cinematique.ipynb

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   "source": [
    "Markdown macros\n",
    "\n",
    "$\\newcommand{\\I}{\\mathrm{i}} \n",
    "\\newcommand{\\E}{\\mathrm{e}} \n",
    "\\newcommand{\\D}{\\mathop{}\\!\\mathrm{d}}\n",
    "\\newcommand{\\bra}[1]{\\langle{#1}|}\n",
    "\\newcommand{\\ket}[1]{|{#1}\\rangle}\n",
    "\\newcommand{\\braket}[1]{\\langle{#1}\\rangle}\n",
    "\\newcommand{\\bm}[1]{\\boldsymbol{#1}}$"
   ]
  },
  {
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   "metadata": {},
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    "# Rotation d'un corps solide\n",
    "\n",
    "Pour décrire le mouvement de rotation d'un solide autour d'un point \"origine\" $O$, on défini un système de coordonnées *fixe* (celui du laboratoire, $XYZ$), et un repère attaché au solide, qu'on désigne comme le repère *mobile* $(xyz$).\n",
    "\n",
    "Pour passer du repère fixe au mobile (ou vice versa) on n'a besoin que de trois rotations, par exemple paramétrées par les angles $(\\phi,\\theta,\\psi)$ d'Euler.\n",
    "\n",
    "Quant à son mouvement, le solide, de masse $M$, est caractérise par son centre de masse $\\bm R_M$ et son moment d'inertie $I^G$ (rapporté à son centre de masse). Sa rotation est décrite par le vecteur de vitesse angulaire $\\bm \\Omega$. L'énergie cinétique du solide ne dépend que de ces paramètres. \n",
    "\n",
    "Le moment cinétique $\\bm L$ est la somme du moment cinétique du mouvement du centre de masse et de la rotation:\n",
    "$$\n",
    "\\bm L = M \\bm R_M \\wedge \\dot{\\bm R}_M + \\bm \\Omega \\wedge \\bm L\n",
    "$$\n",
    "(Dans le repère mobile, le premier terme s'annule.) Le moment cinétiques est proportionnel a la vitesse angulaire:\n",
    "$$\n",
    "L = I\\bm \\Omega\n",
    "$$\n",
    "où le coefficient de proportionnalité est le moment d'inertie (qui est une matrice  $3\\times3$).\n",
    "\n",
    "La *dynamique* est régie par l'équation de Newton, qui tient compte de forces appliquées $\\bm F$, \n",
    "$$\n",
    "M\\frac{\\D^2 \\bm R_M}{\\D t^2} = F,\n",
    "$$\n",
    "et par l'équation d'Euler, qui relie la variation du moment cinétique aux couples $\\bm K$ appliqués:\n",
    "$$\n",
    "\\frac{\\D \\bm L}{\\D t}\\Big|_{XYZ} = \\frac{\\D \\bm L}{\\D t}\\Big|_{xyz} + \\bm \\Omega \\wedge \\bm L\\Big|_{xyz} = \\bm K\n",
    "$$\n",
    "où on calcule la partie droite de l'équation dans le repère mobile.\n",
    "\n",
    "### Méthode de résolution des problèmes\n",
    "\n",
    "Cinématique:\n",
    "* Définir les repères et la géométrie du système solide \n",
    "* Calculer la matrice de passage $R$ (ou de changement de coordonnées $R^T=C$)\n",
    "* Calculer le centre de masse et ses dérivées, sa vitesse et son accélération\n",
    "* Calculer, dans le repère mobile, la matrice d'inertie\n",
    "* Calculer la vitesse angulaire dans le repère mobile\n",
    "* Calculer le moment cinétique dans le repère mobile\n",
    "\n",
    "Dynamique:\n",
    "* Écrire l'équation de Newton\n",
    "* Écrire l'équation d'Euler\n",
    "\n",
    "Pour un système compliqué (plusieurs axes de rotation d'un système articulé) il est nécessaire de faire plusieurs transformations de coordonnées, dans ce cas il est possible d'utiliser la formule transformation du moment d'inertie\n",
    "$$\n",
    "I = RI^G R^T\n",
    "$$\n",
    "ou $R$ est la matrice de passage.\n"
   ]
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  {
   "cell_type": "markdown",
   "id": "f422f45e-0183-458c-aa09-b02b255abeac",
   "metadata": {},
   "source": [
    "# Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "67990397-9f07-4111-b707-e696edca7ceb",
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   "source": [
    "# numpy\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2db7197f-78d1-4bb5-b730-164071243bd6",
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   "source": [
    "# matplotlib\n",
    "import matplotlib\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "12ca916e-8991-447a-a701-f20c673f0101",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c5cc2285-a369-49ce-a2e7-6f67e3eb8947",
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   "source": [
    "# sympy\n",
    "import sympy as sy\n",
    "from sympy import symbols\n",
    "from sympy import simplify, expand, collect, cancel, apart, trigsimp\n",
    "from sympy import integrate, diff, series\n",
    "sy.init_printing(use_unicode = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ec68268-cdb2-4579-8d9e-13954bc4e590",
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   },
   "source": [
    "# Centre de masse\n",
    "\n",
    "Pour un ensemble de $N$ particules de masse $m_n$ la défintion de $\\bm R$ est:\n",
    "$$\n",
    "\\sum_{n = 1}^N m_n (\\bm r_n - \\bm R) = 0\n",
    "$$\n",
    "où $\\bm r_n$ sont les positions:\n",
    "$$\n",
    "\\bm R = \\frac{1}{M}\\sum_{n = 1}^N m_n \\bm r_n, \\quad M = \\sum_{n = 1}^N m_n\n",
    "$$\n",
    "\n",
    "Pour une distribution de masse continue de densité $\\rho(\\bm r)$, la formule équivalente est\n",
    "$$\n",
    "\\bm R = \\frac{1}{M}\\int_V \\D \\bm r\\, \\bm r \\rho(\\bm r), \\quad  M = \\int_V \\D \\bm r\\, \\rho(\\bm r)$$\n",
    "avec $V$ le volume du corps."
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "# Aire et volume\n",
    "\n",
    "L'élément $\\D V$ d'un volume $V$ défini par les formules paramétriques\n",
    "$$\n",
    "x = x(u, v, w), \\; y = y(u, v, w), \\; z = z(u, v, w)\n",
    "$$\n",
    "est,\n",
    "$$\n",
    "\\D V = \\D x \\D y \\D z= J \\D u \\D v \\D w\n",
    "$$\n",
    "avec le jacobien,\n",
    "$$\n",
    "J = \\begin{vmatrix} \n",
    "\\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} & \\frac{\\partial x}{\\partial w} \\\\ \n",
    "\\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v} & \\frac{\\partial y}{\\partial w} \\\\\n",
    "\\frac{\\partial z}{\\partial u} & \\frac{\\partial z}{\\partial v} & \\frac{\\partial z}{\\partial w} \n",
    "\\end{vmatrix}\n",
    "$$\n",
    "\n",
    "On obtient l'élément d'aire $\\D A$ d'une façon équivalente:\n",
    "$$\n",
    "dA = \\D x \\D y = J \\D u \\D v, \\quad J = \\left| \\frac{\\partial \\bm r}{\\partial u} \\times \\frac{\\partial \\bm r}{\\partial v} \\right|\n",
    "$$\n",
    "où $\\bm r = (x(u,v), y(u,v), z(u,v))$.\n",
    "\n",
    "### Surface, forme explicite\n",
    "\n",
    "Soit $\\bm r = (x, y, f(x,y)$ (les deux paramètres sont les coordonnées $(x,y)$):\n",
    "$$\n",
    "A = \\int_S \\D x \\D y\\, \\left| (1,0,f_x) \\times (0,1,f_y) \\right|  = \\int_S \\D x \\D y\\, \\sqrt{1 + f_x^2 + f_y^2}\n",
    "$$ \n"
   ]
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   "source": [
    "# Moment d'inertie\n",
    "\n",
    "On consirère un corps de densité $\\rho = \\rho(\\bm r$, en général non homogène, de volume $V$. Le moment d'inertie est construit sur les moments d'ordre deux de la densité:\n",
    "$$\n",
    "I_{ij} = \\int_V \\D \\bm r \\, (r^2 \\delta_{i,j} - x_i x_j) \\rho(\\bm r)\n",
    "$$\n",
    "avec $i,j = 1, 2, 3$ correspondant aux coordonnées $\\bm r = (x, y, z)$.\n",
    "\n",
    "On peut représenter le moment d'inertie par une matrice:\n",
    "$$\n",
    "I = \\int_V \\D \\bm r \\, \\rho \\begin{pmatrix} y^2 + z^2 & -xy & -xz \\\\  -yx & x^2 + z^2 & -yz \\\\  -zx & -zy & x^2 + y^2 \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "Un transformation de coordonnées permet de rendre la matrice diagonale. Les éléments de la diagonale:\n",
    "$$I = \\mathrm{diag}(I_x, I_y, I_z)$$\n",
    "sont les moments \"principaux\" d'inertie."
   ]
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   "source": [
    "## Choix des coordonnées\n",
    "\n",
    "L'avantage d'écrire $I$ comme un tenseur c'est qu'on peut calculer ses élément dans n'importe quelle base. Cependant le calcul de $I_{ij}$ est grandement simplifié si on choisi un système de coordonnées qui respecte les symétries de l'objet.\n",
    "\n",
    "Si on note $I^G$ le tenseur d'inertie calculé sur le centre de masse, le tenseur $I$ calculé sur un point $\\bm d = (d_x, d_y, d_z)$, avec $d$ la distance du nouvel origine au centre de masse, on a la formule de changement de coordonnées:\n",
    "$$I_{ij} = I^G_{ij} + M(d^2 \\delta_{ij} - d_id_j)$$\n",
    "\n",
    "Dans le cas d'une rotation $R_{\\bm n}(\\varphi)$ du solide autour de centre de masse, la matrice $I^G$ change selon la formule:\n",
    "$$I = R_{\\bm n}(\\varphi) I^G R_{\\bm n}(\\varphi)^T$$\n",
    "\n",
    "Considérons un exemple. Un carré de côté $a$, avec l'origine en $(0,0)$ et en $(a/2,0)$:"
   ]
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   "source": [
    "## Axes principaux\n",
    "\n",
    "Comme le tenseur d'inertie est symétrique, on peut toujours diagonaliser sa matrice, et donc l'ecrire comme\n",
    "$$I = \\mathrm{diag}(I_x, I_y, I_z)$$\n",
    "Les trois valeurs de la diagonale sont les racines du polynôme caractéristique:\n",
    "$$P(\\lambda) = \\det\\left(I - \\lambda 1_3\\right) = 0$$\n",
    "($1_3$ est la matrice identité de dimension 3); les vecteurs propres de la transformation sont les axes principaux du corps. Ils coincident avec les axes de symétrie du corps, si la densité est uniforme."
   ]
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   "source": [
    "# Cinématique du solide\n",
    "\n",
    "## Vitesse angulaire\n",
    "\n",
    "Dans un repère fixe $(X,Y,Z)$ un point $\\bm X$ du solide $\\mathcal{V}$ varie selon le mouvement du centre de masse $\\bm R$ et de sa propre rotation d'angle $\\varphi$ et d'axe $\\bm n$, $\\D \\bm \\varphi = \\bm n \\D \\varphi$:\n",
    "$$\n",
    "\\D \\bm X = \\D \\bm R + \\D \\bm \\varphi \\times \\bm r\n",
    "$$\n",
    "où $\\bm r$ est un point appartenant à $\\mathcal{V}$, mesuré dans le repère $(x,y,z)$ d'origine $O$ (le centre de masse) fixé au solide: $\\bm X = \\bm R + \\bm r$ à chaque instant $t$.\n",
    "\n",
    "Les coordonnées $(X,Y,Z)$ sont fixes, tandis que les coordonnées $(x,y,z)$ évoluent dans le temps:\n",
    "$$\n",
    "\\frac{\\D \\bm X}{\\D t} = \\bm v, \\; \\frac{\\D \\bm R}{\\D t} = \\bm V, \\; \\frac{\\D \\bm \\varphi}{\\D t} = \\bm \\Omega\n",
    "$$\n",
    "On a donc la transformation de la vitesse entre le repère fixe $\\bm v$ et le mouvement du corps:\n",
    "$$\n",
    "\\bm v = \\bm V + \\bm \\Omega \\times \\bm r\n",
    "$$\n",
    "Le vecteur $\\bm r$ ne dépend pas du temps puisqu'il est un point d'un corps solide (indéformable).\n",
    "\n",
    "Si l'origine de coordonnées du corps n'est pas sur son centre de masse, mais il est déplacé de $\\bm d$, on a la transformation suivante:\n",
    "$$\n",
    "\\bm v = \\bm V + \\bm \\Omega \\times \\bm d + \\bm \\Omega \\times \\bm r'\n",
    "$$\n",
    "avec $\\bm r = \\bm r' + \\bm d$,\n",
    "$$\n",
    "\\bm V' = \\bm V + \\bm \\Omega \\times \\bm d, \\; \\bm \\Omega' = \\bm \\Omega\n",
    "$$\n",
    "le centre de masse tourne, mais la vitesse angulaire du corps est invariante."
   ]
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    "## Énergie cinétique\n",
    "\n",
    "Elle est donnée par la formule habituelle\n",
    "$$T = \\frac{1}{2} \\sum_n m_n \\bm v_n^2$$\n",
    "En y substituant l'expression de $\\bm v$ on obtient:\n",
    "$$T = \\frac{1}{2} \\sum_n m_n \\bm V^2 + \\sum_n m_n \\bm \\Omega \\times \\bm r_n + \\frac{1}{2} \\sum_n m_n \\left( \\Omega \\times \\bm r_n \\right)^2$$\n",
    "On note que le terme du milieu est nul, par définition du centre de masse, on a donc:\n",
    "$$T = \\frac{M V^2}{2} + \\frac{1}{2} \\sum_n m_n \\left( \\Omega \\times \\bm r_n \\right)^2=  \\frac{M V^2}{2} +  \\frac{1}{2} \\sum_n m_n (\\Omega_l^2 x_{nl}^2 - \\Omega_i x_{ni} \\Omega_j x_{nj})$$\n",
    "où nous avons introduit la notation en coordonnées (avec la convention que l'on fait la somme sur les indices répétés). En notant que $\\Omega_l^2 x_l^2 = \\Omega_i \\Omega_j \\delta_{ij} x_l^2$, on peut écrire:\n",
    "$$T = \\frac{M V^2}{2} + \\frac{1}{2} \\sum_n m_n (x_{nl}^2 \\delta_{ij} - x_{ni} x_{nj}) \\Omega_i \\Omega_j$$\n",
    "On trouve donc\n",
    "$$T = \\frac{M V^2}{2} + \\frac{1}{2} I_{ij} \\Omega_i \\Omega_j$$\n",
    "avec \n",
    "$$I_{ij} =  \\sum_n m_n (x_{nl}^2 \\delta_{ij} - x_{ni} x_{nj})$$\n",
    "le tenseur d'inertie."
   ]
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   "source": [
    "### Angles d'Euler"
   ]
  },
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    "Les angles d'Euler sont définis par la matrice de rotation entre le repère fixe $(\\bm e_X, \\bm e_Y, \\bm e_Z)$ et le repère du corps (mobile)  $(\\bm e_x, \\bm e_y, \\bm e_z)$:\n",
    "* Rotation d'angle $\\phi$ et d'axe $Z$: $(XYZ) \\rightarrow (Ny_1Z)$\n",
    "\\begin{align}\n",
    "\\bm e_N &= R_Z(\\phi) \\bm e_X \\\\\n",
    "\\bm e_{y_1} &= R_Z(\\phi) \\bm e_Y\n",
    "\\end{align}\n",
    "\n",
    "* Rotation d'angle $\\theta$ et d'axe $N$: $(Ny_1Z) \\rightarrow (Ny_2z)$\n",
    "\\begin{align}\n",
    "\\bm e_{y_2} &= R_N(\\theta) \\bm e_{y_1} \\\\\n",
    "\\bm e_z &= R_N(\\theta) \\bm e_Z\n",
    "\\end{align}\n",
    "\n",
    "* Rotation d'angle $\\psi$ et d'axe $z$: $(Ny_2z) \\rightarrow (xyz)$\n",
    "\\begin{align}\n",
    "\\bm e_x &= R_z(\\psi) \\bm e_N \\\\\n",
    "\\bm e_y &= R_z(\\psi) \\bm e_{y_2}\n",
    "\\end{align}\n",
    "\n",
    "La composition de trois rotations s'écrit\n",
    "\\begin{align}\n",
    "\\bm e_x &= R_z(\\psi)R_x(\\theta)R_z(\\phi) \\bm e_X \\\\\n",
    "\\bm e_y &= R_z(\\psi)R_x(\\theta)R_z(\\phi) \\bm e_Y \\\\\n",
    "\\bm e_z &= R_z(\\psi)R_x(\\theta) \\bm e_Z\n",
    "\\end{align}\n",
    "ou aussi comme:\n",
    "$$ \\begin{pmatrix} \\bm e_x & \\bm e_y & \\bm e_z \\end{pmatrix} = \\begin{pmatrix} \\bm e_X & \\bm e_Y & \\bm e_Z \\end{pmatrix} R(\\phi,\\theta,\\psi) ,$$\n",
    "avec,\n",
    "$$R(\\phi,\\theta,\\psi) = R_Z(\\phi) R_X(\\theta) R_Z(\\psi),$$\n",
    "soit\n",
    "$$R(\\phi, \\theta, \\psi)=\\begin{pmatrix}- \\cos\\theta  \\sin\\psi \\sin\\phi + \\cos\\psi \\cos\\phi & -\\cos\\theta \\sin\\psi \\cos\\phi - \\cos\\psi \\sin\\phi & \\sin\\theta \\sin\\psi \\\\ \\cos\\theta \\cos\\psi \\sin\\phi + \\sin\\psi \\cos\\phi & \\cos\\theta \\cos\\psi \\cos\\phi - \\sin\\psi \\sin\\phi & - \\sin\\theta \\cos\\psi \\\\ \\sin\\theta \\sin\\phi & \\sin\\theta \\cos\\phi & \\cos\\theta \\end{pmatrix}$$\n",
    "Les **colonnes** de cette matrice sont les vecteurs du repère mobile.\n",
    "\n",
    "Si les coordonnées d'un vecteur dans le repère mobile sont $(x,y,z)$, dans le repère fixe elles sont données par:\n",
    "$$\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = C(\\phi,\\theta, \\psi) \\begin{pmatrix} X \\\\ Y \\\\ Z \\end{pmatrix},$$\n",
    "où \n",
    "$$C = R^T = C_z(\\psi) C_x(\\theta) C_z(\\phi)$$\n",
    "est la matrice de changement de coordonnées (transposée de la rotation d'Euler)."
   ]
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   "source": [
    "## Vitesse angulaire en fonction des angles d'Euler\n",
    "\n",
    "Dans le repère du centre de masse $(x,y,z)$:\n",
    "\\begin{align*}\n",
    "\\Omega_x &= \\dot{\\psi} \\sin \\theta \\sin \\psi + \\dot{\\theta} \\cos \\psi \\\\\n",
    "\\Omega_y &= \\dot{\\psi} \\sin \\theta \\cos \\psi - \\dot{\\theta} \\sin \\psi \\\\\n",
    "\\Omega_z &= \\dot{\\psi} \\cos \\theta + \\dot{\\psi}\n",
    "\\end{align*}"
   ]
  },
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   "source": [
    "phi, theta, psi = symbols(\"phi, theta, psi\", real=True)\n",
    "\n",
    "def R_x(phi):\n",
    "    c = sy.cos(phi)\n",
    "    s = sy.sin(phi)\n",
    "    return sy.Matrix([[1,  0, 0],\n",
    "                  [0, c, -s],\n",
    "                  [0, s,  c]])\n",
    "def R_y(phi):\n",
    "    c = sy.cos(phi)\n",
    "    s = sy.sin(phi)\n",
    "    return sy.Matrix([[ c, 0, s],\n",
    "                  [ 0, 1, 0],\n",
    "                  [-s, 0, c]])\n",
    "\n",
    "def R_z(phi):\n",
    "    c = sy.cos(phi)\n",
    "    s = sy.sin(phi)\n",
    "    return sy.Matrix([[c, -s, 0],\n",
    "                  [s,  c, 0],\n",
    "                  [0,  0, 1]])"
   ]
  },
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    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\phi_{t} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\theta \\right)} + \\theta_{t} \\cos{\\left(\\psi \\right)}\\\\\\phi_{t} \\sin{\\left(\\theta \\right)} \\cos{\\left(\\psi \\right)} - \\theta_{t} \\sin{\\left(\\psi \\right)}\\\\\\phi_{t} \\cos{\\left(\\theta \\right)} + \\psi_{t}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡φₜ⋅sin(ψ)⋅sin(θ) + θₜ⋅cos(ψ)⎤\n",
       "⎢                            ⎥\n",
       "⎢φₜ⋅sin(θ)⋅cos(ψ) - θₜ⋅sin(ψ)⎥\n",
       "⎢                            ⎥\n",
       "⎣       φₜ⋅cos(θ) + ψₜ       ⎦"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# d phi/d t est dirigé selon ez(mobile)\n",
    "# d theta/d t est dirigé selon la ligne nodale eN\n",
    "# d psi/d t est dirigé selon eZ (fixe)\n",
    "psi_t, theta_t, phi_t = sy.symbols(\"psi_t, theta_t, phi_t\", real=True)\n",
    "#\n",
    "ex = sy.Matrix([1,0,0])\n",
    "ey = sy.Matrix([0,1,0])\n",
    "ez = sy.Matrix([0,0,1])\n",
    "\n",
    "eZ = R_z(-psi)*R_x(-theta)*ez\n",
    "eN = R_z(-psi)*ex\n",
    "Omega = psi_t*ez + theta_t*eN + phi_t*eZ\n",
    "Omega"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc2517e3-05ca-471e-a77f-bc76489d1937",
   "metadata": {},
   "source": [
    "### Équation d'Euler de changement de repère\n",
    "\n",
    "La variation d'un vector quelconque $\\bm A$ se décompose dans une partie de variation intrinsèque (dans le repère mobile) qu'on note avec une prime, et une autre partie due à la rotation de vitesse angulaire $\\bm \\Omega$:\n",
    "$$\n",
    "\\frac{\\D \\bm A}{\\D t} = \\frac{\\D' \\bm A}{\\D t} + \\Omega \\times \\bm A\n",
    "$$\n",
    "On peut aussi noter explicitement les repères:\n",
    "$$\n",
    "\\left. \\frac{\\D \\bm A}{\\D t} \\right|_{(X,Y,Z)} = \\left. \\frac{\\D \\bm A}{\\D t} \\right|_{(x,y,x)} + \\left. \\Omega \\times \\bm A \\right|_{(x,y,z)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2883f78f-cb62-4e6d-a15c-addc68650fd1",
   "metadata": {},
   "source": [
    "## Moment cinétique\n",
    "\n",
    "Pour un ensemble de points matériels le moment cinétique (ou moment angulaire) $\\bm L$:\n",
    "$$\n",
    "\\bm L = \\sum_n m_n \\bm r_n \\times \\bm v_n.\n",
    "$$\n",
    "Un changement du centre de coordonnées par une translation $\\bm d$, $\\bm r = \\bm r' + \\bm d$ transforme le moment \n",
    "$$\n",
    "\\bm L = \\sum_n (\\bm r'_n \\times \\bm p_n + \\bm d \\times \\bm p_n) = \\bm L' + \\bm d \\times \\bm P.\n",
    "$$\n",
    "Si deux systèmes ont une vitesse relative $\\bm V$: $\\bm v_n = \\bm v'_n + \\bm V$, le moment cinétique change selon la loi:\n",
    "$$\n",
    "\\bm L = \\bm L' + M \\bm R \\times V.\n",
    "$$\n",
    "\n",
    "### Solide\n",
    "\n",
    "La généralisation à un corps solide consiste à remplacer la vitesse des particules par la vitesse angulaire du solide $\\bm \\Omega \\times \\bm r$:\n",
    "$$\n",
    "L_i = I_{ij} \\Omega_j,\n",
    "$$\n",
    "(toujours on somme sur les indices répétés).\n",
    "Notez:\n",
    "$$\n",
    "\\bm r \\times (\\bm \\Omega \\times \\bm r) = \\bm \\Omega r^2 - \\bm r (\\Omega \\cdot \\bm r),\n",
    "$$\n",
    "ce qui s'écrit comme le produit du tenseur d'inertie par le vecteur de vitesse angulaire. \n",
    "\n",
    "Il convient de faire un changement de coordonnées pour exprimer le tenseur d'inertie selon ses axes principales:\n",
    "$$\n",
    "L_x = I_x \\Omega_x,\\; L_y = I_y \\Omega_y, \\; L_z = I_z \\Omega_z.\n",
    "$$\n",
    "\n",
    "### Variation de $\\bm L$: Équations d'Euler\n",
    "\n",
    "La variation \"instantanée\" du moment $\\bm L$ est donnée par la formule générale, applicable à tout vecteur:\n",
    "$$\n",
    "\\left. \\frac{\\D \\bm L}{\\D t} \\right|_{(X,Y,Z)} = \\left. \\frac{\\D \\bm L}{\\D t} \\right|_{(x,y,x)} + \\left. \\Omega \\times \\bm L \\right|_{(x,y,z)},\n",
    "$$\n",
    "où $(X,Y,Z)$ est le repère fixe, et $(x,y,z)$ le repére fixé au solide, mobile par rapport à $(X,Y,Z)$.\n",
    "\n",
    "Dans le cas où le tenseur $I$ est diagonal, on a\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\dot{L}_x &= I_x \\frac{\\D \\Omega_x}{\\D t} + (I_z-I_y)\\Omega_y \\Omega_z, \\\\\n",
    "\\dot{L}_y &= I_y \\frac{\\D \\Omega_y}{\\D t} + (I_x-I_z)\\Omega_x \\Omega_z, \\\\\n",
    "\\dot{L}_z &= I_z \\frac{\\D \\Omega_z}{\\D t} + (I_y-I_x)\\Omega_x \\Omega_y. \\\\\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "3668c8a0-7242-4b2d-8cce-48952ee92154",
   "metadata": {},
   "outputs": [],
   "source": [
    "ox, oy, oz = sy.symbols(\"Omega_x, Omega_y, Omega_z\")\n",
    "oxt, oyt, ozt = sy.symbols(\"\\dot\\Omega_x, \\dot\\Omega_y, \\dot\\Omega_z\")\n",
    "Ix, Iy, Iz, Ig = sy.symbols(\"I_x, I_y, I_z, I^G\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "446cf781-0bdc-4d1a-bce9-1f709df95eb5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pxVv18JZ8q2sPU7NSVWf87M01lc9ychRIWOyJ3K7cwXqsIhl3dcGUHNaYijHMeFof6y4kuIc14CqACzyyi3rgk63Uyg2ZeNMPsQ2scL8objqVuzSoU4mHDHvyZRrHW8YPXatSVB/vvR9osIpUqODCW5jyQLxpFmruMX3tLK7FqGS9BhD8PMI9hceSdR9r4NK3JTHLP2vahbfw3IxlvMZMgcjGr/vJMsVgz9sRSCEgu5r92khlt/CYeIm5NUD8FJ3iv+ftCGxFYPEp/f/uu/zncpqhWgAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\Omega_{y} \\Omega_{z} \\left(- I_{y} + I_{z}\\right)\\\\\\Omega_{x} \\Omega_{z} \\left(I_{x} - I_{z}\\right)\\\\\\Omega_{x} \\Omega_{y} \\left(- I_{x} + I_{y}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡Ω_y⋅Ω_z⋅(-I_y + I_z)⎤\n",
       "⎢                    ⎥\n",
       "⎢ Ωₓ⋅Ω_z⋅(Iₓ - I_z)  ⎥\n",
       "⎢                    ⎥\n",
       "⎣ Ωₓ⋅Ω_y⋅(-Iₓ + I_y) ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "O = sy.Matrix([[ox],[oy],[oz]])\n",
    "Ig = sy.Matrix([[Ix,0,0],[0,Iy,0],[0,0,Iz]])\n",
    "L = Ig*O\n",
    "oL = sy.simplify( O.cross(L) )\n",
    "oL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "b7b8c983-d7bb-4419-9af7-e2318df3b832",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left( I_{x} \\dot\\Omega_{x} + \\Omega_{y} \\Omega_{z} \\left(- I_{y} + I_{z}\\right), \\  I_{y} \\dot\\Omega_{y} + \\Omega_{x} \\Omega_{z} \\left(I_{x} - I_{z}\\right), \\  I_{z} \\dot\\Omega_{z} + \\Omega_{x} \\Omega_{y} \\left(- I_{x} + I_{y}\\right)\\right)$"
      ],
      "text/plain": [
       "(Iₓ⋅\\dot\\Omegaₓ + Ω_y⋅Ω_z⋅(-I_y + I_z), I_y⋅\\dot\\Omega_y + Ωₓ⋅Ω_z⋅(Iₓ - I_z), \n",
       "I_z⋅\\dot\\Omega_z + Ωₓ⋅Ω_y⋅(-Iₓ + I_y))"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ix*oxt + oL[0], Iy*oyt + oL[1], Iz*ozt + oL[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "900d0f60-cdad-4006-b352-272e6482c5cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\left(- I_{y} + I_{z}\\right) \\left(\\phi_{t} \\cos{\\left(\\theta \\right)} + \\psi_{t}\\right) \\left(\\phi_{t} \\sin{\\left(\\theta \\right)} \\cos{\\left(\\psi \\right)} - \\theta_{t} \\sin{\\left(\\psi \\right)}\\right)\\\\\\left(I_{x} - I_{z}\\right) \\left(\\phi_{t} \\cos{\\left(\\theta \\right)} + \\psi_{t}\\right) \\left(\\phi_{t} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\theta \\right)} + \\theta_{t} \\cos{\\left(\\psi \\right)}\\right)\\\\\\left(- I_{x} + I_{y}\\right) \\left(\\phi_{t} \\sin{\\left(\\psi \\right)} \\sin{\\left(\\theta \\right)} + \\theta_{t} \\cos{\\left(\\psi \\right)}\\right) \\left(\\phi_{t} \\sin{\\left(\\theta \\right)} \\cos{\\left(\\psi \\right)} - \\theta_{t} \\sin{\\left(\\psi \\right)}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡      (-I_y + I_z)⋅(φₜ⋅cos(θ) + ψₜ)⋅(φₜ⋅sin(θ)⋅cos(ψ) - θₜ⋅sin(ψ))       ⎤\n",
       "⎢                                                                         ⎥\n",
       "⎢       (Iₓ - I_z)⋅(φₜ⋅cos(θ) + ψₜ)⋅(φₜ⋅sin(ψ)⋅sin(θ) + θₜ⋅cos(ψ))        ⎥\n",
       "⎢                                                                         ⎥\n",
       "⎣(-Iₓ + I_y)⋅(φₜ⋅sin(ψ)⋅sin(θ) + θₜ⋅cos(ψ))⋅(φₜ⋅sin(θ)⋅cos(ψ) - θₜ⋅sin(ψ))⎦"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LO = Ig*Omega\n",
    "oLO = sy.simplify( Omega.cross(LO) )\n",
    "oLO"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82bb3611-b59f-4c8c-930c-7dd999c2702b",
   "metadata": {},
   "source": [
    "# Brouillon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "36a44608-5e32-4a61-9d07-b0d3c1533883",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\phi \\right)} \\sin{\\left(\\psi \\right)} \\cos{\\left(\\theta \\right)} + \\cos{\\left(\\phi \\right)} \\cos{\\left(\\psi \\right)}\\\\\\sin{\\left(\\phi \\right)} \\cos{\\left(\\psi \\right)} \\cos{\\left(\\theta \\right)} + \\sin{\\left(\\psi \\right)} \\cos{\\left(\\phi \\right)}\\\\\\sin{\\left(\\phi \\right)} \\sin{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-sin(φ)⋅sin(ψ)⋅cos(θ) + cos(φ)⋅cos(ψ)⎤\n",
       "⎢                                     ⎥\n",
       "⎢sin(φ)⋅cos(ψ)⋅cos(θ) + sin(ψ)⋅cos(φ) ⎥\n",
       "⎢                                     ⎥\n",
       "⎣            sin(φ)⋅sin(θ)            ⎦"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_z(psi)*R_x(theta)*R_z(phi)*ex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "e2c93c84-0d06-4c73-88af-7a72d5305be9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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fct5TmJ1C+3lskVKohuiof2gjnCvfq97HuLD2WQrBQJ7jxkHwCA0H19SSR6zm93gnik/Z9kRl76LyIar2OUvirL8IifeztfiSh7qpEwSdkG0bnNpaw+xmph4rzvqNjxPwLdPxEsq7KH8cLeKnsPRxck5xR5+Q706YxCNh5sx7kk9E6fDNCWgwbpSOI3fOXHHyOam+1zYQ5aGbNJbrWA5b0j97jacWsMNdDn+tsZUdV+qv1jmAH8ZfWPolwdv9UtH4eCPLJIvZCsThY+38m0+wMM53cTV6p0it567R81obIDgLMQP6TVvqAFb2RQQn/c7jzDVIZ0DuWHKDOC7XFGbDqZA+4EAYO+tgu8qQP15GCfmKTOKXjmTf+vZZkhk44Vh5Jm5XOqXHGa0MYzcn9ClSu3+2HE+Gq0ns6kvHt+/bNWOrNK5a5wDcCJPTWOzBhtzxVsJsev7dIreK/J/fsdDyqofqUEe6QmY7zPAeqREO7J8Uf1a9waFCZldG6jDnfoYW1tllC+WCw/L7zBzHyzvooSy6k6K6rWLGXpZMmIXeseNtHc9UuSrJDTaqFfFTYAOxk2juJipNWhnsywmzxKX9s8t4EhDDtSd2+CzhJ3/t2CqNq6Nw8BEiIuGkzkTVHW9RukVLmK3Mf1sE531JwUHbQr5rV0BwDFzqsva4pdDh7kBNNALJ8UGOow7rvoq7G3qy1V2Sj+o/0b4NplGW220Vs9nKQWgywC287AfnboVG4RT+UfH1u37w02fJMaN8vmhCPgfJ4EYxrSsPXAjrpkv7Z5fxJNvBtDd2uCvhJx9ZM7ay46p1Drx99wz4X6VxlQumeCl2UMbnl3zJoPylnTeNs4AfO0rS2I+dpaURxgL4pcKA57HClLxSYnxpHz6WKls5wD8pjPNjHVxyo7skLWLGXrBxA9lOalwN4RxuFLqTrMJxv5Adyxz8cflacdbDWXIZTwZYirP+eK44f6LAQcdCGZ4iMmxz+yfWsed42hs7PJTwk792bE2Nq9Y5YKLn/JzGGSFj7rni7vhSPCVTmAd1LvoZNBnOjIeZULxe91T7PLrF4j4HGevhNjPi4MKx4lxsnRzwpL/x6ThVyrt06aBTT8Tr5o5v8nLT56OL5RJsZHBC+lvlmTPQ7lCUR3megzZHMCig9GYxY6jsM8454xO3dbrUUxpUGYjqF/EPCo92VJfZyJI1b6dJ9ekfDgrElrkGdqsMY4fZDuMDYZ8+df2lsNg/rkbiR/Xg6uLjyUxR+7thx4Yp/FEZjtuzx5b0T46rljmIbAMHPod7e8VZ9UzMTFYYv48u6rzV6K7iyTl5CiY2SmU4uF8oLK3zuiq+LCee5Akh1rtXXLYxeIqYsU3lcEY40tTSEEVORGXhajF+1V/lvE8MunCC7J/k1nN0NeMppngOfsqr3Fljy3O2eFzFNm4dn8vBlB1zMatccN63U0qvLJ9ZWm75w6Ay0JKzaCsQhczobOYXJTcVnYMZg5mBzsVtANfiZzZsM2LTeaRwDrfXNp7i/pmDn/Lnjq214yq2cev4XA6m7Dgb84Ny3jprMZtmdsmMMSdhvTtXgHSvA12TM/SSnq3zvH1TmDGDd5zPdqQ18EsHS2VN81fqH2/7FLdXNZ5iPmbiP2tsSSfHZvPHlfFwBgdW5SRcivlBOW/PGmviqUf+jNR4Pd7SUiE6pmbxqXp7pE1hxqa5uM3+I+E3m7cIp7idy+tR+ZzCD+dzOaDsEXmYwwHYcrII84Na8zbm/JmOx8nO/XOIU6F6PysyuDFmulsN12KOcR0Rf2x/7fhabo/O51r81h9H5mEpB+diVvmw5v0gnbcNlh52BjoDnYEjMRA774e4bHKkvuq2dgY6A52BJAPdeSdp6Ymdgc5AZ6BtBrrzbrt/unWdgc5AZyDJQHfeSVp64iUY8Dd5zmpqSZ2zGti48Br719TdGFZXvwMD3XnvQHpv0j0nzxM7/IHlXOEZYOoeTrzdSzAb1sNiNwA9rMdAd971uLwaTczwtPHBjMFLn2oBlF7+Fs+fgrJ/xVfez9p4H4R98cY1r317J8kmttXCONYjuycxx3VU/gTfUbHHuHq8HgPdedfj8po0fSkwvPiIt5xVFTkg9PIWP/4OnBTl8XIsPr1GmTfaeENdEKXzfD4vA0NX8+LtLGJOgEi+L+do2BO4elIlBrrzrkTkNamRg2B2+7nCLf5Byrsg+EdZUtQm+TcK3d/mFfKXfa4ExssN6HBlKd+4FDFnbC/9K/FI2DPwevJaBrrzXsvgldb3TnMLdLxfO/k+E++gWc9OOWWuBoJ4Heg6wuw7izkAiiLCxIucSq8hhr+jYI+Q9WhNBrrzrslm11VkQE6Jdd/SO41fk69y4e2GkXN+klCOrh8S6c0kzcCcsvVZzEGqgNKax56xuydXYuBxJT1dzYEYkGNgCYLZHUsSNnPlS+pfa2OGyxozYfj6ta/D5+BIf6eN2THOGPlW23uVmXpXDB+5CI6Ziiaqix3oG6+Ff+PLpJw+utCZnMn7ejdeN/aC1wR7ww3TqAzv6EZYc+ZjC8FexbO8uRrpnyzmuLjX/aPSWC75TvtkYwPfe43tJh2Zhf2+aP+9Rga6877GXi1gkiPASfIEB04liPbd+rZCnCRO/EPIVET7XMaTjnNjFszs0DlrhTi1DwqnXtaFI86td9vTFcFZqiyCbiTlvLGFk1BWZBMnG7C8wj4KKqQOjplZPt/opA1OWDjN0I7ifBDA4VRY5E11c1LC7OpIt+NEofXBjeJ8ps3ZrnBgl29oEnvOoJ5+HQzcXgeMjuIMBnAmfLcSZxTL2Kl+ijOjOM7therHs1Zbny06UtWjzZxeZp0IT2XgWN2mfTdj1n5wqq7U/Q+6cHAlwSnjpIO92jfsd74iZVInHq4C7FHFubx5lSEoYb4BJyUVmuOGQ3cCUxqYiafuAczBrqpdrpWBx9cKrONKMyCHwFIIBz7PceMYeCwPxzW15BEr/D3eieLmFKOkQZQZuznMQYZ2mP2GZRoyZRP6Xmp7w35CcG7ZNn199A6wKR1H7py54uRzAnivbSDKY1ZOGktKLF8s4S2LWfq42sBZf0EjXphlx0tH8EWZsRSxjwv3/etjoM+8r69P5yDi+W0cGk6LWd0/5TDGM28lX0bUtjlg+/ixNWw3I3Nr2lnH6BXYrLz02J2VwUnmBAeP1OYNJ/1O+KfaTl11TGF3Bvef62WgO+/r7dskMjkK56wUMpP8StsjFeSS/SfFzUkl61ZIZOZqjjql7uMoEef2m+zKOTd0oTMn5vSSf3jxlaxMyS6WXZbyVsKMznBvQW1gg90wvfH7zLpTJ9Yp7B5eD66Vge68r7Vn87hw0HZz0JWSk2BmyzIC67pbCo7SOcG4EbV/F+8TVxrLJcwu4yUEsmIh35xvnO7iXi8nhCQu2tBGPu0PbuCiQHm2XMG69FLekpjRL6FtnLAJ7cU3bHm6h5OXW+KxQj4sYh+V7btXyEB33lfYqTMgcVMwdhpUYT92HJZGGAtOY6ngrHisMCU4KGeTt43lnOeK36UK+zSWMdBZEr4vyD80xycs9Fvd54rzpxccdCyU4akP42Uub7GOEuZXKuhuVPoK4WPFapMTxieFcX6sdw72uHyPXxkDj68MT4czzQDOkJk2yyRW+qkiPFXB8gAO7LU2N1vVPk4EJ8OMmXTyee8I6dxIJN0cDM4NB4TDTMl/KpF6KaENbEAfyxypx+PG9ZipWtvjPLcvfWDihiC6WX6wZYm35FFIITcmcYbxM9XYwb457iJv6MlIFrNv93uFZhePJXLC4CT2hvyMTpInsRfq9qwrYKB/w/IKOvFIEOSQcJ44rJJjmoSk+jhXnsMurWdP6rlEgTmYPZ4XCnM3Z4OpvuwhsAeje6QKA+p7riCZeDy6raKxK+kMzGeAmWVxtjxTFWvh6DqCzMHMFY3N8qcwHQn7FJaev5CB7rwXEterLWPAzyxZg2bmvEh8XXRMzlIXNVC50kzMYb271PzRsJew9Lx1DHTnvY6/XnsZA6yJs867VKhbY/a+tP0l9aYwl55Fj9s7IvbY/h6vxEB33pWI7GrmM6DZIzf/+JjC2Z8z83Wom31EcL4llys5hVn5g3+Bpiw7KvYUlp62noF+w3I9h11DZ6Az0Bm4CAM6gYcblvGjgvxFemwA77zIPfY1Ltv3OwOdgc5AZ6ASA/K9PJmVvTeE8+byM7d+eKhL00qcdTWdgc5AZ6AFBopPU/0/XSxYSOP2dmoAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\phi \\right)} \\cos{\\left(\\psi \\right)} - \\sin{\\left(\\psi \\right)} \\cos{\\left(\\phi \\right)} \\cos{\\left(\\theta \\right)}\\\\- \\sin{\\left(\\phi \\right)} \\sin{\\left(\\psi \\right)} + \\cos{\\left(\\phi \\right)} \\cos{\\left(\\psi \\right)} \\cos{\\left(\\theta \\right)}\\\\\\sin{\\left(\\theta \\right)} \\cos{\\left(\\phi \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-sin(φ)⋅cos(ψ) - sin(ψ)⋅cos(φ)⋅cos(θ)⎤\n",
       "⎢                                     ⎥\n",
       "⎢-sin(φ)⋅sin(ψ) + cos(φ)⋅cos(ψ)⋅cos(θ)⎥\n",
       "⎢                                     ⎥\n",
       "⎣            sin(θ)⋅cos(φ)            ⎦"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_z(psi)*R_x(theta)*R_z(phi)*ey"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "ffb20132-b7e4-4e18-8e40-3b3c651abe08",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAABLCAYAAACFg+7aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAANfklEQVR4Ae2d25EdNRCGd108U8au4p0lA8ARGDLwJQKbDEz5yX6jIANDBDZkgInA2BlAAFQBW0TA/8nqQaMjaS5HczlzpqvmSKNLq7vVarU00u7ls2fPPru4uHirJwU/P3/+/H4qY0/bJZCSgPTld6VfZfIubwQZ3yv+dfS8CPJPKirGk0wPYaIGjiHt1So7hu4xdTL0fqf0WI9+trIfWEThCzX6R/B+slHx8UTEw8ux/FwJ1z09DMaTgCN4r8Kr2v8hFpTSSLrHT2jheJ8c1DiM/aPn8RSNCS+M3VbYjKqudlQWBT0Apb8mUWEVWoVncd5FwxM93/kHd8qB3qvyanjjcHaFEwFMdTf1fB4Tc+y7hAbepwq/GYjrdq68cDlXw+POFeubvijv4uEXEXrt5fOt4j+FhFfmNUTdxGdXODHFSPpIIfN8bcB/GOR3ig6U9K8OQsAJ7qNgSd7VtqNfoZvyFF6LGSxuY+U8c1V4zQlqdoWDEM9sjqZj0h8I94EP0YHwS+W76SRXzuMEN8p5FCzBu9pEqXAbUoMGq9tATV4bpEFkEYUL2q8WlaDw3cYsEu6o7rsehID7QY9ysxfpwftTEfWHyjUDS3EbPLcSBE/Ga7hKTbQ7PEmMMJqwGphsY+qh0j/Xw2jCbyB8rXe3x+fr/OjTXylkJLpVjcI7et6oTNdK8SuVawSqeAOqCx04/p/qeav3pBVUOlbga4WUiwHctJGsS2HVWyvvyDL2a7+AZkFqkHby+r7q8N+qCieB07GsgOiYBvTu/DWFMIfitTaa9Y6FIZ1NQ0bcZ4o7BVNIJ6IkbEKnhKNsBwgw57+5hYTqI3gU2/wY6KXNEPBrbuq5DhMVpxwDKQnUUcYaebcVdjwYkSuQkmmR1/fVxv3WnlLp9C+88EOKYkX4O8wM4jD/peo3WxqK23SX7Wxfnw4/wKv6CPaNL8NACMswRTYdobIoOau4WNmoTj0scw5Wx7sn9KEPGXQ/2aM0ZhF4TSlcF68e5fCgtoVjmoRY9tnoSJbhWKau6TCk/LfwJYijUCXAMqYUBd/lna+IgrEdYPCp8uIp8pVlRiEdk6UBfvWsjXdYYMA17gsJohM+sPahLMgyKPJqhcaEtS0cNLC/hoJhDRhFv4vB2MIpeR5Q204JFSJgBB0rWEOI74h4irX8nEJbPuHaeLcBwsAPwRY/OVn04TXE1zteVeHUYW7KUfiNHqzHpSjBf3usOCNtSsC6mIBT7TCdvhMdpoDQalOtlcffyXUCuGkjCcK7Zt7NwhvtLCB+MFlYYhAWeQ3KDY5WVTi1jlKZk+qIEVN0ID4ZPs6UwDTgOr3QSKgw+ISh/8Z7o5AJHIz6lL9jRVfHe0qhlIalh5d41Wp8EHbxGpYdFK+tcDSOcxpbGt6bzvUUxmVIhtGxwChmCyUHbMewoDGlxAJfU1ihGyQKYxpDXEyXsaUI84mvkXcGu5O1+CPEzbmruONd8RT04TVVrzOt6qJBrcEEFo0pVIGD2/pluwDnHSvAJqSzdnpHCR7pQQlIJ5+VE+k4tKQzJQN0Jpu0ufN5L1WGeklQPRznu8p0tChk64V9N6Bry4UyWECjhfcY1so78oVnZMn+4leKlyw1fHXxSplRcBkcwGTEdxEyqpG5Kol+HP77CouWSPkoNvttzfZLiUaVo7N+UZjaEC5VnS1PtPXivYugKXgVTmaQFwovb3QRcGL5TBclK2TsMIJL06eVsxB/B9xrhr68d/EwKa+bUjiNIKZzLBcWqQSclyv5ME1djwucudVrU3bJiKevD+9ZMufgdVMK5yWJj9e179d1HCnsFHD1sZphnaXifXgv0TY5r5tTOI1SLBcf4G1BcCBg5fX68uFxgOskfFvR2cn7gTB8wly8bmrRkBPmnr6sBKTMm100LCvZvfVOCWxuSu3keC+wqAR2hVtU/OfX+NkqnPyKqzHdPbbemLamqFOD/mNwnKXCSWCsYPnaMAbY68qugMcgnKvOkXyHZI6WweoUjtGjZ9GL0nSMHr4/8rQUU+/uC4XC1qmYsDfWGBe9nBJhw7vX5zx4UNnkwFL6aBmsTuHEJ1PdTT2cWKgKEhR4OQSQPZqjPA4rcoCAMt/qOTgQoDz28difA9/qwdNZ5DvDBAcvkjBWBqtTODHC6FnkorTadt9LFbrPWArZSMXitqyc7wF25df+fdWT6ujs+vpiZV0onhlMXV9kBstgdQoHt76jHeOVf7IXpb1SMYWklAir2wKVRymrXI5uIZ7mJct3obnOAw5jZLBKhSsIYXSWhIMPU/pE9ZR8lWtOkShuU+atTMPge5DJW0VyD75zdE5yQfyDXGtTpUsATE+MHqYr69A1XJRGIWPfzh0UVXpOUVFO7kpkT5J4pcVqwq8BF7sb5z0oYxd4OHfHlb5Q+bNyM6SZEPoaPGEZ3y6LH9qb7IJ42OasCmeCVYgQGtC7O42hkI5d4qK0rTjjjjHfLadwKAiDJwnih6mYS9+PFHcKppDyKJO7vqiQNliYtE7iKp0Dn+5CuEIGJivmpNyUVwIGTc5/cwsJ4WWwTXJBPCZs7ikV5ld1UdoL5KEP6YC+l4WpwqWcA//O4yJAkVCsxprp3ay6WTzKpI64Y21tW6av3FTlAGgPOlsgmlD0Nz4RRQ7L4CY0g09lWZWPvSDum3gf9LJwahCif9VjwmohybwcHPUWni1dFoZtLF9SJl5mdGrrKJTSUT6zduSjsNbxir4HleMGGS+4G1y7RCHGXDC/pXrXIIrAWVifhoKxBWTAdYPYTXhlmVGYlUFUzr2icB/6DAsPyqlxCK61LwYeHHTMOL4No5g7koscclS7pjBDLwuLdHfLLNWZ5JnlK20tWJkcDvCglEBVufk+vVBIPyCDWMFo04GXkfmXlmxhTqEtn/Bje7mhyL/+xULLqx6KcCdghWu8KP0uYpgprXRZmOJ0FJYnBeb3lS7eWBnw5ABLdIzcoK+En+m0uY/r24otLj5uTiFLMjCe/rQICjcnMFrNQXftikEYYYrBT5kS6FzXcWEjav86fCeuNEY9IzdetZIdAmVMacJ0cIAXJU7yRRt6yKdcazGgd+qzuADw8Y6RW5Jvh/n/n3DQ0G7ov/HeKOT/VZpYVgZNiSAyt8LRNI45oyIE3hsmfUZchmSYGwt0bu6iNArv2vO0MdV3XRaGDqY58OaAOwZ8qYgHGfit3l3F2ZhFqUKgzPdKN7n0lVuIg3iJb/JR6KkviNOOA3y4OeFajWHRHkuQ1u5tRfDjmDoQOv6dswp6RxiP9GCZSCd/iovStAENtMMU2Nqi0HsOGP1Z39Pz9InKgJutCfODXpAHUoVYDxQXNwP5ANDBuylbUW6uRv7npbKQYxJoQw9K7/pAYe0L4q12z+pOgwRLhx+snlsS6fkiXCjFqi9HGyt9+VY5BjQW2a2irX4u7CsDlcPCM8gub+SQbTSdaSprkQby7PbJBtZZqnhfvrHYZlX70DpYBmelcBphTOeMYKzTaPD1wZNbuY3GPUXFAXxPfkH8rBTOd+axl4VBgz9Wy1J6siYP+vBd2jOMCRwlg7NTOI12HPDiRelYsuG76j/x9ZPbIWHZNcX78K0yra8iOfqPkcFZLRpyAtzTp5WAFPRsFw3TSnbH3imBs5tSOyWyF5hUArvCTSreHXksgV3hYolUepffcjUG1dh6Y9paos6ucBNI3a/i4m+jfVtif4+V8CZhV7jK3Spl4aRJ8cIxCqWHb5c8LcXUu9vpV9j64F+ZzMXQ7QpXUfRSkptCx6mO7LEm5W3uovUQEe4KN0Ra3WX5ZskOfBKkbORfKHSfxBSyCc0U2rJyvjJ4XHn/volgV7i63Zi9cOyVCt8spURXMRleKcGH1dwM7ApXqSulGPhupc9dT8lXueY0RqBMtzJkgO9BJu8kk+c+gDmpkHwHYkGYqgzmunTMMfFGmaxxQk8XChn7du6gqdJzigo+8J7EqRTR2QmbUTh1KtPSkpeOUZ6c/2YrzlghzXfLKRwHRjmjthnYjMKpRzhGzZQVnlY1/8csHmVyl475UwcoBNObu6ytd6unpKwykQfQ1t8udvjz0Cexgg1zsXocmc8pHPgO/LsQwanFN6Fw6jA6G2vROl6jdJTPKaDi5NN5b/S0QHk1Lh2jqKGChm3Q9tD/ykx9FBHeNgNbWTSYFSgdILQyOaWgU1EMgEstKC918AlH/1drPxiEwv07dkIDWwyU/LOSEhuekwq3onA2JS156Zjpr2SNuK4XAguIYy5ah7hOJr4JhZMVwWrRobbqa3WA8ue4dIzSmxVt2ve0Ne9EoEcB1itetZIdAmVsMIXpJxvfhMJ56S996RiFv5PRBPxIZ/2kbIRM0zUuWmeaW2/yJhYNiFcdyQp1yUvHL0VG7sLxJBet16tWecr2Ow152QzOkcKf5UXrLkFJLvudhi4hjcxnqqx1fRD/Dnybgi35cIt3jEYyWxxnd9F6iOB3hRsirX5l+1w47sLEJ7JalrKrrVnzd4WrLG5ZObZozu6idV8xbmaV2pfhOcpJ6dg7a31m69uu6o6q1xf/0uVChePzTUwPH7qZInbYJdBLAtIXVupXucIoHKMx5y9sapc7J4Q9vaoEiivr/wDfREkULRvRaAAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\sin{\\left(\\psi \\right)} \\sin{\\left(\\theta \\right)}\\\\- \\sin{\\left(\\theta \\right)} \\cos{\\left(\\psi \\right)}\\\\\\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡sin(ψ)⋅sin(θ) ⎤\n",
       "⎢              ⎥\n",
       "⎢-sin(θ)⋅cos(ψ)⎥\n",
       "⎢              ⎥\n",
       "⎣    cos(θ)    ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_z(psi)*R_x(theta)*ez"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "e49d9a82-4ff9-4a6b-b7ba-259093fbcc35",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\phi_{t}^{2} \\left(- I_{y} + I_{z}\\right) \\sin{\\left(2 \\theta \\right)}}{2}\\\\\\phi_{t} \\theta_{t} \\left(I_{x} - I_{z}\\right) \\cos{\\left(\\theta \\right)}\\\\\\phi_{t} \\theta_{t} \\left(- I_{x} + I_{y}\\right) \\sin{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  2                      ⎤\n",
       "⎢φₜ ⋅(-I_y + I_z)⋅sin(2⋅θ)⎥\n",
       "⎢─────────────────────────⎥\n",
       "⎢            2            ⎥\n",
       "⎢                         ⎥\n",
       "⎢ φₜ⋅θₜ⋅(Iₓ - I_z)⋅cos(θ) ⎥\n",
       "⎢                         ⎥\n",
       "⎣φₜ⋅θₜ⋅(-Iₓ + I_y)⋅sin(θ) ⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.simplify( oLO.subs([(psi_t,0),(psi,0)]) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "50ffd1e7-4a26-47f6-89c9-9cb9a5e727e2",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}