examples/LieDerivatives2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: Recompiling stale cache file /home/hofi/.julia/lib/v0.4/Flows.ji for module Flows.\n"
     ]
    }
   ],
   "source": [
    "using Flows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(F,G,H)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@funs F,G,H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(u,v,w)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@x_vars u,v,w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(t,s,r)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@t_vars t,s,r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{L}\\underbrace{\\mathrm{e}^{sD_{G}}}_\\mathrm{R}\\mathrm{e}^{rD_{H}}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)*exp(s,D(G),R)*exp(r,D(H))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = exp(t,D(F),L)*exp(s,G,R)*exp(r,H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_exponential_labels(false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}\\mathrm{e}^{sD_{G}}\\mathrm{e}^{rD_{H}}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)*exp(s,D(G),R)*exp(r,D(H))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"exp(t,D(F),L)*exp(s,D(G),R)*exp(r,D(H))\""
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "string(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_exponential_labels()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{L}\\underbrace{\\mathrm{e}^{sD_{G}}}_\\mathrm{R}\\mathrm{e}^{rD_{H}}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)*exp(s,D(G),R)*exp(r,D(H))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{L}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1=exp(t,D(F),L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{R}$"
      ],
      "text/plain": [
       "exp(t,D(F),R)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex2=exp(t,D(F),R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1==ex2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{L}+\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{R}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)+exp(t,D(F),R)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = ex1+ex2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Flows.Label(\"R\")"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_L = Flows.L\n",
    "_R = Flows.R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{L}D_{F}+D_{F}\\underbrace{\\mathrm{e}^{tD_{F}}}_\\mathrm{R}$"
      ],
      "text/plain": [
       "exp(t,D(F),L)*D(F)+D(F)*exp(t,D(F),R)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_derivative(ex, t, to_the_right=[R])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{H}})$"
      ],
      "text/plain": [
       "exp(t,D(F))*(exp(t,D(G))+exp(t,D(H)))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = exp(t,F)*(exp(t,G)+exp(t,H))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(D_{H}\\mathrm{e}^{tD_{H}}+D_{G}\\mathrm{e}^{tD_{G}})+D_{F}\\mathrm{e}^{tD_{F}}(\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{H}})$"
      ],
      "text/plain": [
       "exp(t,D(F))*(D(H)*exp(t,D(H))+D(G)*exp(t,D(G)))+D(F)*exp(t,D(F))*(exp(t,D(G))+exp(t,D(H)))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_derivative(ex, t, to_the_right=[default])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(D_{F}(\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{H}})+\\mathrm{e}^{tD_{G}}D_{G}+\\mathrm{e}^{tD_{H}}D_{H})$"
      ],
      "text/plain": [
       "exp(t,D(F))*(D(F)*(exp(t,D(G))+exp(t,D(H)))+exp(t,D(G))*D(G)+exp(t,D(H))*D(H))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_derivative(ex, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}\\mathrm{e}^{tD_{G}}D_{G}+\\mathrm{e}^{tD_{F}}D_{F}\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{F}}\\mathrm{e}^{tD_{H}}D_{H}+\\mathrm{e}^{tD_{F}}D_{F}\\mathrm{e}^{tD_{H}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*exp(t,D(G))*D(G)+exp(t,D(F))*D(F)*exp(t,D(G))+exp(t,D(F))*exp(t,D(H))*D(H)+exp(t,D(F))*D(F)*exp(t,D(H))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(t_derivative(ex, t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{H}})\\mathrm{e}^{tD_{F}}$"
      ],
      "text/plain": [
       "(exp(t,D(G))+exp(t,D(H)))*exp(t,D(F))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = (exp(t,G)+exp(t,H))*exp(t,F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\mathrm{e}^{tD_{G}}+\\mathrm{e}^{tD_{H}})\\mathrm{e}^{tD_{F}}D_{F}+(\\mathrm{e}^{tD_{G}}D_{G}+\\mathrm{e}^{tD_{H}}D_{H})\\mathrm{e}^{tD_{F}}$"
      ],
      "text/plain": [
       "(exp(t,D(G))+exp(t,D(H)))*exp(t,D(F))*D(F)+(exp(t,D(G))*D(G)+exp(t,D(H))*D(H))*exp(t,D(F))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_derivative(ex, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}\\mathrm{e}^{tD_{G}}D_{F}\\mathrm{e}^{tD_{G}}D_{F}\\mathrm{e}^{tD_{G}}D_{F}$"
      ],
      "text/plain": [
       "D(F)*exp(t,D(G))*D(F)*exp(t,D(G))*D(F)*exp(t,D(G))*D(F)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = D(F)*exp(t,G)*D(F)*exp(t,G)*D(F)*exp(t,G)*D(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}\\mathrm{e}^{tD_{G}}(D_{G}D_{F}\\mathrm{e}^{tD_{G}}D_{F}\\mathrm{e}^{tD_{G}}+D_{F}\\mathrm{e}^{tD_{G}}(D_{F}\\mathrm{e}^{tD_{G}}D_{G}+D_{G}D_{F}\\mathrm{e}^{tD_{G}}))D_{F}$"
      ],
      "text/plain": [
       "D(F)*exp(t,D(G))*(D(G)*D(F)*exp(t,D(G))*D(F)*exp(t,D(G))+D(F)*exp(t,D(G))*(D(F)*exp(t,D(G))*D(G)+D(G)*D(F)*exp(t,D(G))))*D(F)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1 = t_derivative(ex, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}D_{F}D_{G}$"
      ],
      "text/plain": [
       "D(F)*D(F)*D(G)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1 = D(F)*D(F)*D(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}D_{G}D_{G}$"
      ],
      "text/plain": [
       "D(F)*D(G)*D(G)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex2 = D(F)*D(G)*D(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}(D_{F}+D_{G})D_{G}$"
      ],
      "text/plain": [
       "D(F)*(D(F)+D(G))*D(G)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_factorized(ex1, ex2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}(\\mathrm{Id}+D_{G}D_{G})$"
      ],
      "text/plain": [
       "D(F)*(lie_id+D(G)*D(G))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_factorized(D(F), ex2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$(\\mathrm{Id}+D_{F}D_{G})D_{G}$"
      ],
      "text/plain": [
       "(lie_id+D(F)*D(G))*D(G)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_factorized(D(G), ex2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: imported binding for R overwritten in module Main\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(R,)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@funs R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(τ,)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@t_vars τ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_exponential_labels()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\underbrace{\\mathrm{e}^{τD_{F+R}}}_\\mathrm{L}\\underbrace{\\mathrm{e}^{(-τ+t)D_{F}}}_\\mathrm{R}$"
      ],
      "text/plain": [
       "exp(τ,D(F+R),L)*exp(-τ+t,D(F),R)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = exp(τ,D(F+R),_L)exp(t-τ,D(F),_R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_exponential_labels(false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{τD_{F+R}}(D_{F+R}-D_{F})\\mathrm{e}^{(-τ+t)D_{F}}$"
      ],
      "text/plain": [
       "exp(τ,D(F+R),L)*(D(F+R)-D(F))*exp(-τ+t,D(F),R)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = t_derivative(ex,τ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{τD_{F+R}}D_{R}\\mathrm{e}^{(-τ+t)D_{F}}$"
      ],
      "text/plain": [
       "exp(τ,D(F+R),L)*D(R)*exp(-τ+t,D(F),R)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = expand_lie_derivatives(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{2tD_{F}}\\mathrm{e}^{3tD_{G}}$"
      ],
      "text/plain": [
       "exp(2t,D(F))*exp(3t,D(G),L)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex=exp(2t,D(F))exp(3t,D(G),L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{2tD_{F}}(2D_{F}\\mathrm{e}^{3tD_{G}}+3\\mathrm{e}^{3tD_{G}}D_{G})$"
      ],
      "text/plain": [
       "exp(2t,D(F))*(2D(F)*exp(3t,D(G),L)+3exp(3t,D(G),L)*D(G))"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t_derivative(ex,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{F}(2\\mathrm{Id}-D_{G})D_{H}$"
      ],
      "text/plain": [
       "D(F)*(2lie_id-D(G))*D(H)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_factorized(2D(F)D(H), -D(F)D(G)D(H))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$30D_{F}D_{G}D_{H}$"
      ],
      "text/plain": [
       "30D(F)*D(G)*D(H)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex=2*D(F)*3*D(G)*5*D(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}D_{G}\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*D(G)*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = exp(t,F)D(G)exp(-t,F,_R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(D_{F}D_{G}-D_{G}D_{F})\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*(D(F)*D(G)-D(G)*D(F))*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = t_derivative(ex,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F)))*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = t_derivative(ex,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{G}\\mathrm{e}^{-tD_{F}}-2\\mathrm{e}^{tD_{F}}D_{F}D_{G}D_{F}\\mathrm{e}^{-tD_{F}}+\\mathrm{e}^{tD_{F}}D_{G}D_{F}D_{F}\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*D(F)*D(F)*D(G)*exp(-t,D(F),R)-2exp(t,D(F))*D(F)*D(G)*D(F)*exp(-t,D(F),R)+exp(t,D(F))*D(G)*D(F)*D(F)*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(-(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))D_{F}+D_{F}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F})))\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*(-(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F)))*D(F)+D(F)*(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F))))*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = t_derivative(ex,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$-\\mathrm{e}^{tD_{F}}D_{G}D_{F}D_{F}D_{F}\\mathrm{e}^{-tD_{F}}-3\\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{G}D_{F}\\mathrm{e}^{-tD_{F}}+3\\mathrm{e}^{tD_{F}}D_{F}D_{G}D_{F}D_{F}\\mathrm{e}^{-tD_{F}}+\\mathrm{e}^{tD_{F}}D_{F}D_{F}D_{F}D_{G}\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "-exp(t,D(F))*D(G)*D(F)*D(F)*D(F)*exp(-t,D(F),R)-3exp(t,D(F))*D(F)*D(F)*D(G)*D(F)*exp(-t,D(F),R)+3exp(t,D(F))*D(F)*D(G)*D(F)*D(F)*exp(-t,D(F),R)+exp(t,D(F))*D(F)*D(F)*D(F)*D(G)*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$-(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))D_{F}+D_{F}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))$"
      ],
      "text/plain": [
       "-(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F)))*D(F)+D(F)*(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F)))"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1=expand_lie_commutators(commutator(D(F),commutator(D(F),commutator(D(F),D(G)))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\mathrm{e}^{tD_{F}}(-(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F}))D_{F}+D_{F}(-(D_{F}D_{G}-D_{G}D_{F})D_{F}+D_{F}(D_{F}D_{G}-D_{G}D_{F})))\\mathrm{e}^{-tD_{F}}$"
      ],
      "text/plain": [
       "exp(t,D(F))*(-(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F)))*D(F)+D(F)*(-(D(F)*D(G)-D(G)*D(F))*D(F)+D(F)*(D(F)*D(G)-D(G)*D(F))))*exp(-t,D(F),R)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex1 = exp(t,F)*ex1*exp(-t,F,_R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex==ex1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$D_{G}D_{F}\\mathrm{e}^{2tD_{F}}D_{F}D_{F}\\mathrm{e}^{3sD_{5F}}D_{7F}D_{G}$"
      ],
      "text/plain": [
       "D(G)*D(F)*exp(2t,D(F))*D(F)*D(F)*exp(3s,D(5F))*D(7F)*D(G)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex = D(G)*D(F)*exp(2t,F)*D(F)^2*exp(3s,5F)*D(7F)*D(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$7D_{G}\\mathrm{e}^{(15s+2t)D_{F}}D_{F}D_{F}D_{F}D_{F}D_{G}$"
      ],
      "text/plain": [
       "7D(G)*exp(15s+2t,D(F))*D(F)*D(F)*D(F)*D(F)*D(G)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normalize_lie_products(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$7D_{G}D_{F}D_{F}D_{F}D_{F}\\mathrm{e}^{(15s+2t)D_{F}}D_{G}$"
      ],
      "text/plain": [
       "7D(G)*D(F)*D(F)*D(F)*D(F)*exp(15s+2t,D(F))*D(G)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normalize_lie_products(ex, to_the_right=[default])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "Julia 0.4",
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  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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