{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9d7ffe9f-441f-4119-b5e4-0e872922a78d",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "406dc83e-b114-46be-baf8-073a72b47a36",
   "metadata": {},
   "outputs": [],
   "source": [
    "t, a = symbols('t a')\n",
    "theta = Function('theta')(t)\n",
    "r = Function('r')(t)\n",
    "U_c = Function('U_c')(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "df8fe246-6986-4ffd-9c39-6bd259176e42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{a^{2} \\operatorname{U_{c}}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)}}{r{\\left(t \\right)}}$"
      ],
      "text/plain": [
       "-a**2*U_c(t)*cos(theta(t))/r(t)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi = - (a**2*U_c)/r * cos(theta)\n",
    "phi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "a6e55454-52c6-493b-a753-f901cee357bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} \\operatorname{U_{c}}{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\theta{\\left(t \\right)}}{r{\\left(t \\right)}} + \\frac{a^{2} \\operatorname{U_{c}}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} r{\\left(t \\right)}}{r^{2}{\\left(t \\right)}} - \\frac{a^{2} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\operatorname{U_{c}}{\\left(t \\right)}}{r{\\left(t \\right)}}$"
      ],
      "text/plain": [
       "a**2*U_c(t)*sin(theta(t))*Derivative(theta(t), t)/r(t) + a**2*U_c(t)*cos(theta(t))*Derivative(r(t), t)/r(t)**2 - a**2*cos(theta(t))*Derivative(U_c(t), t)/r(t)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi.diff(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "52970004-5d18-4905-a61c-01c5d9a7f81c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} \\operatorname{U_{c}}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)}}{r^{2}{\\left(t \\right)}}$"
      ],
      "text/plain": [
       "a**2*U_c(t)*cos(theta(t))/r(t)**2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi.diff(r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3bc6b6e8-cdba-4eb9-b754-e07cc74c6063",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{a^{2} \\operatorname{U_{c}}{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{r^{2}{\\left(t \\right)}}$"
      ],
      "text/plain": [
       "a**2*U_c(t)*sin(theta(t))/r(t)**2"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi.diff(theta)/r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bcae9e54-d38d-4128-b872-e4bcb11d19c5",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
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