obsidian_backup/补课/多体动力学/sympy_start.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "a, b, th, gama, x, t, y, z = sm.symbols(\"a, b, theta, gamma, x, t, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a, b, theta, gamma, x, t, y, z)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, b, th, gama, x, t, y, z\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sm.Function('f')\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(f{\\left(t \\right)} \\right)} - \\frac{\\tan{\\left(\\frac{a}{b} \\right)}}{\\log{\\left(\\gamma \\right)}}$"
      ],
      "text/plain": [
       "sin(f(t)) - tan(a/b)/log(gamma)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr3 = sm.sin(f(t)) - sm.tan(a/b)/sm.log(gama)\n",
    "expr3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{2 a \\left(\\tan^{2}{\\left(\\frac{a}{b} \\right)} + 1\\right) \\tan{\\left(\\frac{a}{b} \\right)}}{b^{3} \\log{\\left(\\gamma \\right)}} + \\frac{\\tan^{2}{\\left(\\frac{a}{b} \\right)} + 1}{b^{2} \\log{\\left(\\gamma \\right)}}$"
      ],
      "text/plain": [
       "2*a*(tan(a/b)**2 + 1)*tan(a/b)/(b**3*log(gamma)) + (tan(a/b)**2 + 1)/(b**2*log(gamma))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part1 = sm.diff(expr3, a)\n",
    "part2 = sm.diff(part1, b)\n",
    "part2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices & Linear Algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 2],\n",
       "[3, 4]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat1 = sm.Matrix([[1, 2],[3, 4]])\n",
    "mat1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear systems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "lin_expr_1 = a * x +b**2*y + sm.sin(gama) *z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\sin{\\left(f{\\left(t \\right)} \\right)} + z \\log{\\left(f{\\left(t \\right)} \\right)}$"
      ],
      "text/plain": [
       "x*sin(f(t)) + z*log(f(t))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_expr_2 = sm.sin(f(t)) *x + sm.log(f(t)) * z\n",
    "lin_expr_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a x + b^{2} y + z \\sin{\\left(\\gamma \\right)} = 0$"
      ],
      "text/plain": [
       "Eq(a*x + b**2*y + z*sin(gamma), 0)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.Eq(lin_expr_1, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\sin{\\left(f{\\left(t \\right)} \\right)} + z \\log{\\left(f{\\left(t \\right)} \\right)} = 0$"
      ],
      "text/plain": [
       "Eq(x*sin(f(t)) + z*log(f(t)), 0)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.Eq(lin_expr_2, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{x: -b**2*y*log(f(t))/(a*log(f(t)) - sin(gamma)*sin(f(t))),\n",
       "  z: b**2*y*sin(f(t))/(a*log(f(t)) - sin(gamma)*sin(f(t)))}]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = sm.solve([lin_expr_1, lin_expr_2], x, z, dict = True)\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: -b**2*y*log(f(t))/(a*log(f(t)) - sin(gamma)*sin(f(t))),\n",
       " z: b**2*y*sin(f(t))/(a*log(f(t)) - sin(gamma)*sin(f(t)))}"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res_dict = res[0]\n",
    "res_dict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x = - \\frac{b^{2} y \\log{\\left(f{\\left(t \\right)} \\right)}}{a \\log{\\left(f{\\left(t \\right)} \\right)} - \\sin{\\left(\\gamma \\right)} \\sin{\\left(f{\\left(t \\right)} \\right)}}$"
      ],
      "text/plain": [
       "Eq(x, -b**2*y*log(f(t))/(a*log(f(t)) - sin(gamma)*sin(f(t))))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.Eq(x, res_dict[x])"
   ]
  }
 ],
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