicMZzJN2SnDsdeBXwYkl3dV9TVh6n6Kd55xll5OHA+TdsmUzKY2fd3NaPc5Ly45HX5XnDBmUQP7Z9G3CqpFNT5+YAvwLmpl+4dxAt1lllGdxN0il9I7CzYzWRlwNnSvpNRvR+9sxJjlnXpplF/Nxfndgw9Bwr25slf3ZPhsf2LkQF+cNkfliHStJTB2Vo1sH2q4gW8PXAHsBlxMICp0v6XVf0ujQrW9OOXpv1jDWR84GjiP7Rd9o+guhbzHzZVkTfeqDEPJyTHAvnX0llck1ybP24+L1GXpcXCBuUQfz4SmLaUJqlxMj9G9YtfJG0Si4DNrP97KHM7M0vgScAnwfuJiqXCeSwp9Na+lef53VaYD/rZ5jtw5JC1BNJdwM3AjvZfkHq+q2JeVz3Eh3t6WtqT08d5NUsibsr8VK5Ddg7eYkcQ3xpOaE7fo2alaqpYmL/tcDzHIujT8D2rp44gX8M+CiwwPYiok9sX0l/LMOuPPTTvOQ8rKTMt35cnCJ+nKJJdXnpeg7ox+NeuI65uzsAgonzcJcD+xN9FzeVZXgXlxMLZW8MHC7pnh5xe9nTWf5rse3ZwH3A9ZK+2RVvb2L+1wU5bBu3LZXtJcB2kvbNiHs8McLvEtvnAE8khtePEZXkzQ1ITx3k0sz2DsCFRCW2l6Q7ACSdb3sl8Frbu0n6Rdf969CsCk2PJPr+ltu+mHDcmUS/4U7ABpKyRltelIwAPR54taSVJdqUl0zNK8jDqsp868fFKaJZhybV5VXpWdSPryK27XsKkY6TgE9I+jtMXEt5OXAnUOVuJp1v7CuAfrs7TGqPpMuIEWT3JUfRtQ6m7ScDrwMulPSXHLZ17wP5QmIniglI+hqwgOifeC/x+ek8YLakKxqSnjroq1nSqv0xUYntk/GLrTPc/uSM+1eqWVWaSrqImC6wnOjz/wAxZ3U74GLW9w9227MHoelM1vd91c0EzavIwwrLfOvHxcmtWYpG1OVV6jmAH68iFknZmRhEtpbYJAKAGWNj4/u3bR9NLDK94yTf44fC9neJyea7SFqRI/7A9th+PzHBeX53q9vZ+0BeSqw881Mi8zdIXZI5daAoVaWnDqaiZqPWtMuWHYiVdRYSmj5e0itHZEup9cCg98uRf1OuTFZNWZo1qC5vjA8n9lxFNEIOAg6QdGEnLGu3oKXEEOdPVGDIAcBriM71vhk0jD2ONVKPBpZnZFBnH8jriHlUixi/D+QjrJ8QPw94OuWtCFN6eupgKmo2ak27bNka+AFwsqRlRP/tPrZ3631lZZRdDxS+X478m3JlsmrK0qxBdXljfDjFlcRORFemX7aQ8cKV9ADxk3mlY2L4UNjeyvYi22cRCz1fz/iFqnsyhD3bEIuRH5ER1nMfSMVKOVsA/wZWSFojqedCAHmpKD11MBU16xdeC0l/z4+IT2KLARTL4p1PLL1XO2XXAwPebxt6589ULJNVM7BmDa3Le4WNitXA/4ilTscx4ZNy2dh+FzGK7Z/EotQfVLIU2Chw7AN5K/CidMvM9heAZ3Y+4dk2sIekUf3CaAytZi1Noy2TxRlWs6bV5U3F9k+A30s6tDuse5Ry6SgWop5sMepR0GsfyItS/8+h/7yw6UKrWUvTaMtkcYbSrIF1eWNwLLyxOdEfPpsY6T6BrD7cqc4Y6/eBBMDr94FcnYq3PTEEvKXVrKV5tGWyOK1m1TGfWODjQGKf6Myui8p/4TaQvPtAzgReYHsL4H5J/6zTyIbRatbSNNoyWZxWs4pQLK/Z9wfstPuFm/Q5HExsHrySmNi8DLif8ftAfgzYj1hRZ3HNZjaKVrOWptGWyeK0mo2eygdNtbS0tLS0tMD/AT5b87SHoQrwAAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle (- y{\\left(t \\right)} \\frac{d}{d t} \\theta{\\left(t \\right)} + \\frac{d}{d t} x{\\left(t \\right)})\\; {}^A \\hat e_x + (x{\\left(t \\right)} \\frac{d}{d t} \\theta{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)})\\;{}^A \\hat e_y$"
      ],
      "text/plain": [
       "⎛       d          d       ⎞       ⎛     d          d       ⎞\n",
       "⎜- y(t)⋅──(θ(t)) + ──(x(t))⎟ a_x + ⎜x(t)⋅──(θ(t)) + ──(y(t))⎟ a_y\n",
       "⎝       dt         dt      ⎠       ⎝     dt         dt      ⎠"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(vecP.diff(t, A) + spv.cross(theta(t).diff(t) * F.z, vecP)).express(A)  # .simplify()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Translation\n",
    "\n",
    "\\begin{split}\n",
    "    \\mathbf{r}^{OP} \\textrm{, the position vector from } O\\textrm{ to }P\\\\\n",
    "    ^{\\mathbf{N}}\\mathbf{v}^P \\textrm{, the velocity of } P\n",
    "        \\textrm{ in the reference frame } \\mathbf{N}\\\\\n",
    "    ^{\\mathbf{N}}\\mathbf{a}^P \\textrm{, the acceleration of } P\n",
    "        \\textrm{ in the reference frame } \\mathbf{N}\\\\\n",
    "\\end{split}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = RAOreference_frame(\"N\", x=r\"n_x\", y=r\"n_y\", z=r\"n_z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "q_x, q_y = spv.dynamicsymbols(\"q_x, q_y\", level=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle  q_{x}\\; {}^N \\hat n_x + q_{y}\\;{}^N \\hat n_y\\quad   \\dot{q}_{x}\\; {}^N \\hat n_x + \\dot{q}_{y}\\;{}^N \\hat n_y\\quad   \\ddot{q}_{x}\\; {}^N \\hat n_x + \\ddot{q}_{y}\\;{}^N \\hat n_y\\quad $"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r_OP = q_x * N.x + q_y * N.y\n",
    "v_NP = r_OP.dt(N)\n",
    "a_NP = v_NP.diff(sp.Symbol('t'), N)\n",
    "\n",
    "ipd.display(\n",
    "    ipd.Math( # type: ignore\n",
    "        rf\"$ {spv.vlatex(r_OP)}\\quad \"\n",
    "        rf\"  {spv.vlatex(v_NP)}\\quad \"\n",
    "        rf\"  {spv.vlatex(a_NP)}\\quad $\"\n",
    "    )\n",
    ")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Translating, rotating disc__\n",
    "\n",
    "![](./images/kin_1.png)\n",
    "\n",
    "__Solution to rotating disc using Vectors directly and not using \n",
    "point theory__\n",
    "\n",
    "see example in [this document](https://docs.sympy.org/latest/modules/physics/vector/kinematics.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_1, u_2, u_3 = spv.dynamicsymbols(\"u_1, u_2, u_3\", level=0)\n",
    "o_1, o_2, o_3 = spv.dynamicsymbols(\"o_1, o_2, o_3\", level=0)\n",
    "R = sp.symbols(\"R\", real=True, positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = reference_frame(\"N\", x=r\"x\", y=r\"y\", z=r\"z\")\n",
    "B = reference_frame(\"B\", x=r\"x\", y=r\"y\", z=r\"z\")\n",
    "B.set_ang_vel(N, u_3 * B.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle o_{1}\\; {}^\\mathcal N \\hat x + o_{2}\\;{}^\\mathcal N \\hat y + o_{3}\\: {}^\\mathcal N \\hat {z}$"
      ],
      "text/plain": [
       "o₁ n_x + o₂ n_y + o₃ n_z"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_O = o_1 * N.x + o_2 * N.y + o_3 * N.z\n",
    "r_O"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle o_{1}\\; {}^\\mathcal N \\hat x + o_{2}\\;{}^\\mathcal N \\hat y + o_{3}\\: {}^\\mathcal N \\hat {z} + R\\; {}^\\mathcal B \\hat x$"
      ],
      "text/plain": [
       "o₁ n_x + o₂ n_y + o₃ n_z + R b_x"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_P = r_O + R * B.x\n",
    "r_P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle u_{1}\\; {}^\\mathcal N \\hat x + u_{2}\\;{}^\\mathcal N \\hat y + R u_{3}\\;{}^\\mathcal B \\hat y$"
      ],
      "text/plain": [
       "u₁ n_x + u₂ n_y + R⋅u₃ b_y"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(r_P.dt(N).subs(o_1.diff(), u_1).subs(o_2.diff(), u_2).subs(o_3.diff(), 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\dot{u}_{1}\\; {}^\\mathcal N \\hat x + \\dot{u}_{2}\\;{}^\\mathcal N \\hat y -  R u_{3}^{2}\\; {}^\\mathcal B \\hat x + R \\dot{u}_{3}\\;{}^\\mathcal B \\hat y$"
      ],
      "text/plain": [
       "                       2\n",
       "u₁̇ n_x + u₂̇ n_y + -R⋅u₃  b_x + R⋅u₃̇ b_y"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(\n",
    "    r_P.dt(N)\n",
    "    .dt(N)\n",
    "    .subs(o_1.diff().diff(), u_1.diff())\n",
    "    .subs(o_2.diff().diff(), u_2.diff())\n",
    "    .subs(o_3.diff().diff(), 0)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intrinsic coordinates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = RAOreference_frame(\"F\", x=r\"E_x\", y=r\"E_y\", z=r\"E_z\")\n",
    "m, g, R = sp.symbols(\"m, g, R\", real=True, positive=True)\n",
    "r = spv.dynamicsymbols(\"r\", real=True, level=0)\n",
    "y = (r**2) / 2 / R\n",
    "P = r * F.x + y * F.y\n",
    "vtP = P.dt(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "hat_vtP = vtP.normalize()\n",
    "hat_vnP = F.z.cross(hat_vtP)\n",
    "assert hat_vtP.magnitude().simplify() == 1\n",
    "assert hat_vnP.magnitude().simplify() == 1  # Unit vector\n",
    "assert hat_vtP.dot(hat_vnP) == 0  # Orthogonal"
   ]
  }
 ],
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