{ "nbformat": 4, "nbformat_minor": 0, "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": "3.8.5" }, "colab": { "name": "Интегрирование уравнений движения.ipynb", "provenance": [], "collapsed_sections": [] } }, "cells": [ { "cell_type": "markdown", "metadata": { "id": "ggaOOzeSKJCf" }, "source": [ "# Интегрирование уравнений движения" ] }, { "cell_type": "markdown", "metadata": { "id": "R85WF0AKKJCk" }, "source": [ "На предыдущих лекциях было показано, что уравнения, определяющие движение тела, могут быть представлены в виде (смотри первую лекцию):\n", "\n", "$$\n", "\\begin{cases}\n", "\\frac{d\\vec{r}}{dt} = \\vec{v}(t)\\\\\n", "\\frac{d\\vec{v}}{dt} = m^{-1}\\vec{R}(t, \\vec{r})\n", "\\end{cases}\n", "\\label{equations}\\tag{1}\n", "$$\n", "\n", "Начальные условия при этом:\n", "\n", "$$\n", "\\begin{cases}\n", "\\vec{r}(t_0) = \\vec{r}_0\\\\\n", "\\vec{v}(t_0) = \\vec{v}_0\n", "\\end{cases}\n", "\\label{init_cond}\\tag{2}\n", "$$\n", "\n", "Данная запись эквивалентна следующей:\n", "\n", "$$\n", "\\begin{cases}\n", "\\frac{d\\vec{y}}{dt} = \\vec{f}(t, \\vec{y})\\\\\n", "\\vec{y}(t_0) = \\vec{y}_0\n", "\\end{cases}\n", "\\label{normal_task}\\tag{3}\n", "$$\n", "\n", "где\n", "\n", "$$\n", "\\vec{y}(t) = \\begin{pmatrix}\n", "\\vec{r}(t)\\\\\n", "\\vec{v}(t)\\\\\n", "\\end{pmatrix} = \n", "\\begin{pmatrix}\n", "x(t)\\\\\n", "y(t)\\\\\n", "z(t)\\\\\n", "v_x(t)\\\\\n", "v_y(t)\\\\\n", "v_z(t)\\\\\n", "\\end{pmatrix}\n", "$$\n", "\n", "$$\n", "\\vec{f}(t, \\vec{y}) = \\begin{pmatrix}\n", "\\vec{v}(t)\\\\\n", "m^{-1}\\vec{R}(t, \\vec{x})\\\\\n", "\\end{pmatrix} = \\begin{pmatrix}\n", "v_x(t)\\\\\n", "v_y(t)\\\\\n", "v_z(t)\\\\\n", "m^{-1}R_x(t, \\vec{x})\\\\\n", "m^{-1}R_y(t, \\vec{x})\\\\\n", "m^{-1}R_z(t, \\vec{x})\\\\\n", "\\end{pmatrix}\n", "$$\n", "\n", "$$\n", "\\vec{y}_0 = \\begin{pmatrix}\n", "\\vec{r}_0\\\\\n", "\\vec{v}_0\\\\\n", "\\end{pmatrix} = \\begin{pmatrix}\n", "\\vec{x}_0\\\\\n", "\\vec{y}_0\\\\\n", "\\vec{z}_0\\\\\n", "\\vec{v_x}_0\\\\\n", "\\vec{v_y}_0\\\\\n", "\\vec{v_z}_0\\\\\n", "\\end{pmatrix}\n", "$$\n", "\n", "\n", "Для решения данной задачи был предложен \"Метод Эйлера\", позволяющий рекурсивно выразить решение в последующий момент вемени $t_{i + 1}$ через предыдущий $t_{i}$ с помощью данных формул: \n", "\n", "$$\n", "\\begin{cases}\n", " \\vec{y}_{i + 1} = \\vec{y}_i + h * \\vec{f}(t_i, \\vec{y}_i)\\\\\n", " \\vec{y}_0 = \\vec{y}_0\n", "\\end{cases}\n", "$$\n", "\n", "где h - шаг по времени, $t_0, t_1, ..., t_{N - 1}$ - разбиение времени, $t_i = t_0 + i * h$. Здесь для расчета последующего шага используется информация лишь о состоянии объекта на предыдущем шаге. А что, если учитывать больше информации?" ] }, { "cell_type": "markdown", "metadata": { "id": "VnCd2ku2KJCl" }, "source": [ "## Метод Симпсона" ] }, { "cell_type": "markdown", "metadata": { "id": "EBKtcJt4KJCm" }, "source": [ "Как и в предыдущих случаях, будем полагать, что дана система уравнений и начальное условие (3), а также некоторое разбиение времени $t_i = t_0 + h*i$, где h - шаг разбиеня.\n", "Метод сипсона выглядит следующим образом:\n", "\n", "\n", "$$\n", "\\vec{y}_{i + 1} = \\vec{y}_i + h * (\\frac{1}{6}\\vec{k_1} + \\frac{2}{3}\\vec{k_2} + \\frac{1}{6}\\vec{k_3})\\\\\n", "\\vec{k_1} = \\vec{f}(t_i, \\vec{y_i})\\\\\n", "\\vec{k_2} = \\vec{f}(t_i + \\frac{h}{2}, \\vec{y_i} + \\frac{1}{2}h\\vec{k_1})\\\\\n", "\\vec{k_3} = \\vec{f}(t_i + h, \\vec{y_i} + h * (-\\vec{k_1} + 2\\vec{k_2}))\\\\\n", "\\vec{y}_0 = \\vec{y}(0)\n", "$$\n", "\n", "Данный метод гораздо точнее, нежели разобранный ранее метод Эйлера. Реализуем его для уравнения $y' = 4 t^3$ при условии $y(0) = 0$. Решение данной задачи $y = t^4$" ] }, { "cell_type": "code", "metadata": { "id": "47uYP4daKJCn", "outputId": "e035c9f9-b9dd-49f0-95bf-b5626cadbf8d" }, "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "h = 0.1\n", "\n", "def f(t, y): # Функция правой части уравнения\n", " return 4 * t**3 # Здесь может быть другая функция\n", "\n", "\"\"\"\n", "Метод Эйлера\n", "\"\"\"\n", "t_eul = [0]\n", "y_eul = [0]\n", "for i in range(11):\n", " y_eul.append(y_eul[-1] + h * f(t_eul[-1], y_eul[-1]))\n", " t_eul.append(i * h)\n", " \n", "\"\"\"\n", "Метод Симпсона\n", "\"\"\"\n", "t_sim = [0]\n", "y_sim = [0]\n", "for i in range(11):\n", " k_1 = f(t_sim[-1], y_sim[-1])\n", " k_2 = f(t_sim[-1] + h / 2, y_sim[-1] + h / 2 * k_1)\n", " k_3 = f(t_sim[-1] + h, y_sim[-1] + h * (2 * k_2 - k_1))\n", " y_sim.append(y_sim[-1] + h * (k_1 / 6 + 2 / 3 * k_2 + 1 / 6 * k_3))\n", " t_sim.append(i * h)\n", " \n", "plt.plot(np.linspace(0, 1, 100),np.linspace(0, 1, 100)**4)\n", "plt.scatter(t_eul, y_eul, c='r')\n", "plt.scatter(t_sim, y_sim, c='b')\n", "plt.show()\n", " " ], "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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\n", 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