{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 17. One-parameter groups of transformations\n",
"\n",
"This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland)."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'SageMath version 9.6, Release Date: 2022-05-15'"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"version()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let $M$ be a smooth manifold. A **one-parameter group of\n",
"transformations $ϕ$, on $M$** is a smooth map $\\phi: M × R\\to M$ such that\n",
"\n",
"\\begin{equation}\n",
"ϕ(x, 0) = x,\n",
"\\label{}\\tag{17.1}\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"ϕ(ϕ(x, t), s) = ϕ(x, t + s)\\quad \\mbox{for all } x ∈ M, \\ t, s ∈ R.\n",
"\\label{}\\tag{17.2}\n",
"\\end{equation}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we put $$ϕ_t (x) = ϕ(x, t),$$ \n",
"\n",
"then $\\phi_t$, for $t\\in R$ is a smooth map $M\\to M$ and according to (17.2)\n",
"\n",
"$$\\displaystyle \\phi_{t+s}(x)=\\phi(x,t+s)=\\phi(\\phi(x,t),s)\\\\=\\phi(\\phi_t(x),s)=\\phi_s(\\phi_t(x))=(\\phi_s\\circ\\phi_t)(x),$$\n",
"i.e.,\n",
"\n",
"\\begin{equation}\n",
"ϕ_{t+s} = ϕ_s ◦ ϕ_t = ϕ_t ◦ ϕ_s,\n",
"\\label{}\\tag{17.3}\n",
"\\end{equation}\n",
"\n",
"since $\\ t+s=s+t$.\n",
"\n",
"The map $\\phi_0$ is the identity on $M$ since from (17.1) it follows\n",
"$ϕ_0 (x) = ϕ(x, 0) = x$ for all $x ∈ M$. From (17.3) it follows $ϕ_t ◦ ϕ_{−t} = ϕ_{−t} ◦ ϕ_t = ϕ_0$ , which means that each map $ϕ_t$ has\n",
"an inverse, $ϕ_{−t}$ , which is also smooth. Therefore, each $ϕ_t$ is a diffeomorphism\n",
"of $M$ onto itself. Thus, the set of transformations $\\{ϕ_t : t ∈ R\\}$ is an Abelian **group of\n",
"diffeomorphisms of** $M$ onto $M$, and the map $t → ϕ_t$ is a homomorphism from the\n",
"additive group of the real numbers into the group of diffeomorphisms of $M$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.1**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us check that the formula\n",
"$$\\phi((x,y,t)=\\frac{(2x,2y\\cos t+(1-x^2-y^2)\\sin t)}{1+x^2+y^2+(1-x^2-y^2)\\cos t-2y\\sin t}$$\n",
"\n",
"defines a one-parameter group of transformations i.e., the conditions (17.1), (17.2) are satisfied."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"var('t,s,x,y,x0,y0') # symbolic variables\n",
" # components ϕ1,ϕ2 of ϕ:\n",
"ϕ1(x,y,t)=2*x/(1+x^2+y^2+(1-x^2-y^2)*cos(t)-2*y*sin(t))\n",
"ϕ2(x,y,t)=(2*y*cos(t)+(1-x^2-y^2)*sin(t))/(1+x^2+y^2+(1-x^2-y^2)*cos(t)-2*y*sin(t))\n",
" # ϕ=(ϕ1,ϕ2):\n",
"ϕ(x,y,t)=(ϕ1(x,y,t),ϕ2(x,y,t))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First let us check (17.1), i.e. that $ϕ(x,y,t)|_{t=0}=(x,y)$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x, y)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ϕ(x,y,t).subs(t=0) # substitute t by 0 in ϕ"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now check that (17.2) holds for the first component of $ϕ$:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a1=ϕ1(x,y,t+s).trig_expand() # first comp. of the right hand side of (17.2)\n",
" # first comp. of the left hand side of (17.2):\n",
"b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s).normalize()\n",
" # first component of the difference between\n",
" # the left and right hand side of (17.2) (numerator)\n",
"c1=(a1-b1).numerator() # take numerator of a1-b1\n",
"c1.full_simplify() # simplify"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and for the second component of $ϕ$:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a2=ϕ2(x,y,t+s).trig_expand() # second component of the right hand side of (17.2)\n",
" # second component of the left hand side of (17.2)\n",
"b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s).normalize()\n",
" # second component of the difference between\n",
" # the left and right side of (17.2) (numerator):\n",
"c2=(a2-b2).numerator() # take numerator of a2-b2\n",
"c2.full_simplify()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We have checked the condition (17.2)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"### Infinitesimal generator of $\\phi_t$\n",
"\n",
"
\n",
"\n",
"\n",
"Each one-parameter group of transformations $ϕ$ on $M$ defines a family of curves\n",
"in $M$. The map $ϕ_x : R → M$ given by $ϕ_x (t) = ϕ(x, t)$ is a\n",
"smooth curve in M for each $x ∈ M$. Since $ϕ_x (0) = ϕ(x, 0) = x$, the tangent\n",
"vector to the curve $ϕ_x$ at $t = 0$ (defined in (8.9)) belongs to $T_x M$. The **infinitesimal generator of** $ϕ$ is\n",
"the vector field $X$ such that \n",
"\n",
"$$X_x = (ϕ_x )'_0.$$ \n",
"\n",
"The infinitesimal generator of $ϕ$ is a vector\n",
"field tangent to the curves generated by the one-parameter group of transformations."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.2**\n",
"\n",
"Let us compute the infinitesimal generator $X$ of the one-parameter group $\\phi_t$ from the previous example in Cartesian coordinates of $R^2$."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x*y"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# continuation according to (8.10)\n",
"# the first component of X is the derivative of ϕ1 w.r. to t\n",
"X1=diff(ϕ1(x,y,t),t).subs(t=0);X1"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-1/2*x^2 + 1/2*y^2 + 1/2"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# the second component of X is the derivative of ϕ2 w.r. to t\n",
"X2=diff(ϕ2(x,y,t),t).subs(t=0);X2"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-x^2 + y^2 + 1, 2)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X2.numerator_denominator() # numerator and denominator of X2 "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus the infinitesimal generator is \n",
"$\\ \\ X=xy\\frac{\\partial}{\\partial x}+\\frac{-x^2+y^2+1}{2}\\frac{\\partial}{\\partial y}.$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.3**\n",
"\n",
"Plot the infinitesimal generator from the previous example."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"Graphics object consisting of 100 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"M = Manifold(2, 'M') # manifold M=R^2\n",
"X. = M.chart() # Cartesian coordinates \n",
"X = M.vector_field(x*y,(-x^2+y^2+1)/2, name='v') # vector field \n",
" # with components X1,X2\n",
"p=X.plot(scale=0.2,arrowsize=1.5,number_values={x:10, y:10}, \n",
" ranges={x: (0.2,1.7), y: (-0.8,0.8)},color='grey') # plot X\n",
"p.show(aspect_ratio=1) # show plot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.4**\n",
"\n",
"\n",
"For $ϕ$ from Example 17.1,\n",
" plot the curves $\\phi_{(x,y)}$ for some selected points $(x,y)$, where \n",
"$ϕ_{(x,y)} (t) = ϕ(x,y, t)$. "
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 8 graphics primitives"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var('t,s,x,y,x0,y0') # symbolic variables\n",
" # components ϕ1,ϕ2 of ϕ:\n",
"ϕ1(x,y,t)=2*x/(1+x^2+y^2+(1-x^2-y^2)*cos(t)-2*y*sin(t))\n",
"ϕ2(x,y,t)=(2*y*cos(t)+(1-x^2-y^2)*sin(t))/(1+x^2+y^2+(1-x^2-y^2)*cos(t)-2*y*sin(t)) \n",
"ϕ(x,y,t)=(ϕ1(x,y,t),ϕ2(x,y,t)) # ϕ=(ϕ1,ϕ2):\n",
"# plot curves passing through (1,1),(1,1/2),(1,1/3),(1,3/4)\n",
"s1=[parametric_plot( ϕ(x,y,t).subs({x:1,y:y0}), (t, 0, 2*pi),color='grey') \n",
" for y0 in [1,1/2,1/3,3/4]] \n",
"# plot curves passing through (-1,1),(-1,1/2),(-1,1/3),(-1,3/4)\n",
"s2=[parametric_plot( ϕ(x,y,t).subs({x:-1,y:y0}), (t, 0, 2*pi),color='grey') \n",
" for y0 in [1,1/2,1/3,3/4]]\n",
"sum(s1)+sum(s2) # combine plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.5**\n",
"\n",
"Check that the formulas\n",
"$$\\left\\{\\begin{matrix}\n",
"ϕ_1(x,y,t)=x\\cos t-y\\sin t ,\\\\\n",
"ϕ_2(x,y,t)=x\\sin t +y\\cos t,\n",
"\\end{matrix}\n",
"\\right.\n",
"$$\n",
"\n",
"define a one-parameter group of transformations."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
"forget()\n",
"var('t,s,x,y') # symbolic variables\n",
"ϕ1(x,y,t)=x*cos(t)-y*sin(t) # component ϕ1\n",
"ϕ2(x,y,t)=x*sin(t)+y*cos(t) # component ϕ2 \n",
"a1=ϕ1(x,y,t+s).trig_expand() # first components of the right\n",
"b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s) # and left hand sides of (17.2)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c1=(a1-b1) # difference between left and right\n",
"c1.full_simplify() # sides of (17.2)for the first component"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
"a2=ϕ2(x,y,t+s).trig_expand() # second components of the right\n",
"b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s) # and left hand sides of (17.2)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c2=(a2-b2) # difference between left and right\n",
"c2.full_simplify() # hand sides for the second component"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.6**\n",
"\n",
"Compute components of the infinitesimal generator $X$ of $\\phi_t$ from the previous example. Plot $X$."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-y"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# continuation\n",
"# the first component of X is the derivative of ϕ1 w.r. to t\n",
"X1=diff(ϕ1(x,y,t),t).subs(t=0);X1 # cf. (8.10)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# the second component of X is the derivative of ϕ2 w.r. to t\n",
"X2=diff(ϕ2(x,y,t),t).subs(t=0);X2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus $\\ \\ X=-y\\frac{\\partial}{\\partial x}+x\\frac{\\partial}{\\partial y}$"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 100 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"M = Manifold(2, 'M') # manifold M=R^2\n",
"X. = M.chart() # Cartesian coordinates\n",
"X = M.vector_field(X1,X2, name='v') # vector field with comp X1,X2\n",
"p=X.plot(scale=0.3,arrowsize=1.5,number_values={x:10, y:10},\n",
" ranges={x: (-1,1), y: (-1,1)},color='grey') # plot X\n",
"p.show(aspect_ratio=1) # show plot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.7**\n",
"\n",
"For $ϕ$ from Example 17.5\n",
"plot the curves $\\phi_{(x,y)}$ for some selected points $(x,y)$, where \n",
"$ϕ_{(x,y)} (t) = ϕ(x,y, t)$. "
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 4 graphics primitives"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var('t,s,x,y') # symbolic variables\n",
"ϕ1(x,y,t)=x*cos(t)-y*sin(t) # component ϕ1\n",
"ϕ2(x,y,t)=x*sin(t)+y*cos(t) # component ϕ2 \n",
"ϕ(x,y,t)=(ϕ1(x,y,t),ϕ2(x,y,t)) # define ϕ=(ϕ1,ϕ2)\n",
"# plot curves passing through (1,1),(1,1/2),(1,3/2),(1,2)\n",
"s1=[parametric_plot( ϕ(x,y,t).subs({x:1,y:y0}), (t, 0, 2*pi),color='grey') \n",
" for y0 in [1,1/2,3/2,2]]\n",
"sum(s1) # combine plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.8**\n",
"\n",
"Consider $ϕ=(ϕ_1,ϕ_2)$ defined by\n",
"$$\\left\\{\\begin{matrix}\n",
"ϕ_1(x,y,t)=x+at,\\\\\n",
"ϕ_2(x,y,t)=y+bt,\n",
"\\end{matrix}\n",
"\\right.\n",
"$$\n",
"\n",
"for fixed $a,b\\in R$. Show that $ϕ$ defines a one-parametric group of transformations."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"forget()\n",
"var('t,s,x,y,a,b') # symbolic variables\n",
"ϕ1(x,y,t)=x+a*t # define ϕ1\n",
"ϕ2(x,y,t)=y+b*t # define ϕ2"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
"a1=ϕ1(x,y,t+s).expand() # right hand side of (17.2) for ϕ1\n",
"b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (17.2) for ϕ1"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c1=(a1-b1) # difference between lhs and rhs\n",
"c1.full_simplify() # simplification"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [],
"source": [
"a2=ϕ2(x,y,t+s).expand() # right hand side of (17.2) for ϕ2\n",
"b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (17.2) for ϕ2"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c2=(a2-b2) # difference between lhs and rhs\n",
"c2.full_simplify() # simplification"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.9**\n",
"\n",
"Compute components of the infinitesimal generator $X$ of $\\phi_t$ from the previous example for $a=1,\\ b=2$."
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# continuation\n",
"# the first component of X is the derivative of ϕ1 w.r. to t\n",
"X1=diff(x+t,t).subs({t:0});\n",
"X1"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# the second component of X is the derivative of ϕ2 w.r. to t\n",
"X2=diff(y+2*t,t).subs(t=0);X2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus $\\ \\ X=\\frac{\\partial}{\\partial x}+2\\frac{\\partial}{\\partial y}.$\n",
"\n",
"\n",
"
\n",
"\n",
"**Example 17.10**\n",
"\n",
"Plot the vector field $X$ from the previous example."
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"Graphics object consisting of 36 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"M = Manifold(2, 'M') # manifold M\n",
"X. = M.chart() # Cartesian coordinates\n",
"X = M.vector_field(1,2, name='v') # vector field X\n",
"p=X.plot(scale=0.1,arrowsize=1.5,number_values={x:6, y:6},\n",
" ranges={x: (-1,1), y: (-1,1)},color='grey') # plot X\n",
"p.show(aspect_ratio=1) # show plot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.11**\n",
"\n",
"For $ϕ$ from Example 17.8\n",
"plot the curves $\\phi_{(x,y)}$ for some selected points $(x,y)$, where \n",
"$ϕ_{(x,y)} (t) = ϕ(x,y, t)$. "
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 30 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# continuation of Example 17.8\n",
"# selected curves ϕ_(x,y) \n",
"ϕ(x,y,t)=(ϕ1(x,y,t).subs(a=1),ϕ2(x,y,t).subs(b=2)) # a=1, b=2\n",
"s1=[parametric_plot( ϕ(x,y,t).subs({x:1,y:y0}), \n",
" (t,-5,5),ymax=5,ymin=-5,color='grey') \n",
" for y0 in range(-15,15)] # x=1,y=-15,-14,...,14\n",
"sum(s1).show(aspect_ratio=1) # combine plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.12**\n",
"\n",
"Consider $ϕ=(ϕ_1,ϕ_2)$ defined by\n",
"$$\\left\\{\\begin{matrix}\n",
"ϕ_1(x,y,t)=x\\exp(at),\\\\\n",
"ϕ_2(x,y,t)=y\\exp(bt),\n",
"\\end{matrix}\n",
"\\right.\n",
"$$\n",
"\n",
"for fixed $a,b\\in R$. Show that $ϕ$ defines a one-parameter group of transformations."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var('t,s,x,y,a,b') # symbolic variables\n",
"ϕ1(x,y,t)=exp(a*t)*x # define ϕ1\n",
"ϕ2(x,y,t)=exp(b*t)*y # define ϕ2\n",
"\n",
"a1=ϕ1(x,y,t+s).expand() # right hand side of (17.2) for ϕ1\n",
"b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (17.2) for ϕ1\n",
"\n",
"c1=(a1-b1) # difference between lhs and rhs\n",
"c1.full_simplify() # simplification"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [],
"source": [
"a2=ϕ2(x,y,t+s).expand() # right hand side of (17.2) for ϕ2\n",
"b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (17.2) for ϕ2"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c2=(a2-b2) # difference between lhs and rhs\n",
"c2.full_simplify() # simplification"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.13**\n",
"\n",
"Compute components of the infinitesimal generator $X$ of $\\phi_t$ from the previous example for $a=1,\\ b=2$."
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# first component of X is the derivative of ϕ1 w.r. to t\n",
"X1=diff(exp(t)*x,t).subs({t:0}); # cf. (8.10)\n",
"X1"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*y"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# second component of X is the derivative of ϕ2 w.r. to t\n",
"X2=diff(exp(2*t)*y,t).subs({t:0});\n",
"X2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus $\\ \\ X=x\\frac{\\partial}{\\partial x}+2y\\frac{\\partial}{\\partial y}.$\n",
"\n",
"
\n",
"\n",
"**Example 17.14**\n",
"\n",
"Plot the vector field $X$ from the previous example."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 36 graphics primitives"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"M = Manifold(2, 'M') # manifold M=R^2\n",
"X. = M.chart() # Cartesian coordinates\n",
"X = M.vector_field(x,2*y, name='v') # vector field X\n",
"p=X.plot(scale=0.1,arrowsize=1.5,number_values={x:6, y:6},\n",
" ranges={x: (0.1,3), y: (-3,3)},color='grey') # plot X\n",
"p.show(aspect_ratio=0.3) # show plot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Example 17.15**\n",
"\n",
"For $ϕ$ from Example 17.12\n",
"plot the curves $\\phi_{(x,y)}$ for some selected points $(x,y)$, where \n",
"$ϕ_{(x,y)} (t) = ϕ(x,y, t)$. "
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 10 graphics primitives"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ϕ(x,y,t)=(exp(t)*x,exp(2*t)*y) # define ϕ\n",
"# plot curves passing through (1,y0), for y0 in [-5,...,4]\n",
"s1=[parametric_plot( ϕ(x,y,t).subs({x:1,y:y0}), (t, -0.5,0.5 ),\n",
" color='grey',ymax=3,ymin=-3,aspect_ratio=0.15) \n",
" for y0 in range(-5,5)] # x=1, y=-5,-4,...,4\n",
"sum(s1) # combine plots"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## What's next?\n",
"\n",
"Take a look at the notebook [Integral curves](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/18Manifold_Integral_Curves.ipynb)."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.6",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.10"
}
},
"nbformat": 4,
"nbformat_minor": 4
}