LieOperator.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import sympy\n",
    "from operator import mul\n",
    "from functools import reduce\n",
    "from mpmath import fac\n",
    "import functools\n",
    "from sympy import init_printing\n",
    "from IPython.display import display_latex\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Expr, Symbol, Eq, Mul, Add, Pow, expand, sympify, Tuple\n",
    "from sympy.core.basic import Basic\n",
    "from sympy.core.operations import AssocOp\n",
    "\n",
    "from sympy.core.singleton import (Singleton, S)\n",
    "from sympy.core.decorators import _sympifyit, call_highest_priority\n",
    "from sympy.core.cache import cacheit\n",
    "\n",
    "from sympy.core.function import diff\n",
    "from sympy.series.order import Order\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "property"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(Expr.func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "class LieOperatorExpr(Expr):\n",
    "    _op_priority = 12.0\n",
    "    \n",
    "    is_LieOperator = True\n",
    "    is_Identity = False\n",
    "    \n",
    "    def __neg__(self):\n",
    "        return DOMul(S.NegativeOne, self)\n",
    "    \n",
    "    def __abs__(self):\n",
    "        raise NotImplementedError\n",
    "\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__radd__')\n",
    "    def __add__(self, other):\n",
    "        return DOAdd(self, other)\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__add__')\n",
    "    def __radd__(self, other):\n",
    "        return DOAdd(other, self)\n",
    "    \n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__rmul__')\n",
    "    def __mul__(self, other):\n",
    "        return DOMul(self, other)\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__mul__')\n",
    "    def __rmul__(self, other):\n",
    "        return DOMul(other, self)\n",
    "\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__rpow__')\n",
    "    def __pow__(self, other):\n",
    "        return DOPow(self, other)\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__pow__')\n",
    "    def __rpow__(self, other):\n",
    "        raise NotImplementedError(\"DiffOperator Power is not defined\")\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__rdiv__')\n",
    "    def __div__(self, other):\n",
    "        return DOMul(self, other**S.NegativeOne)\n",
    "    @_sympifyit('other', NotImplemented)\n",
    "    @call_highest_priority('__div__')\n",
    "    def __rdiv__(self, other):\n",
    "        return DOMul(other, self**S.NegativeOne)\n",
    "\n",
    "class LieOperatorOne(LieOperatorExpr):\n",
    "    \"\"\"\n",
    "    Represents the identic operator ( DiffOperator**0).\n",
    "    \"\"\"\n",
    "\n",
    "    __metaclass__ = Singleton\n",
    "\n",
    "    is_Identity = True\n",
    "\n",
    "class LieOperator(LieOperatorExpr):\n",
    "    \"\"\"\n",
    "    >>> from sympy.abc import x, a\n",
    "    >>> from sympy.core.function import Function\n",
    "    >>> from sympy.tensor.diff_operator import DiffOperator\n",
    "    >>> f = Function(\"f\")\n",
    "\n",
    "    >>> Lx = LieOperator(x)\n",
    "\n",
    "\n",
    "    >>> Lx + Lx\n",
    "    DiffOperator(x) + DiffOperator(x)\n",
    "\n",
    "    >>> Lx(f(x))\n",
    "    Derivative(f(x), x)\n",
    "\n",
    "    >>> Dx + Dx**2\n",
    "    DiffOperator(x) + DiffOperator(x)**2\n",
    "\n",
    "    >>> (Dx + Dx**2)(f(x))\n",
    "    Derivative(f(x), x) + Derivative(f(x), x, x)\n",
    "\n",
    "    >>> (a*Dx + Dx**3)(x**4)\n",
    "    4*a*x**3 + 24*x\n",
    "\n",
    "    \"\"\"\n",
    "    def __new__(cls,ham, *coordmomenta):\n",
    "        obj = LieOperatorExpr.__new__(cls,ham,*coordmomenta)\n",
    "        obj._ham = obj._args[0]\n",
    "        return obj\n",
    "\n",
    "    @property\n",
    "    def coords(self):\n",
    "        return self._args[1:(len(self._args)+1)//2]\n",
    "    \n",
    "    @property\n",
    "    def momenta(self):\n",
    "        return self._args[(len(self._args)+1)//2:]\n",
    "    \n",
    "    @property\n",
    "    def hamiltonian(self):\n",
    "        return self._ham\n",
    "\n",
    "    def _hashable_content(self):\n",
    "        return tuple(sorted(self._args, key=hash))\n",
    "\n",
    "    def __call__(self, e):\n",
    "        terms = []\n",
    "        for c,m in zip(self.coords, self.momenta):\n",
    "            terms.append(Mul(diff(self.hamiltonian,c),diff(e,m)) -Mul(diff(self.hamiltonian,m),diff(e,c)))\n",
    "        return reduce(Add,terms)\n",
    "\n",
    "    def _eval_nseries(self, x, n, logx):\n",
    "        return self + Order(n)\n",
    "\n",
    "    def _latex(self, p):\n",
    "        return \"e^{:%s:}\" % self.hamiltonian.func\n",
    "#         rep = []\n",
    "\n",
    "#         for c,m in zip(self.coords,self.momenta):\n",
    "#             str1 = \"\\\\frac{\\partial %s}{\\partial %s}  \\\\frac{\\partial }{\\partial %s}\" % (p._print(self.hamiltonian),p._print(c),p._print(m))\n",
    "#             str2 = \"\\\\frac{\\partial %s}{\\partial %s}  \\\\frac{\\partial }{\\partial %s}\" % (p._print(self.hamiltonian),p._print(m),p._print(c))\n",
    "#             rep.append(str1 + ' - ' + str2)\n",
    "#         return ' + '.join(rep)\n",
    "        \n",
    "\n",
    "    def _needs_brackets(self):\n",
    "        return False\n",
    "\n",
    "class DOAdd(LieOperatorExpr, Add):\n",
    "    \"\"\"\n",
    "    A Sum of the LieOperator expressions.\n",
    "    \"\"\"\n",
    "\n",
    "    def __new__(cls, *args):\n",
    "\n",
    "        expr = Add.__new__(cls, *args)\n",
    "\n",
    "        if expr.is_Mul:\n",
    "            return DOMul(*expr.args)\n",
    "        return expr\n",
    "\n",
    "    def __call__(self, e):\n",
    "        return Add(*tuple(d(e) for d in self.args))\n",
    "\n",
    "\n",
    "    @classmethod\n",
    "    def flatten(cls, args):\n",
    "        new_seq = []\n",
    "        i = 0\n",
    "        while args:\n",
    "            o = args.pop()\n",
    "            if o.__class__ is cls:\n",
    "                args.extend(o.args)\n",
    "            else:\n",
    "                new_seq.append(o)\n",
    "        new_seq.reverse()\n",
    "        return new_seq, [], None\n",
    "    \n",
    "    \n",
    "    \n",
    "class DOMul(LieOperatorExpr, Mul):\n",
    "    def __new__(cls, *args):\n",
    "        if any(arg.is_zero for arg in args):\n",
    "            return S.Zero\n",
    "\n",
    "        expr = Mul.__new__(cls, *args)\n",
    "        print(args)\n",
    "        return expr\n",
    "\n",
    "    # is it needed?\n",
    "    @classmethod\n",
    "    def flatten(cls, args_seq):\n",
    "        return args_seq, [], None\n",
    "\n",
    "    def as_ordered_terms(self, order=None):\n",
    "        return self.args\n",
    "\n",
    "    @property\n",
    "    def coords(self):\n",
    "        c = ()\n",
    "        for d in self.args:\n",
    "            c += d.coords\n",
    "        return tuple(set(c))\n",
    "    \n",
    "    @property\n",
    "    def momenta(self):\n",
    "        m = ()\n",
    "        for d in self.args:\n",
    "            m += d.momenta\n",
    "        return tuple(set(m))\n",
    "    \n",
    "    def p(self,n):\n",
    "        F = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "        for i in range(n+1):\n",
    "            for j in range(n+1):\n",
    "                if (j-i)>=0:\n",
    "        #             F[i,j] = 1/(np.math.factorial(j-i))\n",
    "                    F[i,j] = sympy.Rational(1,int(fac(j-i))) # keep it rational !!!!\n",
    "\n",
    "        G = sympy.Matrix(np.zeros((n+1,n+1)))\n",
    "        for i in range(n+1):\n",
    "            for j in range(n+1):\n",
    "                if (j-i)>=0:\n",
    "                    if i==j:\n",
    "                        sigprod = 1\n",
    "                    else:\n",
    "                        l = sympy.symbols('s{}:{}'.format(i,j))\n",
    "                        sigprod = reduce(sympy.Mul,l)\n",
    "                    G[i,j] = sympy.Mul(sympy.Rational(1,int(np.math.factorial(j-i))),sigprod, evaluate=False)\n",
    "\n",
    "        qthpower = sympy.Matrix(np.eye(n+1))\n",
    "        FGm1 = sympy.nsimplify(F*G - qthpower)\n",
    "\n",
    "        summ = []\n",
    "\n",
    "        for q in range(1,n+1):\n",
    "            qthpower = sympy.nsimplify(sympy.MatMul(qthpower, FGm1, evaluate=False))\n",
    "\n",
    "            summ.append(sympy.nsimplify((-1)**(q-1)/q * qthpower)[0,n]) # q-1 because mathematica starts at 1 and not 0, there for sign change\n",
    "\n",
    "        return sympy.expand(reduce(functools.partial(sympy.Add,evaluate=False),summ))\n",
    "    \n",
    "    def translated(self,n,ham1_symbol,ham2_symbol):\n",
    "        l = sympy.symbols('s{}:{}'.format(0,n))\n",
    "        temp = reduce(functools.partial(sympy.Mul,evaluate=False),[k*k for k in l])\n",
    "        temp = sympy.expand(sympy.Mul(temp,  self.p(n), evaluate=False),evaluate=False)\n",
    "        with sympy.evaluate(False):\n",
    "            ln = sympy.symbols('r{}:{}'.format(0,n), commutative=False)\n",
    "            for i, s in enumerate(l):\n",
    "                temp = temp.subs(l[i],ln[i])\n",
    "            for i in range(n):\n",
    "                g = sympy.Symbol('g',commutative=False)\n",
    "                f = sympy.Symbol('f',commutative=False)\n",
    "                temp = temp.subs(ln[i]**3,g)\n",
    "                temp = temp.subs(ln[i]**2,f)\n",
    "        \n",
    "        temp = temp.subs(f,ham1_symbol)\n",
    "        temp = temp.subs(g,ham2_symbol)\n",
    "\n",
    "        return temp\n",
    "    \n",
    "    def SumTranslated(self,n,ham1_symbol,ham2_symbol):\n",
    "        return reduce(functools.partial(sympy.Add,evaluate=False),[self.translated(i,ham1_symbol,ham2_symbol) for i in range(1,n+1)])\n",
    "    \n",
    "    @property\n",
    "    def hamiltonian(self):\n",
    "        return self.SumTranslated(4,self.args[0].hamiltonian,self.args[1].hamiltonian).doit()\n",
    "\n",
    "    \n",
    "    \n",
    "    def __call__(self, e):\n",
    "        for e in self.args:\n",
    "            print('ok')\n",
    "        return e\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "px = Symbol('px')\n",
    "y = Symbol('y')\n",
    "py = Symbol('py')\n",
    "f = sympy.Function('f',commutative=False)(x,px)\n",
    "g = sympy.Function('g',commutative=False)(y,py)\n",
    "H = sympy.Function('H',commutative=False)(x,px,y,py)\n",
    "H1 = sympy.Function('H1',commutative=False)(x,px,y,py)\n",
    "H2 = sympy.Function('H2',commutative=False)(x,px,y,py)\n",
    "h = Symbol('h',commutative=False)\n",
    "loe = LieOperator(H,x,y,px,py)\n",
    "loe1 =LieOperator(H1,x,y,px,py)\n",
    "loe2 =LieOperator(H2,x,y,px,py)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H1:} + e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H1(x, px, y, py), x, y, px, py) + LieOperator(H2(x, px, y, py), x,\n",
       " y, px, py)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe1+loe2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(LieOperator(H1(x, px, y, py), x, y, px, py), LieOperator(H2(x, px, y, py), x, y, px, py))\n"
     ]
    },
    {
     "data": {
      "image/png": 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UFFfyJKu1TPCchc1rgHKESve2LWCz1qzbHN6NvZSwwZcw1+NxTWD8AC1RA+c5sd8zXd0Y0YaqD3j6Sh14/Xb3ATb0K3H+MNl1LH1KmCdxWE5rfSM6LV3Kx2Zz6wvlAOursWAoMcytvlCOjbTqpyyXzd8Ev0VGWqubMKH7y/pXVDodbeo6ICY4ZfHHFz1vOOGbbmz9+KFp3/4xw/Z5zYGvo6cQj7rKF70wMSrj58azf/0XX6632+1MS4OVUaOeo3IGLvfXaaVgSWIzYDr1AX3k/a51CPb0rdiuRhRIsYl8iTvS4L/Kovjnf0Wf7btKf8pCT7IVlQmes7B5DVCMUPHetgVs1pq1m0N67KVkDb6kuR6Pa/LiB2iNR1tpbFw8aFO+vNoxQkv6gKdjKNgZnTgMjW5mD24CP9KP7Hq/EOt7EnckJ1Zc8AywWaX1hXKA9dVYMH1ihFt9oRybE52va0ZrdRMm0YJrTsnILYT8hFXSvEu8pSZyzP9A9aOMtJ2jWbhXl/nSK3MNbX22nkzUd67S/WtxGS93fBburzGbraeEGjVnUvDnaSVgyWJjfcgmPqGPvJ+1DrmevhnbVYgCIZtUzAwcSWVHKc9V+hi82JOsv2fLWhDKU9jah/JYV1NVqK1cKeHethFsDmi23+U3B+Z2ybHX4O8ip3ytFB98Jb9TgOsxfUFcy7bjgvjB1MRJ0BI1cJ7TZQamq91lfsLTMRTsWixxWBAFWDiPooAsTwocyamtETznYLNKawi1letlnERNPzBhfTJ0NzpDFjZSpWdgcudvBgZfIz9F2IRJcrA1dHeZE7pW6sVu00o1gLC++Nb+7snr26G5uukwVqrL1gfMGkX0zGFchufGxmDf5yVLqBnwDU9D6kMJfbJobVVNbMVVbNd7P28dcj19q7ZbxVJqXkwqEjO5q6zmSdbTpWOrKtRWHiWAjXXk00OG45gNML85sNaaHHsN3trvlK+V4oOvOd+ZTW6zg68c/yKioEUk4n0NNqwRxoKCI9n68mNEoIB/zFbD291qHfgqamYOdzgNP11DzXY9ybKTjq2qUFt5lKiBLRIx5YBUodm64sA69qOWTZikBltzHmyfQnzsnOvD5syJG92rFm3RTyTGF2uNa/v80rFqjpxI4Of1zZEHNZGBEgdAaxzOz7PZcMy0Vq1gxFnYagq1dccJYLNM5nQ+tlCFIQRzu9TYyy0u4cR+JMUGX7O+v4LrJfTJUhMLlaXv59WwRhjbKjzy87RCIOzzz7PZsCdZM1Yw4ixsNYXauuNEBWyxiClHpAotqis12Dr1bxV8nU7H3ePk3eX+eJ53F31cr1E1lK0P7s1dVbs/WsRKf3Bl/NzuDZityfIawBRr4xwQAAEQ+D4CiJlZNgU2YMsisKwQc7vU2EtVGHwtuzCUBoGtEGCNcCuSoVMkAXhSllmALQsbCmUTuDSHvdmOTf+CxYt7uP7c3eL60kv23Q8ndXA5Sj3V8aBXwnqe1WC2zm+XIDu5Rax4GT/37r4YDSDbkigIAiDwgwQQM7OMDmzAlkVgWSHmduHY67k/7+3/BSsMvpZdGEqDwFYIsEa4FcnQKZIAPCnLLMCWhQ2FsgkE/1W87I/utcBmJktvTz34ejYveu1qe2x3U9ejTt2bwWyl89v3P5776TNTyJUJctnrJ9EAWrz4AwIgAAKTCCBmTsIUngRsIZFJn4FtEqbRkxi/YOz11IuzmhcKdVuFwRdVjT0IfDcB1gi/+0JxdR8mAE/KAgxsWdhQKJtAMNjSL7N1E11/3dpZ5p+MVzZhZerSx9pXG+oJsKFsk/9nHl1s2Krwrsw/P/dibvt6Xs12+Gt3Zg5NtW81xJ+YgGbzuh7sdm3X1Y9PwxGBBGA7gUbZpCTtSVHM3OSFFBWtqcXY0NW8tQGwvUWUPmHA7YKxl3n1z6WhVxEVHXxhOJE2HsvVDQG0GI8tJYeC2Jb0Q6sQAtqR4lGEEG2CZQBblnEMNvQ5Wej6QhpgMNgamugynPmElfmsF9zS93jpV0R2b0yMs3V+O26jlxsGZbzcdqKrPUFhprfjgL8gAAIgMIUAYuYUStE5wBYhmXIA2KZQGj+H8QvGXgf9f8VdQy/m6Se6zBfFo6vPDL7GRSMHBL6JAGuE33RZuJbiBOBJWciBLQsbCmUTCAZb3kRXf/e8/u5L401Ymdqe5s6vV9OuVz+Q3T25eOrmwcz5ZnNl2ucaKffhbiJDA+hI4S8IgAAITCGAmDmFUnQOsEVIphwAtimUxs9h/KKxlx59tf8+NKUrDL7GRSMHBL6JAGuE33RZuJbiBOBJWciBLQsbCmUTiAZb7NHFfj3U20u1a8qfzC1cdruaKa67fqnicLY6m8cP2z+2iLJl/FwsRu8QIQUCIAACMwhg0DADljsV2ByLGSlgmwFr4FTGLxp76ektu9RDhcHXgFocAoEvJMAa4RdeHS6pHAF4UhZrYMvChkLZBC792xZpz9foera3az30YlxnvYrWzryD8dXQaqmm3OXvoh9RHMxWe53/aIbLKJarv3SvX+vYb2gARAJ7EAABEHhPADHzPaOBM4BtAMr7Q8D2nlHqDMaPxly017e8tyOutniFwVdKNvJA4HsIsEb4PReFK6lAAJ6UBR3YsrChUCaB3f7QNMeTor3+GnZHV/dI4et4Or1O+7uZs9L/cjx093U9msfptNfLcA1nm+PP075bWjUq4+Xqle7dc5FoAC1l/AEBEACBSQQQMydhCk8CtpDIpM/ANgnT6EmWH425aK9L3N08l6ow+BrVjAwQ+CoCthF+1VXhYsoTgCdlMQe2LGwotB4BNtGlzHuAwu3WHWuX2wrz9Oc+u8tpV+JyB4MylLtzS3TJXIx+t9+fz24ybuCqcegDBJ7P0xPUPwB2U1+JxvfOXLIHDWLtB2zvHGswXyw239EubM5o8DpqHUzwe+g72x+2xys/+KqFJFmvb9bkqchcj8B3D74SjXA9hNv/JrS9tzaU7UliDQhsbz1rUyf4jiZ28MWZ3tj68fw/jHRO/5hhu9wWHWN7egrxqB9xfNErhKIyfq5dl0IXGZhaY99eJfkyq4096VqqSPjBSl9X/f4pmgv9wev/9Uu+6eX/9IbG99YRJMZM/R8P6fYDtreeNXSCTGxRoDjolRNEbuP8Ltfb7Xa2C6GWH3xJ5IX4X8Mq3z74Gm+ENWiLq1N83y2HmExPEm9AYJPjwisoCXppsYMvd6nmPnr9KCNt52je6dVlvvTKXENbn60fcdRPOp77L4rLeLm06tfQ90k4dmzHnn9sAlCCqm/XcDb/kj9L/bny7fQrX9/ruD927x9D46tsiqzqYT9gyyKQX8gPFKe/7fUcf43ZHAEMvjQL36wODlKfJIDB1yfpyv5u9N2y7fNWHQz4FtHQCcA2RGXaMb+X3uDgaxcNF7uVum6H5uqmwxiMLlsfMCt40V34cRmeG4/n2PdJSP61j1YeG7ocCZq+XsOjW97t668TFzhC4NRNdKHxjfCRfhj2y7IQsGVhU8oLFLv7NRq5ZH5vvWIYfGn2nlnr2eK3asbg67fsHV4tOqGQyMY+w4BZBgO2LGxBL73JwVf7qsXMq59W7B7dNDatXLmzru1Pbkx0lSOuazrjBrqivMVV1nc6aHziLDNNEOw3jVNwFrAFQKZ+9ALFU33BRFf3nuup1591HgZfWdi+vRAGX99u4fT1oRNK8xGfCwNmmQjYsrDp1wnyKZLvGHxlkth+sUN3h8n2L2QbV9BcL/qtnXqZLmy/SaDvdLqLR+PbnBPAflkmA7YsbFSoCxR6/uYbJrroorBH/C/qAxh8FcUtrjJ0QuJMMk8QDDiPV382sGVho0IYfBGJDe8fDS2zv+GL2I70V3M1D7z+Da8Et53rgNJcArzTQePLpVivHOyXxR7YsrD1hbpAYdYExUTXEo7CyiL+FzUIBl9FccurDJ2QPJvMUgQDzsJFJwMbkcjZY/CVQ01amYN59SK2UgReTXsD3b57dVupWlGPHAK800Hjk2OXqUpgv6mkvPOAzcMx80MXKMy/pDDRNROd5NMR/4taB4OvorjlVYZOSJ5NZimCAWfhopOBjUjk7DH4yqEmrMzevAMQWzkCTfsGgHuDhxfLMRdVE+t00PhEWWaaGNhvGqfgLGALgMz52AWKm3lrDCa65oCTfS7if2H7YPBVGLiw6tAJCTPIXDkw4Fxi7fnAloWtK4TB1wJ4UoqeMM9V2BSHq6nw3tjXeBauH9VVJuA6HTS+yqbIqh72A7YsAvmFukDxatcYwERXPkdhJRH/SxsEg6/SxGXVh75blj1mq4EBZyMzBYAtC1tbCIOvfHZiSt7MPNdO/OshxfBaQcixfesi7uhaAeU2v8J2Omh8mzQg7JdlNmDLwmYK9YHidjRb83fUK3Vh2z4BxP/iNsTgqzhyURWiExJljvliYMD5zHQJYMvCZgph8JWNTk7BS3s/1wkTXQVNcunX6CpYJaqSRIA6HTQ+SVaZrgX2m86KnQlsDMaspB8o/s6zCuNkqQR8s0pV+V26MPj6LnvOvRp0QnOJCTsfBswyCLBlYdOF/F4ag69cjlXL7Q7P53N/bBeNqirkpyo/6idQXnjr4k/ZnF9s3+mg8XEoG0rDflnGArYsbPqGa7+XbjDRlQlSVrHArLLEfa0aDL6+1rRTLgyd0BRKgs+BAbOMA2xZ2DD4ysQmrNi1aTdMdJW1y3O/P5plhbH9IIH98a+5HvUCbWh8m7Q+7JdlNmDLwmYKeYHieW4aEz2wbZ2AZ9atX8x29GPwtR1bra0UndDaRAt/HwyYBRzYsrCZQl4vjcFXNkcUBAEQAAEQAAEQAAEQAAEQAAEQAAEQAAEQAIEkgf9GctF5SbggqAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\operatorname{H_{1}}{\\left (x,px,y,py \\right )} + \\frac{\\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )}}{2} - \\frac{\\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )}}{6} - \\frac{\\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )}}{12} + \\frac{\\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}^{2}{\\left (x,px,y,py \\right )}}{12} + \\frac{\\operatorname{H_{1}}^{2}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )}}{12} + \\frac{\\operatorname{H_{1}}^{2}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}^{2}{\\left (x,px,y,py \\right )}}{24} + \\operatorname{H_{2}}{\\left (x,px,y,py \\right )} - \\frac{\\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )}}{2} - \\frac{\\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )}}{6} + \\frac{\\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )}}{12} + \\frac{\\operatorname{H_{2}}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}^{2}{\\left (x,px,y,py \\right )}}{12} + \\frac{\\operatorname{H_{2}}^{2}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )}}{12} - \\frac{\\operatorname{H_{2}}^{2}{\\left (x,px,y,py \\right )} \\operatorname{H_{1}}^{2}{\\left (x,px,y,py \\right )}}{24}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                   H₁(x, px, y, py)⋅H₂(x, px, y, py)   H₁(x, px, y, py)⋅H₂(x, \n",
       "H₁(x, px, y, py) + ───────────────────────────────── - ───────────────────────\n",
       "                                   2                                          \n",
       "\n",
       "                                                                              \n",
       "px, y, py)⋅H₁(x, px, y, py)   H₁(x, px, y, py)⋅H₂(x, px, y, py)⋅H₁(x, px, y, p\n",
       "─────────────────────────── - ────────────────────────────────────────────────\n",
       " 6                                                             12             \n",
       "\n",
       "                                         2                   2                \n",
       "y)⋅H₂(x, px, y, py)   H₁(x, px, y, py)⋅H₂ (x, px, y, py)   H₁ (x, px, y, py)⋅H\n",
       "─────────────────── + ────────────────────────────────── + ───────────────────\n",
       "                                      12                                   12 \n",
       "\n",
       "                    2                 2                                       \n",
       "₂(x, px, y, py)   H₁ (x, px, y, py)⋅H₂ (x, px, y, py)                      H₂(\n",
       "─────────────── + ─────────────────────────────────── + H₂(x, px, y, py) - ───\n",
       "                                   24                                         \n",
       "\n",
       "                                                                              \n",
       "x, px, y, py)⋅H₁(x, px, y, py)   H₂(x, px, y, py)⋅H₁(x, px, y, py)⋅H₂(x, px, y\n",
       "────────────────────────────── - ─────────────────────────────────────────────\n",
       "             2                                           6                    \n",
       "\n",
       "                                                                              \n",
       ", py)   H₂(x, px, y, py)⋅H₁(x, px, y, py)⋅H₂(x, px, y, py)⋅H₁(x, px, y, py)   \n",
       "───── + ─────────────────────────────────────────────────────────────────── + \n",
       "                                         12                                   \n",
       "\n",
       "                   2                   2                                    2 \n",
       "H₂(x, px, y, py)⋅H₁ (x, px, y, py)   H₂ (x, px, y, py)⋅H₁(x, px, y, py)   H₂ (\n",
       "────────────────────────────────── + ────────────────────────────────── - ────\n",
       "                12                                   12                       \n",
       "\n",
       "                2              \n",
       "x, px, y, py)⋅H₁ (x, px, y, py)\n",
       "───────────────────────────────\n",
       "             24                "
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(loe1*loe2).hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(LieOperator(H1(x, px, y, py), x, y, px, py), LieOperator(H2(x, px, y, py), x, y, px, py))\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H1:} e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H1(x, px, y, py), x, y, px, py)⋅LieOperator(H2(x, px, y, py), x, y\n",
       ", px, py)"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe1*loe2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(LieOperator(H1(x, px, y, py), x, y, px, py), LieOperator(H2(x, px, y, py), x, y, px, py))\n"
     ]
    }
   ],
   "source": [
    "loe12mul = loe1*loe2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ok\n",
      "ok\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H2(x, px, y, py), x, y, px, py)"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe12mul(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H1:} e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H1(x, px, y, py), x, y, px, py)⋅LieOperator(H2(x, px, y, py), x, y\n",
       ", px, py)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mul(loe1,loe2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H2:} e^{:H1:}$$"
      ],
      "text/plain": [
       "LieOperator(H2(x, px, y, py), x, y, px, py)⋅LieOperator(H1(x, px, y, py), x, y\n",
       ", px, py)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mul(loe2,loe1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 564,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 e^{:H1:} e^{:H2:}$$"
      ],
      "text/plain": [
       "2⋅LieOperator(H1(x, px, y, py), x, y, px, py)⋅LieOperator(H2(x, px, y, py), x,\n",
       " y, px, py)"
      ]
     },
     "execution_count": 564,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mul(loe1,loe2,2)\n",
    "# (loe1*loe2).SumTranslated(3,loe1.hamiltonian,loe2.hamiltonian)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 491,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( x, \\quad y\\right )$$"
      ],
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 491,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe.coords"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 492,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( px, \\quad py\\right )$$"
      ],
      "text/plain": [
       "(px, py)"
      ]
     },
     "execution_count": 492,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe.momenta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 493,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$H{\\left (x,px,y,py \\right )}$$"
      ],
      "text/plain": [
       "H(x, px, y, py)"
      ]
     },
     "execution_count": 493,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe.hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 494,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAUBAMAAADxfUlCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtmzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQYGWNggIG8fxOi/xrAeCCa+QMD11dkAQbuBww8DSgiHAoMLA/gImwFDAzyCQzsBgzpUDGuDQwM811c/BMYVsNVMTCoMzDkI3GBzCtgQWSxP2BBmAjQHJBbfjLAzWFrYFD/EjDvXwJDBlARo7JJAEwthE4PYFVAEWHtZBBZgCLC3WN8EEWAgR/oXFTAr4DKZ2AAepnBAEWQ+QKDCJrtJjYHUJQAABnOH9lyiV0MAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$e^{:H:}$$"
      ],
      "text/plain": [
       "LieOperator(H(x, px, y, py), x, y, px, py)"
      ]
     },
     "execution_count": 494,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 495,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{\\partial}{\\partial px} H{\\left (x,px,y,py \\right )} \\frac{\\partial}{\\partial x} f{\\left (x,px \\right )} + \\frac{\\partial}{\\partial x} H{\\left (x,px,y,py \\right )} \\frac{\\partial}{\\partial px} f{\\left (x,px \\right )}$$"
      ],
      "text/plain": [
       "   ∂                   ∂              ∂                    ∂           \n",
       "- ───(H(x, px, y, py))⋅──(f(x, px)) + ──(H(x, px, y, py))⋅───(f(x, px))\n",
       "  ∂px                  ∂x             ∂x                  ∂px          "
      ]
     },
     "execution_count": 495,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 496,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{d}{d px} x \\frac{\\partial}{\\partial x} f{\\left (x,px \\right )} + \\frac{d}{d x} x \\frac{\\partial}{\\partial px} f{\\left (x,px \\right )}$$"
      ],
      "text/plain": [
       "   d     ∂              d      ∂           \n",
       "- ───(x)⋅──(f(x, px)) + ──(x)⋅───(f(x, px))\n",
       "  dpx    ∂x             dx    ∂px          "
      ]
     },
     "execution_count": 496,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(f).subs(H,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 497,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\partial}{\\partial px} f{\\left (x,px \\right )}$$"
      ],
      "text/plain": [
       " ∂           \n",
       "───(f(x, px))\n",
       "∂px          "
      ]
     },
     "execution_count": 497,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(f).subs(H,x).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 498,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{\\partial}{\\partial py} H{\\left (x,px,y,py \\right )} \\frac{\\partial}{\\partial y} g{\\left (y,py \\right )} + \\frac{\\partial}{\\partial y} H{\\left (x,px,y,py \\right )} \\frac{\\partial}{\\partial py} g{\\left (y,py \\right )}$$"
      ],
      "text/plain": [
       "   ∂                   ∂              ∂                    ∂           \n",
       "- ───(H(x, px, y, py))⋅──(g(y, py)) + ──(H(x, px, y, py))⋅───(g(y, py))\n",
       "  ∂py                  ∂y             ∂y                  ∂py          "
      ]
     },
     "execution_count": 498,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 499,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{d}{d py} x \\frac{\\partial}{\\partial y} g{\\left (y,py \\right )} + \\frac{d}{d y} x \\frac{\\partial}{\\partial py} g{\\left (y,py \\right )}$$"
      ],
      "text/plain": [
       "   d     ∂              d      ∂           \n",
       "- ───(x)⋅──(g(y, py)) + ──(x)⋅───(g(y, py))\n",
       "  dpy    ∂y             dy    ∂py          "
      ]
     },
     "execution_count": 499,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(g).subs(H,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 500,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77urRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QBAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 500,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe(g).subs(H,x).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 501,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H1:} + e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H1(x, px, y, py), x, y, px, py) + LieOperator(H2(x, px, y, py), x,\n",
       " y, px, py)"
      ]
     },
     "execution_count": 501,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe12 = loe1+loe2\n",
    "loe12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 502,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAyYAAAAvCAMAAAAhDi5VAAAAM1BMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADxgEwMAAAAEHRSTlMAEM3dMiKJu1SZZnZE76sg9CJyJAAAAAlwSFlzAAAOxAAADsQBlSsOGwAACINJREFUeAHtnem2pCoMhbGcS09d3/9pb5iDBAccYbU/Wo4UkI8NghhpxnYc36Fp6h2/f+VPs4BANZsFTxYQShTeRYp2RBIlGMwCAtV7FjxZQChRvh8eqP6QRukFs4BA1Z4FTxYQWpRedpDpqy+keM4CAlV8FjxZQCBReLDsZxdS/DMLCFTxWfBkAQGiVN+OsalF8iQYzAIC1XsWPFlASFGqz7eom2pKejTJAgJ3kn+ioNp4QXD8q7gV7TS8wJhYE7KAQPBZ8GQBoUSpVfcop4RXhLOAQL0kC54sILQov6kQwb+UF4SzgNCKwDkLniwgtCiTeGvCiqnRVxI8ZwGB6j0LniwglCiFenIf1KCCtEonmAUEqu4seLKA0KIU+tEk5cEkCwitCJyz4MkCwojS/njwl/ZLkywgjCSw7PhPFFQbbwjWJbw16SPfmdTXO7h8t7guvxxiEwNuDFnwpAlRN03Twtt27+iakIx1X07Tb2Di3PrvVeo7xqDfln7C3g0RYghqkijPrG0lKErNB4xur39jpVxYatKV5bOpBc/qbu+fhVyK25sM/f55iABDnCbstTyozleDL4UYJj6STOWq/c4PqknMkhnZTb6RUzWnhPU/Gn8YW0+Ef/ECCJohThPoJo+LQvPgSl8LvxRi/IO37IVay15jMPGLMOUdgwkYffTF5wsgaIY4TZa7yT2i0Dym2WwIvFYUsD1i0hW+cdU7R6YNdUf/5HPQjWZJkbsgwgy7NVnsJs/z0Bp6V18sSlHufTuyBDPoOVcxDL96HJyBeGzauoLr4AlDRRdDz6dS9Z/0lBGVaJPMYhtdkFfX2y5sgaCMtAaRDHBxO0SQYb8mi91Ei/IYzzZJ4OONhZnjoxB1U/aoVW7iqaay58dPP8JX9oG6VavBBfS9bzmw0kYx1rBfCQ9DTcsC0R2fSg0TssImaZgTK78VRb/cGdwAETBykQEYHTOFVSGIAEOMJtDCXFGavjXCKlGe4tmuzJshGv6AsueY9fmq/9mZFu8G/Gig7zXg69KhR5W6Yx8+XftOoeied6oWrSjbJBByYjtbJi9v97EBIoKBzc3kdgUhwgy7NZnfiPm+DD99i1KiPMizUZ6ZKE9BFJ/SHB/xRQm3/0/Mcqb1Q8HOYODpxjZZ3eX4CPVB7Z0nhWtiZQ26TyD6j0+68DcuNsl/YCeOrfiY4/FsZpg3KwoiYOQSA2d0zHS5ZxA0A0+yW5M5D39oryZ1j1KiPMUTLcpzEEID808t6rEVwpuLq4Et3YRngtu7zHQUvpSt2o7CW9Mdua7VZPovT2STuLGiiclco/7dCLGbAQzeDEEyxGnidROYIdT6ayF974KKup9njzozUcqXQEyy2Z7YTfSkCypn1t55dTV82FFeokS0mFUN7lKvTeLGjnYE4znvPmaKOKOJhSCMtAaRiHJquA2CZIjTZN5NeH102sH7UZ49yniivAPiT0yLSj04byTyYNCkSz/Cd4V8FB/4SG+OD1+e+sIcj44Wzn3Sw89PIl3/TGzg8dekWwtsgKCNXGGYmSnNMGncWJIhThOqm8AKijyUKM/wrAmB4z1RYG+fF0B8+bpU5w/E2HQ/rF0K9Bk/m6g1zhEGqBaeHoQbpH1/yZfGKvjMPhDdQ/QoXGf8JIzZWG5Sv3cZe8ahjddnHyJg5AoDMnMVgmSI0wS6iXwS1GcYus2SuQw9xDOr98U/tfH6/BaIEdZ18Q6oG7Z3dVwfwQUSDjSaqHlE8RsGeIMgJIc7QinHlHEah6GHqTsdDZeboZdDm5eEJzKxUKZdgfAq/iQI2sg1BmzmGgTNEKGJ9EPV/qhSFOQ1JEV5ikfqEyfKyyBUU4vc3hV1E0b6RahlYTGt91o1dDOxkiAjxBMIBOkkOpYtvFi+CII2yNBgBvl8sgqxwGCyhfspv63gWxmKWwiOMNqOejGEEuVenjiKl0Go6pZz5f0bB4sXajoPM9QjDdUESUzR0WUdVNF8nb/QqzNeEieWOW/3dT7ifBWEZ5BTKrxWlIdjppfGjdXTbjcn969InOrTdV2rHwzRTdnk7tlmYkTgXJ44ipdB6AqK296Vz8LkOC/yadHQoDIuZHsoAitqKpqV8KSkP2Dxk+BYeGGibfbOF0H4Bjkla4YdEAsMKOs4HPYnXlaYfHxR7uWJo3gZhKlNGTi2vSvxWZZ0YOnK6UPeP5V/C+MPLnqa4CfBscyXfcZwcPdjH8I3yClRM+yAWGdAJRzThL2E5xjFSyBAlTO2d338I980IEJfh6K+IYNn4MglRy/rMy+s8JxBcX3LWoFQgpg9atf8YM+s35PzQhsHWwrKb/fkci/KjsShnZMvsuCMbC1F+pqg7V2tTyvpVnpGzV2UB4JAHsmU3+5FBpybLY2TsCi2ZSWqCdre1fq0kr6857aEU3NDEGHP3FNLvDQzGidhUWzLonypL63LkzJH27vCQqJYlQr48p5U4BXZIAjkkTzzzL2i4GvypHFIB+trDDglV0RhW1aqmjjbu1rHXKiovd4sp9RtXCYORNC9OC7vB1KFcMCUVEWxLYvypX6gjvcV6W7vuuYHuy/v237tQiCPZNe9+DZ7DhYUxKGdkw+WdlVyl8K2rCQ1cbd3NT6ttFvpVTV6NF8XghkK1zP3aCm3pQ/hpCxK6po4e9SSfrC3NY/4ghwI8V9Hco/kuXtxfP43p6RxHF/fmy2KKc6h0C0rVU3w9q4BP9iYOro3DYaARxPlkTx3L77XpgOl0Tiur++B7G9KiinS1wTtURvy5b2pXo8Ug/aonVEY9+Ij2d+dNoxztyVHyrMUOWhiaiLky2t+kETAUjieuUnYThhpcYjIVC5ZiPQ1CfnypqKFtBNROO7FaVEYaxGOuZZcAEEkr0nQlzcpVTCF416cFIUxFuOYi6kFMMRrNfkfH8iE8e/ql1wAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- 3 \\frac{\\partial}{\\partial px} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )} - \\frac{\\partial}{\\partial py} \\operatorname{H_{1}}{\\left (x,px,y,py \\right )} - 3 \\frac{\\partial}{\\partial px} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )} - \\frac{\\partial}{\\partial py} \\operatorname{H_{2}}{\\left (x,px,y,py \\right )}$$"
      ],
      "text/plain": [
       "     ∂                       ∂                         ∂                      \n",
       "- 3⋅───(H₁(x, px, y, py)) - ───(H₁(x, px, y, py)) - 3⋅───(H₂(x, px, y, py)) - \n",
       "    ∂px                     ∂py                       ∂px                     \n",
       "\n",
       " ∂                   \n",
       "───(H₂(x, px, y, py))\n",
       "∂py                  "
      ]
     },
     "execution_count": 502,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe12(3*x+y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 451,
   "metadata": {},
   "outputs": [],
   "source": [
    "loe12 = Mul(loe1,loe2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 453,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{:H1:} e^{:H2:}$$"
      ],
      "text/plain": [
       "LieOperator(H1(x, px, y, py), x, y, px, py)⋅LieOperator(H2(x, px, y, py), x, y\n",
       ", px, py)"
      ]
     },
     "execution_count": 453,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loe12.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 448,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'Mul' object has no attribute 'hamiltonian'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-448-82f7225ebecd>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mloe12\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhamiltonian\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m: 'Mul' object has no attribute 'hamiltonian'"
     ]
    }
   ],
   "source": [
    "loe12.hamiltonian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:py35]",
   "language": "python",
   "name": "conda-env-py35-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}