{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Problem 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "X1 = Symbol('X_1')\n",
    "X2 = Symbol('X_2')\n",
    "a = Symbol('a')\n",
    "phi = 1/(6*a)*(3*X1**2*X2 - X2**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi_1 = integrate(-simplify(phi.diff(X2)),X1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{X_{1}^{3}}{6 a} + \\frac{X_{1} X_{2}^{2}}{2 a}$$"
      ],
      "text/plain": [
       "    3        2\n",
       "  X₁    X₁⋅X₂ \n",
       "- ─── + ──────\n",
       "  6⋅a    2⋅a  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi_2 = integrate(simplify(phi.diff(X1)),X2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{X_{1} X_{2}^{2}}{2 a}$$"
      ],
      "text/plain": [
       "     2\n",
       "X₁⋅X₂ \n",
       "──────\n",
       " 2⋅a  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psi_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = Symbol('K')\n",
    "psi = psi_2 - X1**3/(6*a) + K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "chi = simplify(psi - 1/2*(X1**2 + X2**2)) #definition of chi ()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$K - \\frac{2 a^{2}}{3}$$"
      ],
      "text/plain": [
       "       2\n",
       "    2⋅a \n",
       "K - ────\n",
       "     3  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nsimplify(chi.subs(X1,a),rational=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ksol = solve(chi.subs(X1,a),K)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "chi = chi.subs(K,Ksol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{X_{1}^{3}}{6 a} - \\frac{X_{1}^{2}}{2} + \\frac{X_{1} X_{2}^{2}}{2 a} - \\frac{X_{2}^{2}}{2} + \\frac{2 a^{2}}{3}$$"
      ],
      "text/plain": [
       "    3     2        2     2      2\n",
       "  X₁    X₁    X₁⋅X₂    X₂    2⋅a \n",
       "- ─── - ─── + ────── - ─── + ────\n",
       "  6⋅a    2     2⋅a      2     3  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nsimplify(chi,rational=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate integral of chi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$2\\mu \\int\\limits_{x_1 = -2a}^a\\quad \\int\\limits_{x_2 = \\frac{-2a-x_1}{\\sqrt{3}}}^{\\frac{2a+x_1}{\\sqrt{3}}}\\ \\chi(x_1, x_2)\\ dx_2\\ dx_1  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "chi_1 = integrate(chi,(X2,-2*a/sqrt(3) - X1/sqrt(3), 2*a/sqrt(3) + X1/sqrt(3)  ))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{9 \\sqrt{3} a^{4} \\mu}{5}$$"
      ],
      "text/plain": [
       "      4  \n",
       "9⋅√3⋅a ⋅μ\n",
       "─────────\n",
       "    5    "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu=Symbol('mu')\n",
    "nsimplify(2*mu*integrate(chi_1,(X1,-2*a,a)),rational=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "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",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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