{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"toc": "true"
},
"source": [
"# Table of Contents\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Presentation\n",
"\n",
"I want to reproduce the symbolic (algebraic) computations done in §5.A of [L.S.'s PhD thesis](http://www.cmap.polytechnique.fr/~sacchelli/)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"I want to only use a free and open-source software, so I'm using [Python 3](https://docs.python.org/3) with the [Sympy](https://sympy.org) module."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPython 3.6.5\n",
"IPython 6.4.0\n",
"\n",
"sympy 1.1.1\n",
"\n",
"compiler : GCC 7.3.0\n",
"system : Linux\n",
"release : 4.15.0-23-generic\n",
"machine : x86_64\n",
"processor : x86_64\n",
"CPU cores : 4\n",
"interpreter: 64bit\n",
"Git hash : 3f62e1141f480e65f685cabe81f7a79b17bcd06d\n"
]
}
],
"source": [
"%load_ext watermark\n",
"%watermark -v -m -p sympy -g"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from sympy import *"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"init_printing(use_latex='mathjax')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Just a small introduction to SymPy: it works by using Python expressions on formal variables.\n",
"First, we define variables, and then we can solve linear equations for example:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left ( x, \\quad y, \\quad z\\right )$$"
],
"text/plain": [
"(x, y, z)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"x y z\")"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ -21\\right ]$$"
],
"text/plain": [
"[-21]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solve(x + 2 + 19)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ \\left \\{ x : - 2 y - 19\\right \\}\\right ]$$"
],
"text/plain": [
"[{x: -2⋅y - 19}]"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solve(x + 2*y + 19)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Computations"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We start by defining $b_1 > 0$ and the other variables."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"b1 = symbols(\"b1\", positive=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We need a lot of variables."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"var(\"h1 h2 h3 h4 k131 k132 k141 k142 k231 k232 k241 k242 rest1 rest2 t s a b\");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then we can start to follow Ludovic's notebook and define the first expressions."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}0 & - b_{1}\\\\b_{1} & 0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡0 -b₁⎤\n",
"⎢ ⎥\n",
"⎣b₁ 0 ⎦"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"J = Matrix([[0, -b1], [b1, 0]])\n",
"J"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}h_{1}\\\\h_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡h₁⎤\n",
"⎢ ⎥\n",
"⎣h₂⎦"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"h0 = Matrix([h1, h2])\n",
"h0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\hat{h}$ is defined as $t \\mapsto \\exp(t J) . h_0$ and $\\hat{x}$ as $t \\mapsto \\int_{s=0}^{s=t} \\hat{h}(s) \\mathrm{d}s$."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left( t \\mapsto \\left[\\begin{matrix}h_{1} \\left(\\frac{1}{2} e^{i b_{1} t} + \\frac{1}{2} e^{- i b_{1} t}\\right) + h_{2} \\left(\\frac{i}{2} e^{i b_{1} t} - \\frac{i}{2} e^{- i b_{1} t}\\right)\\\\h_{1} \\left(- \\frac{i}{2} e^{i b_{1} t} + \\frac{i}{2} e^{- i b_{1} t}\\right) + h_{2} \\left(\\frac{1}{2} e^{i b_{1} t} + \\frac{1}{2} e^{- i b_{1} t}\\right)\\end{matrix}\\right] \\right)$$"
],
"text/plain": [
" ⎡ ⎛ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t⎞ ⎛ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t⎞ ⎤\n",
" ⎢ ⎜ℯ ℯ ⎟ ⎜ⅈ⋅ℯ ⅈ⋅ℯ ⎟ ⎥\n",
" ⎢ h₁⋅⎜─────── + ────────⎟ + h₂⋅⎜───────── - ──────────⎟ ⎥\n",
" ⎢ ⎝ 2 2 ⎠ ⎝ 2 2 ⎠ ⎥\n",
"t ↦ ⎢ ⎥\n",
" ⎢ ⎛ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t⎞ ⎛ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t⎞⎥\n",
" ⎢ ⎜ ⅈ⋅ℯ ⅈ⋅ℯ ⎟ ⎜ℯ ℯ ⎟⎥\n",
" ⎢h₁⋅⎜- ───────── + ──────────⎟ + h₂⋅⎜─────── + ────────⎟⎥\n",
" ⎣ ⎝ 2 2 ⎠ ⎝ 2 2 ⎠⎦"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hhat = Lambda(t, (t * J).exp() * h0)\n",
"hhat"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left( t \\mapsto \\left[\\begin{matrix}- \\frac{h_{2}}{b_{1}} + \\frac{1}{4 b_{1}^{2}} \\left(\\left(- 2 i b_{1} h_{1} + 2 b_{1} h_{2}\\right) e^{i b_{1} t} + \\left(2 i b_{1} h_{1} + 2 b_{1} h_{2}\\right) e^{- i b_{1} t}\\right)\\\\\\frac{h_{1}}{b_{1}} + \\frac{1}{4 b_{1}^{2}} \\left(\\left(- 2 b_{1} h_{1} - 2 i b_{1} h_{2}\\right) e^{i b_{1} t} + \\left(- 2 b_{1} h_{1} + 2 i b_{1} h_{2}\\right) e^{- i b_{1} t}\\right)\\end{matrix}\\right] \\right)$$"
],
"text/plain": [
" ⎡ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t⎤\n",
" ⎢ h₂ (-2⋅ⅈ⋅b₁⋅h₁ + 2⋅b₁⋅h₂)⋅ℯ + (2⋅ⅈ⋅b₁⋅h₁ + 2⋅b₁⋅h₂)⋅ℯ ⎥\n",
" ⎢- ── + ───────────────────────────────────────────────────────────────⎥\n",
" ⎢ b₁ 2 ⎥\n",
" ⎢ 4⋅b₁ ⎥\n",
"t ↦ ⎢ ⎥\n",
" ⎢ ⅈ⋅b₁⋅t -ⅈ⋅b₁⋅t ⎥\n",
" ⎢h₁ (-2⋅b₁⋅h₁ - 2⋅ⅈ⋅b₁⋅h₂)⋅ℯ + (-2⋅b₁⋅h₁ + 2⋅ⅈ⋅b₁⋅h₂)⋅ℯ ⎥\n",
" ⎢── + ──────────────────────────────────────────────────────────────── ⎥\n",
" ⎢b₁ 2 ⎥\n",
" ⎣ 4⋅b₁ ⎦"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xhat = Lambda(t, integrate(hhat(s), (s, 0, t)))\n",
"xhat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\hat{z}$ is slightly more complex:"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{h_{1}^{2}}{4} e^{i b_{1} s} - \\frac{h_{1}^{2}}{2} + \\frac{h_{1}^{2}}{4} e^{- i b_{1} s} + \\frac{h_{2}^{2}}{4} e^{i b_{1} s} - \\frac{h_{2}^{2}}{2} + \\frac{h_{2}^{2}}{4} e^{- i b_{1} s}$$"
],
"text/plain": [
" 2 ⅈ⋅b₁⋅s 2 2 -ⅈ⋅b₁⋅s 2 ⅈ⋅b₁⋅s 2 2 -ⅈ⋅b₁⋅s\n",
"h₁ ⋅ℯ h₁ h₁ ⋅ℯ h₂ ⋅ℯ h₂ h₂ ⋅ℯ \n",
"─────────── - ─── + ──────────── + ─────────── - ─── + ────────────\n",
" 4 2 4 4 2 4 "
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"integrand = simplify(b1 * (hhat(s)[0] * xhat(s)[1] - hhat(s)[1] * xhat(s)[0] )/2)\n",
"integrand"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left( t \\mapsto t \\left(- \\frac{h_{1}^{2}}{2} - \\frac{h_{2}^{2}}{2}\\right) + \\frac{1}{16 b_{1}^{2}} \\left(\\left(- 4 i b_{1} h_{1}^{2} - 4 i b_{1} h_{2}^{2}\\right) e^{i b_{1} t} + \\left(4 i b_{1} h_{1}^{2} + 4 i b_{1} h_{2}^{2}\\right) e^{- i b_{1} t}\\right) \\right)$$"
],
"text/plain": [
" ⎛ 2 2⎞ ⎛ 2 2⎞ ⅈ⋅b₁⋅t ⎛ 2 \n",
" ⎜ h₁ h₂ ⎟ ⎝- 4⋅ⅈ⋅b₁⋅h₁ - 4⋅ⅈ⋅b₁⋅h₂ ⎠⋅ℯ + ⎝4⋅ⅈ⋅b₁⋅h₁ + 4⋅ⅈ⋅\n",
"t ↦ t⋅⎜- ─── - ───⎟ + ────────────────────────────────────────────────────────\n",
" ⎝ 2 2 ⎠ 2 \n",
" 16⋅b₁ \n",
"\n",
" 2⎞ -ⅈ⋅b₁⋅t\n",
"b₁⋅h₂ ⎠⋅ℯ \n",
"────────────────\n",
" \n",
" "
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"zhat = Lambda(t, integrate(integrand, (s, 0, t)))\n",
"zhat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we asked $b_1 > 0$, this won't get too complicated:"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\frac{e^{- i b_{1} t}}{4 b_{1}} \\left(h_{1}^{2} + h_{2}^{2}\\right) \\left(2 b_{1} t e^{i b_{1} t} + i e^{2 i b_{1} t} - i\\right)$$"
],
"text/plain": [
" ⎛ 2 2⎞ ⎛ ⅈ⋅b₁⋅t 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t \n",
"-⎝h₁ + h₂ ⎠⋅⎝2⋅b₁⋅t⋅ℯ + ⅈ⋅ℯ - ⅈ⎠⋅ℯ \n",
"─────────────────────────────────────────────────────────\n",
" 4⋅b₁ "
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"factor(zhat(t))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Two more expressions:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
"exp21 = (3 * a * h1**2 + a * h2**2 + 2 * b * h1 * h2 + k131 * h1 * h3 + k141 * h1 * h4\n",
" + k231 * h2 * h3 + k241 * h2 * h4 + rest1(h3, h4))"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"exp22 = (b * h1**2 + 3 * b * h2**2 + 2 * a * h1 * h2 + k132 * h1 * h3 + k142 * h1 * h4\n",
" + k232 * h2 * h3 + k242 * h2 * h4 + rest2(h3, h4))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Both terms `rest1` and `rest2` are not important, they only depend on $h_3$ and $h_4$ and will vanish as soon as we only differentiate with respect to $h_1$ and $h_2$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So far so good. Next cell:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}\\frac{2 \\pi}{b_{1}} h_{1}\\\\\\frac{2 \\pi}{b_{1}} h_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡2⋅π⋅h₁⎤\n",
"⎢──────⎥\n",
"⎢ b₁ ⎥\n",
"⎢ ⎥\n",
"⎢2⋅π⋅h₂⎥\n",
"⎢──────⎥\n",
"⎣ b₁ ⎦"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C10 = 2 * (pi / b1) * Matrix([h1, h2])\n",
"C10"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}6 a h_{1} + 2 b h_{2} + h_{3} k_{131} + h_{4} k_{141} & 2 a h_{2} + 2 b h_{1} + h_{3} k_{231} + h_{4} k_{241}\\\\2 a h_{2} + 2 b h_{1} + h_{3} k_{132} + h_{4} k_{142} & 2 a h_{1} + 6 b h_{2} + h_{3} k_{232} + h_{4} k_{242}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡6⋅a⋅h₁ + 2⋅b⋅h₂ + h₃⋅k₁₃₁ + h₄⋅k₁₄₁ 2⋅a⋅h₂ + 2⋅b⋅h₁ + h₃⋅k₂₃₁ + h₄⋅k₂₄₁⎤\n",
"⎢ ⎥\n",
"⎣2⋅a⋅h₂ + 2⋅b⋅h₁ + h₃⋅k₁₃₂ + h₄⋅k₁₄₂ 2⋅a⋅h₁ + 6⋅b⋅h₂ + h₃⋅k₂₃₂ + h₄⋅k₂₄₂⎦"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A0 = simplify(Matrix([\n",
" [diff(exp21, h1), diff(exp21, h2)],\n",
" [diff(exp22, h1), diff(exp22, h2)],\n",
"]))\n",
"A0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`rest1` and `rest2` already vanished."
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}6 a h_{1} + 2 b h_{2} + h_{3} k_{131} + h_{4} k_{141} & 2 a h_{2} + 2 b h_{1} + h_{3} k_{231} + h_{4} k_{241} & \\frac{2 \\pi}{b_{1}} h_{1}\\\\2 a h_{2} + 2 b h_{1} + h_{3} k_{132} + h_{4} k_{142} & 2 a h_{1} + 6 b h_{2} + h_{3} k_{232} + h_{4} k_{242} & \\frac{2 \\pi}{b_{1}} h_{2}\\\\- \\frac{2 \\pi}{b_{1}} h_{1} & - \\frac{2 \\pi}{b_{1}} h_{2} & 0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2⋅π\n",
"⎢6⋅a⋅h₁ + 2⋅b⋅h₂ + h₃⋅k₁₃₁ + h₄⋅k₁₄₁ 2⋅a⋅h₂ + 2⋅b⋅h₁ + h₃⋅k₂₃₁ + h₄⋅k₂₄₁ ───\n",
"⎢ b\n",
"⎢ \n",
"⎢ 2⋅π\n",
"⎢2⋅a⋅h₂ + 2⋅b⋅h₁ + h₃⋅k₁₃₂ + h₄⋅k₁₄₂ 2⋅a⋅h₁ + 6⋅b⋅h₂ + h₃⋅k₂₃₂ + h₄⋅k₂₄₂ ───\n",
"⎢ b\n",
"⎢ \n",
"⎢ -2⋅π⋅h₁ -2⋅π⋅h₂ \n",
"⎢ ──────── ──────── 0\n",
"⎣ b₁ b₁ \n",
"\n",
"⋅h₁⎤\n",
"───⎥\n",
"₁ ⎥\n",
" ⎥\n",
"⋅h₂⎥\n",
"───⎥\n",
"₁ ⎥\n",
" ⎥\n",
" ⎥\n",
" ⎥\n",
" ⎦"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"jacobian = simplify(Matrix([\n",
" [diff(exp21, h1), diff(exp21, h2), C10[0]],\n",
" [diff(exp22, h1), diff(exp22, h2), C10[1]],\n",
" [diff(zhat(2 * pi / b1), h1), diff(zhat(2 * pi / b1), h2), 0],\n",
"]))\n",
"jacobian"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We need one more variable to solve an equation."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$dt$$"
],
"text/plain": [
"dt"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var('dt')"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [],
"source": [
"tc = factor(solve(Equality((jacobian + Matrix([[dt,0,0], [0,dt,0], [0,0,0]])).det(), 0), dt)[0])"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\frac{1}{h_{1}^{2} + h_{2}^{2}} \\left(2 a h_{1}^{3} + 2 a h_{1} h_{2}^{2} + 2 b h_{1}^{2} h_{2} + 2 b h_{2}^{3} + h_{1}^{2} h_{3} k_{232} + h_{1}^{2} h_{4} k_{242} - h_{1} h_{2} h_{3} k_{132} - h_{1} h_{2} h_{3} k_{231} - h_{1} h_{2} h_{4} k_{142} - h_{1} h_{2} h_{4} k_{241} + h_{2}^{2} h_{3} k_{131} + h_{2}^{2} h_{4} k_{141}\\right)$$"
],
"text/plain": [
" ⎛ 3 2 2 3 2 2 \n",
"-⎝2⋅a⋅h₁ + 2⋅a⋅h₁⋅h₂ + 2⋅b⋅h₁ ⋅h₂ + 2⋅b⋅h₂ + h₁ ⋅h₃⋅k₂₃₂ + h₁ ⋅h₄⋅k₂₄₂ - h₁\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" 2 \n",
"⋅h₂⋅h₃⋅k₁₃₂ - h₁⋅h₂⋅h₃⋅k₂₃₁ - h₁⋅h₂⋅h₄⋅k₁₄₂ - h₁⋅h₂⋅h₄⋅k₂₄₁ + h₂ ⋅h₃⋅k₁₃₁ + h₂\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 2 2 \n",
" h₁ + h₂ \n",
"\n",
"2 ⎞ \n",
" ⋅h₄⋅k₁₄₁⎠ \n",
"───────────\n",
" \n",
" "
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tc"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"I can compare by copying the result from the document:"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"tc2 = (-1 / (h1**2 + h2**2)) * (\n",
" 2 * a * h1**3\n",
" + 6 * a * h1**2 * h2\n",
" - 4 * b * h1**2 * h2\n",
" + 2 * a * h1 * h2**2\n",
" + 2 * b * h2**3\n",
" + h2**2 * h3 * k131\n",
" - h1 * h2 * h3 * k132\n",
" + h2**2 * h4 * k141\n",
" - h1 * h2 * h4 * k142\n",
" - h1 * h2 * h3 * k231\n",
" + h1**2 * h3 * k232\n",
" - h1 * h2 * h4 * k241\n",
" + h1**2 * h4 * k242 )"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Drat, they are not equal! We might have a mistake somewhere, even though I just CAN'T find it!"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{6 h_{1}^{2} h_{2} \\left(a - b\\right)}{h_{1}^{2} + h_{2}^{2}}$$"
],
"text/plain": [
" 2 \n",
"6⋅h₁ ⋅h₂⋅(a - b)\n",
"────────────────\n",
" 2 2 \n",
" h₁ + h₂ "
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(tc - tc2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's use the one from Ludovic's notebook."
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [],
"source": [
"tc = tc2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next cell."
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [],
"source": [
"A12 = simplify(A0 + Matrix([[tc, 0], [0, tc]]))"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left ( u_{1}, \\quad u_{2}, \\quad u_{5}\\right )$$"
],
"text/plain": [
"(u₁, u₂, u₅)"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"u1 u2 u5\")"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [],
"source": [
"Psi = Lambda((u1, u2, u5), simplify(\n",
" u5 * Matrix(A12.dot(Matrix([h1, h2]))).dot(Matrix([h2, -h1])) + 2 * pi / b1 * ( h1**2 + h2**2) * (h1 * u2 - h2 * u1)\n",
"))"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\frac{1}{b_{1}} \\left(- 2 a b_{1} h_{1}^{2} h_{2} u_{5} - 2 a b_{1} h_{2}^{3} u_{5} + 2 b b_{1} h_{1}^{3} u_{5} + 2 b b_{1} h_{1} h_{2}^{2} u_{5} + b_{1} h_{1}^{2} h_{3} k_{132} u_{5} + b_{1} h_{1}^{2} h_{4} k_{142} u_{5} - b_{1} h_{1} h_{2} h_{3} k_{131} u_{5} + b_{1} h_{1} h_{2} h_{3} k_{232} u_{5} - b_{1} h_{1} h_{2} h_{4} k_{141} u_{5} + b_{1} h_{1} h_{2} h_{4} k_{242} u_{5} - b_{1} h_{2}^{2} h_{3} k_{231} u_{5} - b_{1} h_{2}^{2} h_{4} k_{241} u_{5} - 2 \\pi h_{1}^{3} u_{2} + 2 \\pi h_{1}^{2} h_{2} u_{1} - 2 \\pi h_{1} h_{2}^{2} u_{2} + 2 \\pi h_{2}^{3} u_{1}\\right)$$"
],
"text/plain": [
" ⎛ 2 3 3 2 \n",
"-⎝- 2⋅a⋅b₁⋅h₁ ⋅h₂⋅u₅ - 2⋅a⋅b₁⋅h₂ ⋅u₅ + 2⋅b⋅b₁⋅h₁ ⋅u₅ + 2⋅b⋅b₁⋅h₁⋅h₂ ⋅u₅ + b₁⋅h\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
" 2 2 \n",
"₁ ⋅h₃⋅k₁₃₂⋅u₅ + b₁⋅h₁ ⋅h₄⋅k₁₄₂⋅u₅ - b₁⋅h₁⋅h₂⋅h₃⋅k₁₃₁⋅u₅ + b₁⋅h₁⋅h₂⋅h₃⋅k₂₃₂⋅u₅ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" b₁ \n",
"\n",
" 2 2 \n",
"- b₁⋅h₁⋅h₂⋅h₄⋅k₁₄₁⋅u₅ + b₁⋅h₁⋅h₂⋅h₄⋅k₂₄₂⋅u₅ - b₁⋅h₂ ⋅h₃⋅k₂₃₁⋅u₅ - b₁⋅h₂ ⋅h₄⋅k₂\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
"\n",
" 3 2 2 3 ⎞ \n",
"₄₁⋅u₅ - 2⋅π⋅h₁ ⋅u₂ + 2⋅π⋅h₁ ⋅h₂⋅u₁ - 2⋅π⋅h₁⋅h₂ ⋅u₂ + 2⋅π⋅h₂ ⋅u₁⎠ \n",
"─────────────────────────────────────────────────────────────────\n",
" "
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"my_psi = factor(simplify(Psi(u1, u2, u5)))\n",
"my_psi"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's compare with the value from Ludovic's notebook:"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [],
"source": [
"his_psi = (1 / b1) * (\n",
" 2 * h1**3 * (pi * u2 - b * b1 * u5 ) -\n",
" h1**2 * (2 * h2 * pi * u1 - 2 * a * b1 * h2 * u5 + b1 * h3 * k132 * u5 +\n",
" b1 * h4 * k142 * u5 ) +\n",
" h2**2 * ( -2 * h2 * pi * u1 + 2 * a * b1 * h2 * u5 + b1 * h3 * k231 * u5 +\n",
" b1 * h4 * k241 * u5 ) +\n",
" h1 * h2 * (b1 * (h3 * k131 + h4 * k141 - h3 * k232 - h4 * k242) * u5 +\n",
" 2 * h2 * (pi * u2 - b * b1 * u5))\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$0$$"
],
"text/plain": [
"0"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(my_psi - his_psi)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Ok, we have the same result so far! Great!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next cell."
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\nu_{0}$$"
],
"text/plain": [
"ν₀"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"nu0\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here again, Sympy fails to solve the equation. It can solve on both lines separately, but cannot find a solution that satisfies both lines."
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
"s1 = solve(Eq((Matrix(A12.dot(Matrix([-h2, h1]))) + nu0 * C10)[0]), nu0)[0]"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [],
"source": [
"s2 = solve(Eq((Matrix(A12.dot(Matrix([-h2, h1]))) + nu0 * C10)[1]), nu0)[0]"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ \\left \\{ a : b\\right \\}, \\quad \\left \\{ h_{1} : 0\\right \\}\\right ]$$"
],
"text/plain": [
"[{a: b}, {h₁: 0}]"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solve(Eq(s1, s2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is not possible to impose these constraints. And Sympy fails to solve both equation simultaneously:"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left [ \\right ]$$"
],
"text/plain": [
"[]"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solve(Eq(Matrix(A12.dot(Matrix([-h2, h1]))) + nu0 * C10, Matrix([0,0])), nu0)"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [],
"source": [
"nu = simplify(solve(Eq(Matrix(A12.dot(Matrix([-h2, h1]))) + nu0 * C10, Matrix([0,0])), nu0))\n",
"if nu == []:\n",
" nu = var(\"nu\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can continue by using a formal variable for $\\nu$ (`nu`)."
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [],
"source": [
"v = Matrix([nu, -h2, h1])"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}\\nu\\\\- h_{2}\\\\h_{1}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ ν ⎤\n",
"⎢ ⎥\n",
"⎢-h₂⎥\n",
"⎢ ⎥\n",
"⎣h₁ ⎦"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's finish."
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [],
"source": [
"var(\"eta f F\");"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [],
"source": [
"d = [\n",
" lambda F: -eta**2 * diff(F, eta) + eta * t * diff(F, t),\n",
" lambda F: diff(F, h1),\n",
" lambda F: diff(F, h2)\n",
"]"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(F)>,\n",
" (F)>,\n",
" (F)>]"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next cell."
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [],
"source": [
"var(\"i1 i2 g1 g2 dt1 dt2\");"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\eta^{2} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\eta \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}$$"
],
"text/plain": [
" 2 \n",
"η ⋅g₂(t, h₁, h₂) + η⋅g₁(dt₁⋅η + t, h₁, h₂)"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g = eta * g1(t + eta * dt1, h1, h2) + eta**2 * g2(t, h1, h2)\n",
"g"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have a sum to compute:"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [],
"source": [
"res = 0\n",
"for i1 in range(0, 3):\n",
" for i2 in range(0, 3):\n",
" res += v[i1] * v[i2] * d[i1](d[i2](g))"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\eta \\left(\\eta^{2} \\nu^{2} \\left(2 dt_{1} \\eta \\left. \\frac{\\partial}{\\partial \\xi_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + \\eta \\left(dt_{1}^{2} \\eta \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + 2 dt_{1} \\left. \\frac{\\partial}{\\partial \\xi_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + 2 \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) + 4 \\eta \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} - t \\left(\\eta \\frac{\\partial}{\\partial t} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\left. \\frac{\\partial}{\\partial \\xi_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }}\\right) + t \\left(- dt_{1} \\eta \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} - \\eta \\frac{\\partial}{\\partial t} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + t \\left(\\eta \\frac{\\partial^{2}}{\\partial t^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }}\\right)\\right) - t \\left(dt_{1} \\eta \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + 2 \\eta \\frac{\\partial}{\\partial t} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\left. \\frac{\\partial}{\\partial \\xi_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }}\\right) + 2 \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right) - 2 \\eta h_{1} \\nu \\left(dt_{1} \\eta \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + 2 \\eta \\frac{\\partial}{\\partial h_{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} - t \\left(\\eta \\frac{\\partial^{2}}{\\partial h_{2}\\partial t} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }}\\right) + \\frac{\\partial}{\\partial h_{2}} \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right) + 2 \\eta h_{2} \\nu \\left(dt_{1} \\eta \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }} + 2 \\eta \\frac{\\partial}{\\partial h_{1}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} - t \\left(\\eta \\frac{\\partial^{2}}{\\partial h_{1}\\partial t} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=dt_{1} \\eta + t }}\\right) + \\frac{\\partial}{\\partial h_{1}} \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right) + h_{1}^{2} \\left(\\eta \\frac{\\partial^{2}}{\\partial h_{2}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\frac{\\partial^{2}}{\\partial h_{2}^{2}} \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right) - 2 h_{1} h_{2} \\left(\\eta \\frac{\\partial^{2}}{\\partial h_{1}\\partial h_{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\frac{\\partial^{2}}{\\partial h_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right) + h_{2}^{2} \\left(\\eta \\frac{\\partial^{2}}{\\partial h_{1}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )} + \\frac{\\partial^{2}}{\\partial h_{1}^{2}} \\operatorname{g_{1}}{\\left (dt_{1} \\eta + t,h_{1},h_{2} \\right )}\\right)\\right)$$"
],
"text/plain": [
" ⎛ ⎛ ⎛ ⎛ 2 \n",
" ⎜ 2 2 ⎜ ⎛ ∂ ⎞│ ⎜ 2 ⎜ ∂ \n",
"η⋅⎜η ⋅ν ⋅⎜2⋅dt₁⋅η⋅⎜───(g₁(ξ₁, h₁, h₂))⎟│ + η⋅⎜dt₁ ⋅η⋅⎜────(g₁(ξ₁, \n",
" ⎜ ⎜ ⎝∂ξ₁ ⎠│ξ₁=dt₁⋅η + t ⎜ ⎜ 2 \n",
" ⎝ ⎝ ⎝ ⎝∂ξ₁ \n",
"\n",
" ⎞│ \n",
" ⎟│ ⎛ ∂ ⎞│ \n",
"h₁, h₂))⎟│ + 2⋅dt₁⋅⎜───(g₁(ξ₁, h₁, h₂))⎟│ + 2⋅g₂(t, h₁\n",
" ⎟│ ⎝∂ξ₁ ⎠│ξ₁=dt₁⋅η + t \n",
" ⎠│ξ₁=dt₁⋅η + t \n",
"\n",
" ⎞ \n",
" ⎟ ⎛ ∂ ⎛ ∂ ⎞│ \n",
", h₂)⎟ + 4⋅η⋅g₂(t, h₁, h₂) - t⋅⎜η⋅──(g₂(t, h₁, h₂)) + ⎜───(g₁(ξ₁, h₁, h₂))⎟│ \n",
" ⎟ ⎝ ∂t ⎝∂ξ₁ ⎠│ξ₁\n",
" ⎠ \n",
"\n",
" ⎛ ⎛ 2 ⎞│ \n",
" ⎞ ⎜ ⎜ ∂ ⎟│ ∂ \n",
" ⎟ + t⋅⎜- dt₁⋅η⋅⎜────(g₁(ξ₁, h₁, h₂))⎟│ - η⋅──(g₂(t, h₁, \n",
"=dt₁⋅η + t⎠ ⎜ ⎜ 2 ⎟│ ∂t \n",
" ⎝ ⎝∂ξ₁ ⎠│ξ₁=dt₁⋅η + t \n",
"\n",
" ⎛ 2 ⎛ 2 ⎞│ ⎞⎞ ⎛ \n",
" ⎜ ∂ ⎜ ∂ ⎟│ ⎟⎟ ⎜ \n",
"h₂)) + t⋅⎜η⋅───(g₂(t, h₁, h₂)) + ⎜────(g₁(ξ₁, h₁, h₂))⎟│ ⎟⎟ - t⋅⎜dt\n",
" ⎜ 2 ⎜ 2 ⎟│ ⎟⎟ ⎜ \n",
" ⎝ ∂t ⎝∂ξ₁ ⎠│ξ₁=dt₁⋅η + t⎠⎠ ⎝ \n",
"\n",
" ⎛ 2 ⎞│ \n",
" ⎜ ∂ ⎟│ ∂ ⎛ ∂ \n",
"₁⋅η⋅⎜────(g₁(ξ₁, h₁, h₂))⎟│ + 2⋅η⋅──(g₂(t, h₁, h₂)) + ⎜───(g₁(ξ₁, \n",
" ⎜ 2 ⎟│ ∂t ⎝∂ξ₁ \n",
" ⎝∂ξ₁ ⎠│ξ₁=dt₁⋅η + t \n",
"\n",
" ⎞ ⎞ ⎛ ⎛ 2 \n",
" ⎞│ ⎟ ⎟ ⎜ ⎜ ∂ \n",
"h₁, h₂))⎟│ ⎟ + 2⋅g₁(dt₁⋅η + t, h₁, h₂)⎟ - 2⋅η⋅h₁⋅ν⋅⎜dt₁⋅η⋅⎜───────(\n",
" ⎠│ξ₁=dt₁⋅η + t⎟ ⎟ ⎝ ⎝∂h₂ ∂ξ₁ \n",
" ⎠ ⎠ \n",
"\n",
" ⎞│ ⎛ 2 \n",
" ⎟│ ∂ ⎜ ∂ \n",
"g₁(ξ₁, h₁, h₂))⎟│ + 2⋅η⋅───(g₂(t, h₁, h₂)) - t⋅⎜η⋅──────(g₂(t, h₁,\n",
" ⎠│ξ₁=dt₁⋅η + t ∂h₂ ⎝ ∂t ∂h₂ \n",
" \n",
"\n",
" ⎛ 2 ⎞│ ⎞ ⎞ \n",
" ⎜ ∂ ⎟│ ⎟ ∂ ⎟ \n",
" h₂)) + ⎜───────(g₁(ξ₁, h₁, h₂))⎟│ ⎟ + ───(g₁(dt₁⋅η + t, h₁, h₂))⎟ \n",
" ⎝∂h₂ ∂ξ₁ ⎠│ξ₁=dt₁⋅η + t⎠ ∂h₂ ⎠ \n",
" \n",
"\n",
" ⎛ ⎛ 2 ⎞│ \n",
" ⎜ ⎜ ∂ ⎟│ ∂ \n",
"+ 2⋅η⋅h₂⋅ν⋅⎜dt₁⋅η⋅⎜───────(g₁(ξ₁, h₁, h₂))⎟│ + 2⋅η⋅───(g₂(t, h₁, h\n",
" ⎝ ⎝∂h₁ ∂ξ₁ ⎠│ξ₁=dt₁⋅η + t ∂h₁ \n",
" \n",
"\n",
" ⎛ 2 ⎛ 2 ⎞│ ⎞ \n",
" ⎜ ∂ ⎜ ∂ ⎟│ ⎟ \n",
"₂)) - t⋅⎜η⋅──────(g₂(t, h₁, h₂)) + ⎜───────(g₁(ξ₁, h₁, h₂))⎟│ ⎟ + ─\n",
" ⎝ ∂t ∂h₁ ⎝∂h₁ ∂ξ₁ ⎠│ξ₁=dt₁⋅η + t⎠ ∂\n",
" \n",
"\n",
" ⎞ ⎛ 2 2 \n",
"∂ ⎟ 2 ⎜ ∂ ∂ \n",
"──(g₁(dt₁⋅η + t, h₁, h₂))⎟ + h₁ ⋅⎜η⋅────(g₂(t, h₁, h₂)) + ────(g₁(dt₁⋅η + t, h\n",
"h₁ ⎠ ⎜ 2 2 \n",
" ⎝ ∂h₂ ∂h₂ \n",
"\n",
" ⎞ ⎛ 2 2 ⎞\n",
" ⎟ ⎜ ∂ ∂ ⎟\n",
"₁, h₂))⎟ - 2⋅h₁⋅h₂⋅⎜η⋅───────(g₂(t, h₁, h₂)) + ───────(g₁(dt₁⋅η + t, h₁, h₂))⎟\n",
" ⎟ ⎝ ∂h₂ ∂h₁ ∂h₂ ∂h₁ ⎠\n",
" ⎠ \n",
"\n",
" ⎛ 2 2 ⎞⎞\n",
" 2 ⎜ ∂ ∂ ⎟⎟\n",
" + h₂ ⋅⎜η⋅────(g₂(t, h₁, h₂)) + ────(g₁(dt₁⋅η + t, h₁, h₂))⎟⎟\n",
" ⎜ 2 2 ⎟⎟\n",
" ⎝ ∂h₁ ∂h₁ ⎠⎠"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then a double derivative"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [],
"source": [
"d2f = simplify((diff(res, eta, eta) / 2).subs(eta, 0))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, we should replace $g_1$ and $g_2$ with the following expression:"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [],
"source": [
"g1 = Lambda((t, h1, h2), simplify(Matrix(list(xhat(t)) + [0])))"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}\\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(i h_{1} - 2 h_{2} e^{i b_{1} t} + h_{2} + \\left(- i h_{1} + h_{2}\\right) e^{2 i b_{1} t}\\right)\\\\\\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(2 h_{1} e^{i b_{1} t} - h_{1} + i h_{2} - \\left(h_{1} + i h_{2}\\right) e^{2 i b_{1} t}\\right)\\\\0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡⎛ ⅈ⋅b₁⋅t 2⋅ⅈ⋅b₁⋅t⎞ -ⅈ⋅b₁⋅t⎤\n",
"⎢⎝ⅈ⋅h₁ - 2⋅h₂⋅ℯ + h₂ + (-ⅈ⋅h₁ + h₂)⋅ℯ ⎠⋅ℯ ⎥\n",
"⎢────────────────────────────────────────────────────────────⎥\n",
"⎢ 2⋅b₁ ⎥\n",
"⎢ ⎥\n",
"⎢⎛ ⅈ⋅b₁⋅t 2⋅ⅈ⋅b₁⋅t⎞ -ⅈ⋅b₁⋅t ⎥\n",
"⎢⎝2⋅h₁⋅ℯ - h₁ + ⅈ⋅h₂ - (h₁ + ⅈ⋅h₂)⋅ℯ ⎠⋅ℯ ⎥\n",
"⎢─────────────────────────────────────────────────────────── ⎥\n",
"⎢ 2⋅b₁ ⎥\n",
"⎢ ⎥\n",
"⎣ 0 ⎦"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g1(t, h1, h2)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [],
"source": [
"g2 = Lambda((t, h1, h2), simplify(Matrix([exp21, exp22] + [zhat(t)])))"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}3 a h_{1}^{2} + a h_{2}^{2} + 2 b h_{1} h_{2} + h_{1} h_{3} k_{131} + h_{1} h_{4} k_{141} + h_{2} h_{3} k_{231} + h_{2} h_{4} k_{241} + \\operatorname{rest_{1}}{\\left (h_{3},h_{4} \\right )}\\\\2 a h_{1} h_{2} + b h_{1}^{2} + 3 b h_{2}^{2} + h_{1} h_{3} k_{132} + h_{1} h_{4} k_{142} + h_{2} h_{3} k_{232} + h_{2} h_{4} k_{242} + \\operatorname{rest_{2}}{\\left (h_{3},h_{4} \\right )}\\\\\\frac{e^{- i b_{1} t}}{4 b_{1}} \\left(- 2 b_{1} t \\left(h_{1}^{2} + h_{2}^{2}\\right) e^{i b_{1} t} + i h_{1}^{2} + i h_{2}^{2} - i \\left(h_{1}^{2} + h_{2}^{2}\\right) e^{2 i b_{1} t}\\right)\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 2 \n",
"⎢3⋅a⋅h₁ + a⋅h₂ + 2⋅b⋅h₁⋅h₂ + h₁⋅h₃⋅k₁₃₁ + h₁⋅h₄⋅k₁₄₁ + h₂⋅h₃⋅k₂₃₁ + h₂⋅h₄⋅k₂\n",
"⎢ \n",
"⎢ 2 2 \n",
"⎢2⋅a⋅h₁⋅h₂ + b⋅h₁ + 3⋅b⋅h₂ + h₁⋅h₃⋅k₁₃₂ + h₁⋅h₄⋅k₁₄₂ + h₂⋅h₃⋅k₂₃₂ + h₂⋅h₄⋅k₂\n",
"⎢ \n",
"⎢ ⎛ ⎛ 2 2⎞ ⅈ⋅b₁⋅t 2 2 ⎛ 2 2⎞ 2⋅ⅈ⋅b₁⋅\n",
"⎢ ⎝- 2⋅b₁⋅t⋅⎝h₁ + h₂ ⎠⋅ℯ + ⅈ⋅h₁ + ⅈ⋅h₂ - ⅈ⋅⎝h₁ + h₂ ⎠⋅ℯ \n",
"⎢ ──────────────────────────────────────────────────────────────────────\n",
"⎣ 4⋅b₁ \n",
"\n",
" ⎤\n",
"₄₁ + rest₁(h₃, h₄)⎥\n",
" ⎥\n",
" ⎥\n",
"₄₂ + rest₂(h₃, h₄)⎥\n",
" ⎥\n",
"t⎞ -ⅈ⋅b₁⋅t ⎥\n",
" ⎠⋅ℯ ⎥\n",
"─────────── ⎥\n",
" ⎦"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g2(t, h1, h2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see if the replacement is possible:"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/latex": [
"$$h_{1}^{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{2}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{2}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) - 2 h_{1} h_{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{1}\\partial h_{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) + 2 h_{1} \\nu \\left(t \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} - \\frac{\\partial}{\\partial h_{2}} \\operatorname{g_{1}}{\\left (t,h_{1},h_{2} \\right )}\\right) + h_{2}^{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{1}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) - 2 h_{2} \\nu \\left(t \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} - \\frac{\\partial}{\\partial h_{1}} \\operatorname{g_{1}}{\\left (t,h_{1},h_{2} \\right )}\\right)$$"
],
"text/plain": [
" ⎛ ⎛ 3 ⎞│ 2 ⎞ ⎛ \n",
" 2 ⎜ ⎜ ∂ ⎟│ ∂ ⎟ ⎜ \n",
"h₁ ⋅⎜dt₁⋅⎜────────(g₁(ξ₁, h₁, h₂))⎟│ + ────(g₂(t, h₁, h₂))⎟ - 2⋅h₁⋅h₂⋅⎜dt₁\n",
" ⎜ ⎜ 2 ⎟│ 2 ⎟ ⎝ \n",
" ⎝ ⎝∂h₂ ∂ξ₁ ⎠│ξ₁=t ∂h₂ ⎠ \n",
"\n",
" ⎛ 3 ⎞│ 2 ⎞ ⎛ ⎛ \n",
" ⎜ ∂ ⎟│ ∂ ⎟ ⎜ ⎜ \n",
"⋅⎜───────────(g₁(ξ₁, h₁, h₂))⎟│ + ───────(g₂(t, h₁, h₂))⎟ + 2⋅h₁⋅ν⋅⎜t⋅⎜───\n",
" ⎝∂h₂ ∂h₁ ∂ξ₁ ⎠│ξ₁=t ∂h₂ ∂h₁ ⎠ ⎝ ⎝∂h₂\n",
" \n",
"\n",
" 2 ⎞│ ⎞ ⎛ ⎛ 3 \n",
"∂ ⎟│ ∂ ⎟ 2 ⎜ ⎜ ∂ \n",
"────(g₁(ξ₁, h₁, h₂))⎟│ - ───(g₁(t, h₁, h₂))⎟ + h₂ ⋅⎜dt₁⋅⎜────────(g₁(ξ₁, h\n",
" ∂ξ₁ ⎠│ξ₁=t ∂h₂ ⎠ ⎜ ⎜ 2 \n",
" ⎝ ⎝∂h₁ ∂ξ₁ \n",
"\n",
" ⎞│ 2 ⎞ ⎛ ⎛ 2 ⎞│ \n",
" ⎟│ ∂ ⎟ ⎜ ⎜ ∂ ⎟│ \n",
"₁, h₂))⎟│ + ────(g₂(t, h₁, h₂))⎟ - 2⋅h₂⋅ν⋅⎜t⋅⎜───────(g₁(ξ₁, h₁, h₂))⎟│ \n",
" ⎟│ 2 ⎟ ⎝ ⎝∂h₁ ∂ξ₁ ⎠│ξ₁=\n",
" ⎠│ξ₁=t ∂h₁ ⎠ \n",
"\n",
" ⎞\n",
" ∂ ⎟\n",
" - ───(g₁(t, h₁, h₂))⎟\n",
"t ∂h₁ ⎠\n",
" "
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(d2f)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$h_{1}^{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{2}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{2}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) - 2 h_{1} h_{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{1}\\partial h_{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) + 2 h_{1} \\nu \\left(t \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} - \\frac{\\partial}{\\partial h_{2}} \\operatorname{g_{1}}{\\left (t,h_{1},h_{2} \\right )}\\right) + h_{2}^{2} \\left(dt_{1} \\left. \\frac{\\partial^{3}}{\\partial \\xi_{1}\\partial h_{1}^{2}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} + \\frac{\\partial^{2}}{\\partial h_{1}^{2}} \\operatorname{g_{2}}{\\left (t,h_{1},h_{2} \\right )}\\right) - 2 h_{2} \\nu \\left(t \\left. \\frac{\\partial^{2}}{\\partial \\xi_{1}\\partial h_{1}} \\operatorname{g_{1}}{\\left (\\xi_{1},h_{1},h_{2} \\right )} \\right|_{\\substack{ \\xi_{1}=t }} - \\frac{\\partial}{\\partial h_{1}} \\operatorname{g_{1}}{\\left (t,h_{1},h_{2} \\right )}\\right)$$"
],
"text/plain": [
" ⎛ ⎛ 3 ⎞│ 2 ⎞ ⎛ \n",
" 2 ⎜ ⎜ ∂ ⎟│ ∂ ⎟ ⎜ \n",
"h₁ ⋅⎜dt₁⋅⎜────────(g₁(ξ₁, h₁, h₂))⎟│ + ────(g₂(t, h₁, h₂))⎟ - 2⋅h₁⋅h₂⋅⎜dt₁\n",
" ⎜ ⎜ 2 ⎟│ 2 ⎟ ⎝ \n",
" ⎝ ⎝∂h₂ ∂ξ₁ ⎠│ξ₁=t ∂h₂ ⎠ \n",
"\n",
" ⎛ 3 ⎞│ 2 ⎞ ⎛ ⎛ \n",
" ⎜ ∂ ⎟│ ∂ ⎟ ⎜ ⎜ \n",
"⋅⎜───────────(g₁(ξ₁, h₁, h₂))⎟│ + ───────(g₂(t, h₁, h₂))⎟ + 2⋅h₁⋅ν⋅⎜t⋅⎜───\n",
" ⎝∂h₂ ∂h₁ ∂ξ₁ ⎠│ξ₁=t ∂h₂ ∂h₁ ⎠ ⎝ ⎝∂h₂\n",
" \n",
"\n",
" 2 ⎞│ ⎞ ⎛ ⎛ 3 \n",
"∂ ⎟│ ∂ ⎟ 2 ⎜ ⎜ ∂ \n",
"────(g₁(ξ₁, h₁, h₂))⎟│ - ───(g₁(t, h₁, h₂))⎟ + h₂ ⋅⎜dt₁⋅⎜────────(g₁(ξ₁, h\n",
" ∂ξ₁ ⎠│ξ₁=t ∂h₂ ⎠ ⎜ ⎜ 2 \n",
" ⎝ ⎝∂h₁ ∂ξ₁ \n",
"\n",
" ⎞│ 2 ⎞ ⎛ ⎛ 2 ⎞│ \n",
" ⎟│ ∂ ⎟ ⎜ ⎜ ∂ ⎟│ \n",
"₁, h₂))⎟│ + ────(g₂(t, h₁, h₂))⎟ - 2⋅h₂⋅ν⋅⎜t⋅⎜───────(g₁(ξ₁, h₁, h₂))⎟│ \n",
" ⎟│ 2 ⎟ ⎝ ⎝∂h₁ ∂ξ₁ ⎠│ξ₁=\n",
" ⎠│ξ₁=t ∂h₁ ⎠ \n",
"\n",
" ⎞\n",
" ∂ ⎟\n",
" - ───(g₁(t, h₁, h₂))⎟\n",
"t ∂h₁ ⎠\n",
" "
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(d2f.subs({\n",
" g1: Lambda((t, h1, h2), simplify(Matrix(list(xhat(t)) + [0]))),\n",
" g2: Lambda((t, h1, h2), simplify(Matrix([exp21, exp22] +[zhat(t)]))),\n",
"}))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"OK. This failed. But we can copy and paste this and the replacement of $g_1$ and $g_2$ will be effective:"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [],
"source": [
"d2f = (\n",
" h1**2*(dt1*diff(g1(t, h1, h2), t, h2, h2)\n",
" + diff(g2(t, h1, h2), h2, h2))\n",
" - 2*h1*h2*(dt1*diff(g1(t, h1, h2), t, h1, h2)\n",
" + diff(g2(t, h1, h2), h1, h2))\n",
" + 2*h1*nu*(t*diff(g1(t, h1, h2), t, h2)\n",
" - diff(g1(t, h1, h2), h2))\n",
" + h2**2*(dt1*diff(g1(t, h1, h2), t, h1, h1)\n",
" + diff(g2(t, h1, h2), h1, h1))\n",
" - 2*h2*nu*(t*diff(g1(t, h1, h2), t, h1)\n",
" - diff(g1(t, h1, h2), h1))\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And the same for $t$."
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [],
"source": [
"t = 2 * pi / b1"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}2 a h_{1}^{2} + 6 a h_{2}^{2} - 4 b h_{1} h_{2} + 2 h_{1} \\nu \\left(i t \\left(- \\frac{1}{2} \\left(e^{2 i b_{1} t} - 2 e^{i b_{1} t} + 1\\right) e^{- i b_{1} t} + e^{i b_{1} t} - 1\\right) - \\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(e^{2 i b_{1} t} - 2 e^{i b_{1} t} + 1\\right)\\right) - 2 h_{2} \\nu \\left(t \\left(- \\frac{1}{2} \\left(e^{2 i b_{1} t} - 1\\right) e^{- i b_{1} t} + e^{i b_{1} t}\\right) - \\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(- i e^{2 i b_{1} t} + i\\right)\\right)\\\\- 4 a h_{1} h_{2} + 6 b h_{1}^{2} + 2 b h_{2}^{2} + 2 h_{1} \\nu \\left(t \\left(- \\frac{1}{2} \\left(e^{2 i b_{1} t} - 1\\right) e^{- i b_{1} t} + e^{i b_{1} t}\\right) - \\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(- i e^{2 i b_{1} t} + i\\right)\\right) - 2 h_{2} \\nu \\left(i t \\left(\\frac{1}{2} \\left(e^{2 i b_{1} t} - 2 e^{i b_{1} t} + 1\\right) e^{- i b_{1} t} - e^{i b_{1} t} + 1\\right) - \\frac{e^{- i b_{1} t}}{2 b_{1}} \\left(- e^{2 i b_{1} t} + 2 e^{i b_{1} t} - 1\\right)\\right)\\\\- \\frac{h_{1}^{2}}{2 b_{1}} \\left(2 b_{1} t e^{i b_{1} t} + i e^{2 i b_{1} t} - i\\right) e^{- i b_{1} t} - \\frac{h_{2}^{2}}{2 b_{1}} \\left(2 b_{1} t e^{i b_{1} t} + i e^{2 i b_{1} t} - i\\right) e^{- i b_{1} t}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ ⎛ ⎛ ⎛ 2⋅ⅈ⋅b₁⋅t ⅈ⋅b₁⋅t ⎞ -\n",
"⎢ 2 2 ⎜ ⎜ ⎝ℯ - 2⋅ℯ + 1⎠⋅ℯ \n",
"⎢2⋅a⋅h₁ + 6⋅a⋅h₂ - 4⋅b⋅h₁⋅h₂ + 2⋅h₁⋅ν⋅⎜ⅈ⋅t⋅⎜- ──────────────────────────────\n",
"⎢ ⎝ ⎝ 2 \n",
"⎢ \n",
"⎢ ⎛ ⎛ ⎛ 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t \n",
"⎢ 2 2 ⎜ ⎜ ⎝ℯ - 1⎠⋅ℯ ⅈ⋅b\n",
"⎢-4⋅a⋅h₁⋅h₂ + 6⋅b⋅h₁ + 2⋅b⋅h₂ + 2⋅h₁⋅ν⋅⎜t⋅⎜- ──────────────────────── + ℯ \n",
"⎢ ⎝ ⎝ 2 \n",
"⎢ \n",
"⎢ 2 ⎛ ⅈ\n",
"⎢ h₁ ⋅⎝2⋅b₁⋅t⋅ℯ \n",
"⎢ - ──────────────\n",
"⎣ \n",
"\n",
"ⅈ⋅b₁⋅t ⎞ ⎛ 2⋅ⅈ⋅b₁⋅t ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t⎞ ⎛ ⎛ ⎛\n",
" ⅈ⋅b₁⋅t ⎟ ⎝ℯ - 2⋅ℯ + 1⎠⋅ℯ ⎟ ⎜ ⎜ ⎝\n",
"────── + ℯ - 1⎟ - ────────────────────────────────────⎟ - 2⋅h₂⋅ν⋅⎜t⋅⎜- ─\n",
" ⎠ 2⋅b₁ ⎠ ⎝ ⎝ \n",
" \n",
" ⎞ ⎛ 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t⎞ ⎛ ⎛⎛ 2⋅ⅈ⋅b₁⋅t ⅈ⋅b₁⋅t \n",
"₁⋅t⎟ ⎝- ⅈ⋅ℯ + ⅈ⎠⋅ℯ ⎟ ⎜ ⎜⎝ℯ - 2⋅ℯ + 1\n",
" ⎟ - ────────────────────────────⎟ - 2⋅h₂⋅ν⋅⎜ⅈ⋅t⋅⎜──────────────────────────\n",
" ⎠ 2⋅b₁ ⎠ ⎝ ⎝ 2 \n",
" \n",
"⋅b₁⋅t 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t 2 ⎛ ⅈ⋅b₁⋅t 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ\n",
" + ⅈ⋅ℯ - ⅈ⎠⋅ℯ h₂ ⋅⎝2⋅b₁⋅t⋅ℯ + ⅈ⋅ℯ - ⅈ⎠⋅ℯ \n",
"───────────────────────────────── - ──────────────────────────────────────────\n",
" 2⋅b₁ 2⋅b₁ \n",
"\n",
" 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t ⎞ ⎛ 2⋅ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t⎞ ⎤\n",
"ℯ - 1⎠⋅ℯ ⅈ⋅b₁⋅t⎟ ⎝- ⅈ⋅ℯ + ⅈ⎠⋅ℯ ⎟ ⎥\n",
"─────────────────────── + ℯ ⎟ - ────────────────────────────⎟ ⎥\n",
" 2 ⎠ 2⋅b₁ ⎠ ⎥\n",
" ⎥\n",
"⎞ -ⅈ⋅b₁⋅t ⎞ ⎛ 2⋅ⅈ⋅b₁⋅t ⅈ⋅b₁⋅t ⎞ -ⅈ⋅b₁⋅t⎞⎥\n",
"⎠⋅ℯ ⅈ⋅b₁⋅t ⎟ ⎝- ℯ + 2⋅ℯ - 1⎠⋅ℯ ⎟⎥\n",
"────────── - ℯ + 1⎟ - ──────────────────────────────────────⎟⎥\n",
" ⎠ 2⋅b₁ ⎠⎥\n",
" ⎥\n",
"⋅b₁⋅t ⎥\n",
" ⎥\n",
"───── ⎥\n",
" ⎦"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d2f"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The replacement does not work apparently, so let's do it manually:"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [],
"source": [
"d2f = Matrix([\n",
"[ 2*a*h1**2 + 6*a*h2**2 - 4*b*h1*h2 + 2*h1*nu*(I*t*(-(exp(2*I*b1*t) - 2*exp(I*b1*t) + 1)*exp(-I*b1*t)/2 + exp(I*b1*t) - 1) - (exp(2*I*b1*t) - 2*exp(I*b1*t) + 1)*exp(-I*b1*t)/(2*b1)) - 2*h2*nu*(t*(-(exp(2*I*b1*t) - 1)*exp(-I*b1*t)/2 + exp(I*b1*t)) - (-I*exp(2*I*b1*t) + I)*exp(-I*b1*t)/(2*b1))],\n",
"[-4*a*h1*h2 + 6*b*h1**2 + 2*b*h2**2 + 2*h1*nu*(t*(-(exp(2*I*b1*t) - 1)*exp(-I*b1*t)/2 + exp(I*b1*t)) - (-I*exp(2*I*b1*t) + I)*exp(-I*b1*t)/(2*b1)) - 2*h2*nu*(I*t*((exp(2*I*b1*t) - 2*exp(I*b1*t) + 1)*exp(-I*b1*t)/2 - exp(I*b1*t) + 1) - (-exp(2*I*b1*t) + 2*exp(I*b1*t) - 1)*exp(-I*b1*t)/(2*b1))],\n",
"[ -h1**2*(2*b1*t*exp(I*b1*t) + I*exp(2*I*b1*t) - I)*exp(-I*b1*t)/(2*b1) - h2**2*(2*b1*t*exp(I*b1*t) + I*exp(2*I*b1*t) - I)*exp(-I*b1*t)/(2*b1)]])"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [],
"source": [
"d2Exp = simplify(d2f)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It still works well. And we removed the complex exponential, this result is purely real now!"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}2 a h_{1}^{2} + 6 a h_{2}^{2} - 4 b h_{1} h_{2} - \\frac{4 \\pi}{b_{1}} h_{2} \\nu\\\\- 4 a h_{1} h_{2} + 6 b h_{1}^{2} + 2 b h_{2}^{2} + \\frac{4 \\pi}{b_{1}} h_{1} \\nu\\\\- \\frac{2 \\pi}{b_{1}} \\left(h_{1}^{2} + h_{2}^{2}\\right)\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 2 4⋅π⋅h₂⋅ν ⎤\n",
"⎢2⋅a⋅h₁ + 6⋅a⋅h₂ - 4⋅b⋅h₁⋅h₂ - ──────── ⎥\n",
"⎢ b₁ ⎥\n",
"⎢ ⎥\n",
"⎢ 2 2 4⋅π⋅h₁⋅ν⎥\n",
"⎢-4⋅a⋅h₁⋅h₂ + 6⋅b⋅h₁ + 2⋅b⋅h₂ + ────────⎥\n",
"⎢ b₁ ⎥\n",
"⎢ ⎥\n",
"⎢ ⎛ 2 2⎞ ⎥\n",
"⎢ -2⋅π⋅⎝h₁ + h₂ ⎠ ⎥\n",
"⎢ ───────────────── ⎥\n",
"⎣ b₁ ⎦"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d2Exp"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left( \\left ( u_{1}, \\quad u_{2}, \\quad u_{5}\\right ) \\mapsto 2 a h_{1}^{2} h_{2} u_{5} + 2 a h_{2}^{3} u_{5} - 2 b h_{1}^{3} u_{5} - 2 b h_{1} h_{2}^{2} u_{5} - h_{1}^{2} h_{3} k_{132} u_{5} - h_{1}^{2} h_{4} k_{142} u_{5} + h_{1} h_{2} h_{3} k_{131} u_{5} - h_{1} h_{2} h_{3} k_{232} u_{5} + h_{1} h_{2} h_{4} k_{141} u_{5} - h_{1} h_{2} h_{4} k_{242} u_{5} + h_{2}^{2} h_{3} k_{231} u_{5} + h_{2}^{2} h_{4} k_{241} u_{5} + \\frac{2 \\pi}{b_{1}} h_{1}^{3} u_{2} - \\frac{2 \\pi}{b_{1}} h_{1}^{2} h_{2} u_{1} + \\frac{2 \\pi}{b_{1}} h_{1} h_{2}^{2} u_{2} - \\frac{2 \\pi}{b_{1}} h_{2}^{3} u_{1} \\right)$$"
],
"text/plain": [
" \n",
" 2 3 3 2 2 \n",
"(u₁, u₂, u₅) ↦ 2⋅a⋅h₁ ⋅h₂⋅u₅ + 2⋅a⋅h₂ ⋅u₅ - 2⋅b⋅h₁ ⋅u₅ - 2⋅b⋅h₁⋅h₂ ⋅u₅ - h₁ ⋅h\n",
" \n",
"\n",
" \n",
" 2 \n",
"₃⋅k₁₃₂⋅u₅ - h₁ ⋅h₄⋅k₁₄₂⋅u₅ + h₁⋅h₂⋅h₃⋅k₁₃₁⋅u₅ - h₁⋅h₂⋅h₃⋅k₂₃₂⋅u₅ + h₁⋅h₂⋅h₄⋅k₁\n",
" \n",
"\n",
" 3 \n",
" 2 2 2⋅π⋅h₁ ⋅u₂ 2⋅π⋅\n",
"₄₁⋅u₅ - h₁⋅h₂⋅h₄⋅k₂₄₂⋅u₅ + h₂ ⋅h₃⋅k₂₃₁⋅u₅ + h₂ ⋅h₄⋅k₂₄₁⋅u₅ + ────────── - ────\n",
" b₁ \n",
"\n",
" 2 2 3 \n",
"h₁ ⋅h₂⋅u₁ 2⋅π⋅h₁⋅h₂ ⋅u₂ 2⋅π⋅h₂ ⋅u₁\n",
"───────── + ───────────── - ──────────\n",
" b₁ b₁ b₁ "
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Psi"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So now we can call $\\Psi$ on the three components of this `d2Exp` :"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$- \\frac{4 \\pi}{b_{1}} a h_{1}^{2} h_{2} \\left(h_{1}^{2} + h_{2}^{2}\\right) - \\frac{4 \\pi}{b_{1}} a h_{2}^{3} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{4 \\pi}{b_{1}} b h_{1}^{3} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{4 \\pi}{b_{1}} b h_{1} h_{2}^{2} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{2 \\pi}{b_{1}} h_{1}^{3} \\left(- 4 a h_{1} h_{2} + 6 b h_{1}^{2} + 2 b h_{2}^{2} + \\frac{4 \\pi}{b_{1}} h_{1} \\nu\\right) - \\frac{2 \\pi}{b_{1}} h_{1}^{2} h_{2} \\left(2 a h_{1}^{2} + 6 a h_{2}^{2} - 4 b h_{1} h_{2} - \\frac{4 \\pi}{b_{1}} h_{2} \\nu\\right) + \\frac{2 \\pi}{b_{1}} h_{1}^{2} h_{3} k_{132} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{2 \\pi}{b_{1}} h_{1}^{2} h_{4} k_{142} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{2 \\pi}{b_{1}} h_{1} h_{2}^{2} \\left(- 4 a h_{1} h_{2} + 6 b h_{1}^{2} + 2 b h_{2}^{2} + \\frac{4 \\pi}{b_{1}} h_{1} \\nu\\right) - \\frac{2 \\pi}{b_{1}} h_{1} h_{2} h_{3} k_{131} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{2 \\pi}{b_{1}} h_{1} h_{2} h_{3} k_{232} \\left(h_{1}^{2} + h_{2}^{2}\\right) - \\frac{2 \\pi}{b_{1}} h_{1} h_{2} h_{4} k_{141} \\left(h_{1}^{2} + h_{2}^{2}\\right) + \\frac{2 \\pi}{b_{1}} h_{1} h_{2} h_{4} k_{242} \\left(h_{1}^{2} + h_{2}^{2}\\right) - \\frac{2 \\pi}{b_{1}} h_{2}^{3} \\left(2 a h_{1}^{2} + 6 a h_{2}^{2} - 4 b h_{1} h_{2} - \\frac{4 \\pi}{b_{1}} h_{2} \\nu\\right) - \\frac{2 \\pi}{b_{1}} h_{2}^{2} h_{3} k_{231} \\left(h_{1}^{2} + h_{2}^{2}\\right) - \\frac{2 \\pi}{b_{1}} h_{2}^{2} h_{4} k_{241} \\left(h_{1}^{2} + h_{2}^{2}\\right)$$"
],
"text/plain": [
" \n",
" 2 ⎛ 2 2⎞ 3 ⎛ 2 2⎞ 3 ⎛ 2 2⎞ \n",
" 4⋅π⋅a⋅h₁ ⋅h₂⋅⎝h₁ + h₂ ⎠ 4⋅π⋅a⋅h₂ ⋅⎝h₁ + h₂ ⎠ 4⋅π⋅b⋅h₁ ⋅⎝h₁ + h₂ ⎠ 4\n",
"- ──────────────────────── - ───────────────────── + ───────────────────── + ─\n",
" b₁ b₁ b₁ \n",
"\n",
" 3 ⎛ 2 2 4⋅π⋅h₁⋅ν⎞ \n",
" 2 ⎛ 2 2⎞ 2⋅π⋅h₁ ⋅⎜-4⋅a⋅h₁⋅h₂ + 6⋅b⋅h₁ + 2⋅b⋅h₂ + ────────⎟ \n",
"⋅π⋅b⋅h₁⋅h₂ ⋅⎝h₁ + h₂ ⎠ ⎝ b₁ ⎠ \n",
"─────────────────────── + ─────────────────────────────────────────────────── \n",
" b₁ b₁ \n",
"\n",
" 2 ⎛ 2 2 4⋅π⋅h₂⋅ν⎞ \n",
" 2⋅π⋅h₁ ⋅h₂⋅⎜2⋅a⋅h₁ + 6⋅a⋅h₂ - 4⋅b⋅h₁⋅h₂ - ────────⎟ 2 ⎛ 2\n",
" ⎝ b₁ ⎠ 2⋅π⋅h₁ ⋅h₃⋅k₁₃₂⋅⎝h₁ \n",
"- ───────────────────────────────────────────────────── + ────────────────────\n",
" b₁ b₁ \n",
"\n",
" 2 ⎛ 2 \n",
" 2⎞ 2 ⎛ 2 2⎞ 2⋅π⋅h₁⋅h₂ ⋅⎜-4⋅a⋅h₁⋅h₂ + 6⋅b⋅h₁ + 2⋅b\n",
" + h₂ ⎠ 2⋅π⋅h₁ ⋅h₄⋅k₁₄₂⋅⎝h₁ + h₂ ⎠ ⎝ \n",
"─────── + ─────────────────────────── + ──────────────────────────────────────\n",
" b₁ b₁ \n",
"\n",
" 2 4⋅π⋅h₁⋅ν⎞ \n",
"⋅h₂ + ────────⎟ ⎛ 2 2⎞ ⎛ 2 \n",
" b₁ ⎠ 2⋅π⋅h₁⋅h₂⋅h₃⋅k₁₃₁⋅⎝h₁ + h₂ ⎠ 2⋅π⋅h₁⋅h₂⋅h₃⋅k₂₃₂⋅⎝h₁ + h₂\n",
"──────────────── - ───────────────────────────── + ───────────────────────────\n",
" b₁ b₁ \n",
"\n",
" 3 ⎛\n",
"2⎞ ⎛ 2 2⎞ ⎛ 2 2⎞ 2⋅π⋅h₂ ⋅⎜\n",
" ⎠ 2⋅π⋅h₁⋅h₂⋅h₄⋅k₁₄₁⋅⎝h₁ + h₂ ⎠ 2⋅π⋅h₁⋅h₂⋅h₄⋅k₂₄₂⋅⎝h₁ + h₂ ⎠ ⎝\n",
"── - ───────────────────────────── + ───────────────────────────── - ─────────\n",
" b₁ b₁ \n",
"\n",
" 2 2 4⋅π⋅h₂⋅ν⎞ \n",
"2⋅a⋅h₁ + 6⋅a⋅h₂ - 4⋅b⋅h₁⋅h₂ - ────────⎟ 2 ⎛ 2 2⎞ \n",
" b₁ ⎠ 2⋅π⋅h₂ ⋅h₃⋅k₂₃₁⋅⎝h₁ + h₂ ⎠ 2⋅π⋅\n",
"───────────────────────────────────────── - ─────────────────────────── - ────\n",
" b₁ b₁ \n",
"\n",
" \n",
" 2 ⎛ 2 2⎞\n",
"h₂ ⋅h₄⋅k₂₄₁⋅⎝h₁ + h₂ ⎠\n",
"───────────────────────\n",
" b₁ "
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Psi(d2Exp[0], d2Exp[1], d2Exp[2])"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\frac{4 \\pi^{2}}{b_{1}^{3}} \\left(b_{1} \\left(- 8 a h_{1}^{6} h_{2} - 24 a h_{1}^{4} h_{2}^{3} - 24 a h_{1}^{2} h_{2}^{5} - 8 a h_{2}^{7} + 8 b h_{1}^{7} + 24 b h_{1}^{5} h_{2}^{2} + 24 b h_{1}^{3} h_{2}^{4} + 8 b h_{1} h_{2}^{6} + h_{1}^{6} h_{3} k_{132} + h_{1}^{6} h_{4} k_{142} - h_{1}^{5} h_{2} h_{3} k_{131} + h_{1}^{5} h_{2} h_{3} k_{232} - h_{1}^{5} h_{2} h_{4} k_{141} + h_{1}^{5} h_{2} h_{4} k_{242} + 2 h_{1}^{4} h_{2}^{2} h_{3} k_{132} - h_{1}^{4} h_{2}^{2} h_{3} k_{231} + 2 h_{1}^{4} h_{2}^{2} h_{4} k_{142} - h_{1}^{4} h_{2}^{2} h_{4} k_{241} - 2 h_{1}^{3} h_{2}^{3} h_{3} k_{131} + 2 h_{1}^{3} h_{2}^{3} h_{3} k_{232} - 2 h_{1}^{3} h_{2}^{3} h_{4} k_{141} + 2 h_{1}^{3} h_{2}^{3} h_{4} k_{242} + h_{1}^{2} h_{2}^{4} h_{3} k_{132} - 2 h_{1}^{2} h_{2}^{4} h_{3} k_{231} + h_{1}^{2} h_{2}^{4} h_{4} k_{142} - 2 h_{1}^{2} h_{2}^{4} h_{4} k_{241} - h_{1} h_{2}^{5} h_{3} k_{131} + h_{1} h_{2}^{5} h_{3} k_{232} - h_{1} h_{2}^{5} h_{4} k_{141} + h_{1} h_{2}^{5} h_{4} k_{242} - h_{2}^{6} h_{3} k_{231} - h_{2}^{6} h_{4} k_{241}\\right) + 4 \\pi \\nu \\left(h_{1}^{6} + 3 h_{1}^{4} h_{2}^{2} + 3 h_{1}^{2} h_{2}^{4} + h_{2}^{6}\\right)\\right)$$"
],
"text/plain": [
" 2 ⎛ ⎛ 6 4 3 2 5 7 7 \n",
"4⋅π ⋅⎝b₁⋅⎝- 8⋅a⋅h₁ ⋅h₂ - 24⋅a⋅h₁ ⋅h₂ - 24⋅a⋅h₁ ⋅h₂ - 8⋅a⋅h₂ + 8⋅b⋅h₁ + 24⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" 5 2 3 4 6 6 6 5 \n",
"b⋅h₁ ⋅h₂ + 24⋅b⋅h₁ ⋅h₂ + 8⋅b⋅h₁⋅h₂ + h₁ ⋅h₃⋅k₁₃₂ + h₁ ⋅h₄⋅k₁₄₂ - h₁ ⋅h₂⋅h₃⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" 5 5 5 4 2 \n",
"k₁₃₁ + h₁ ⋅h₂⋅h₃⋅k₂₃₂ - h₁ ⋅h₂⋅h₄⋅k₁₄₁ + h₁ ⋅h₂⋅h₄⋅k₂₄₂ + 2⋅h₁ ⋅h₂ ⋅h₃⋅k₁₃₂ - \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" 4 2 4 2 4 2 3 3 \n",
"h₁ ⋅h₂ ⋅h₃⋅k₂₃₁ + 2⋅h₁ ⋅h₂ ⋅h₄⋅k₁₄₂ - h₁ ⋅h₂ ⋅h₄⋅k₂₄₁ - 2⋅h₁ ⋅h₂ ⋅h₃⋅k₁₃₁ + 2⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 3 \n",
" b₁ \n",
"\n",
" 3 3 3 3 3 3 2 4 \n",
"h₁ ⋅h₂ ⋅h₃⋅k₂₃₂ - 2⋅h₁ ⋅h₂ ⋅h₄⋅k₁₄₁ + 2⋅h₁ ⋅h₂ ⋅h₄⋅k₂₄₂ + h₁ ⋅h₂ ⋅h₃⋅k₁₃₂ - 2⋅\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" 2 4 2 4 2 4 5 \n",
"h₁ ⋅h₂ ⋅h₃⋅k₂₃₁ + h₁ ⋅h₂ ⋅h₄⋅k₁₄₂ - 2⋅h₁ ⋅h₂ ⋅h₄⋅k₂₄₁ - h₁⋅h₂ ⋅h₃⋅k₁₃₁ + h₁⋅h₂\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
"5 5 5 6 6 ⎞ \n",
" ⋅h₃⋅k₂₃₂ - h₁⋅h₂ ⋅h₄⋅k₁₄₁ + h₁⋅h₂ ⋅h₄⋅k₂₄₂ - h₂ ⋅h₃⋅k₂₃₁ - h₂ ⋅h₄⋅k₂₄₁⎠ + 4⋅π\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
"\n",
" ⎛ 6 4 2 2 4 6⎞⎞\n",
"⋅ν⋅⎝h₁ + 3⋅h₁ ⋅h₂ + 3⋅h₁ ⋅h₂ + h₂ ⎠⎠\n",
"───────────────────────────────────────\n",
" \n",
" "
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(expand(Psi(d2Exp[0], d2Exp[1], d2Exp[2]) / (1 / (1/b1 * 2 * (h1**2 + h2**2) * pi))))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We don't have the value for `nu` !"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Conclusion"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We do NOT obtain the same result as the document. Everything failed at the end.\n",
"\n",
"Too bad, but still, it was interesting. I guess?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"> See [here](https://github.com/Naereen/notebooks) for other notebooks I wrote."
]
}
],
"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",
"name": "python",
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