{ "cells": [ { "cell_type": "markdown", "id": "3a4603bd", "metadata": {}, "source": [ "# Le pendule\n", "\n", "On considère l'équation :\n", "$$ \\theta^{\\prime\\prime} = -\\sin{\\theta}$$\n", "\n", "Pour résoudre cette équation, on définit le tableau $y(t) = (\\theta(t), \\theta^\\prime(t))$. \n", "\n", "1. Écrire la fonction Python ``f(t, y)`` qui renvoie la dérivée de $y$\n", "\n", "2. Résoudre et tracer le resultat de l'équation différentielle pour les conditions initiales : $\\theta(0) = 2\\pi/4$ et $\\theta^\\prime(0) = 0$\n", "\n", "3. Vérifier que l'énergie totale est conservée" ] }, { "cell_type": "markdown", "id": "501f2f54", "metadata": { "tags": [ "sol" ] }, "source": [ "--------------------------------------------------------------------------------------\n", "\n", "## Solution" ] }, { "cell_type": "code", "execution_count": 1, "id": "59950058", "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "\n", "import numpy as np\n", "from numpy import pi\n", "\n", "from scipy.integrate import solve_ivp" ] }, { "cell_type": "code", "execution_count": 2, "id": "cd77e54c", "metadata": {}, "outputs": [], "source": [ "def f(t, y):\n", " theta, theta_prime = y\n", " return np.array([theta_prime, -np.sin(theta)])\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "8cb62538", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t_eval = np.linspace(0, 10)\n", "\n", "res = solve_ivp(f, [0, 10], [3*np.pi/4, 0], t_eval=t_eval)\n", "\n", "plt.plot(res.t, res.y[0])\n", "plt.xlabel('t')\n", "plt.ylabel(r'$\\theta$');" ] }, { "cell_type": "code", "execution_count": 4, "id": "23482812", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "energie_potentielle = -np.cos(res.y[0])\n", "energie_cinetique = res.y[1]**2/2\n", "\n", "plt.plot(res.t, energie_potentielle, label='Pot.')\n", "plt.plot(res.t, energie_cinetique, label='Cin.')\n", "plt.plot(res.t, energie_potentielle + energie_cinetique, label='tot.')\n", "plt.legend();" ] }, { "cell_type": "code", "execution_count": null, "id": "165952c3", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.8.8" }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 5 }