{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 18. Integral curves\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 $X$ be a vector field on a smooth manifold $M$. A curve $\\gamma : I → M$ is an **integral\n", "curve of $\\mathbf{X}$** if \n", "\n", "\\begin{equation}\n", "\\displaystyle \\gamma'_t = X_{\\gamma(t)} , \\mbox{ for } t ∈ I.\n", "\\tag{18.1}\n", "\\end{equation}\n", "\n", "Let us recall that $\\gamma'_t\\ $ is the tangent vector to $\\gamma$ at $t$ (defined in (8.9)).\n", "\n", "If $\\gamma(0) = x\\in M$, we say that **the curve $\\gamma$ starts at $x$.**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Let ϕ be a one-parameter group of transformations on $M$ (defined in a previous notebook) and let $X$ be its\n", "infinitesimal generator.\n", "If we put $y = ϕ_{t_0} (x) = ϕ(x, t_0 )$, we get\n", "\n", "$$ϕ_y (t) = ϕ(y, t) = ϕ(ϕ(x, t_0 ), t) = ϕ(x, t_0 + t) = ϕ_x (t_0 + t).$$\n", "\n", "The tangent vector to the curve $\\phi_y$ at $t=0$ satisfies\n", "$$(ϕ_y )'_0 ( f ) =\\frac{d}{dt}(f (ϕ_y (t)))\\Big|_{t=0}=\n", "\\frac{d}{dt}(f (ϕ_x (t+t_0)))\\Big|_{t=0}=\n", "\\frac{d}{ds}(f (ϕ_x (s)))\\Big|_{s=t_0}=(ϕ_x)'_{t_0} ( f ),\n", "$$
\n", "for any $f ∈ C^∞ (M)$, and therefore $(ϕ_x )'_{t_0} = (ϕ_y )'_0 = X_y = X_{ ϕ_x (t_0 )}$. \n", "\n", "We have checked that:
\n", "if $y = ϕ_x (t_0 )$, for some $t_0 ∈ R$ then $(ϕ_x )'_{t_0} = (ϕ_y )'_0$\n", "and, therefore, $(ϕ_x )'_{t_0} = X_{ϕ_x (t_0 )}$ ; that is, **the tangent vector to the curve $ϕ_x$, at any point\n", " of the curve, coincides with the value of the infinitesimal generator of ϕ at that point.**\n", "\n", "Thus\n", "if $ϕ$ is a one-parameter group of transformations and\n", "$X$ is its infinitesimal generator, then **the curve $ϕ_x$ is an integral curve of X that starts at $x$**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us recall that if $\\phi=(x^1,\\ldots,x^n)$ are local coordinates on a smooth manifold $M$, $\\gamma$ is a smooth curve in $M$, then from (8.10) it follows\n", "$$\\gamma'_t=\\frac{d}{dt}(x^i\\circ\\gamma)\\Big|_t\\frac{\\partial}{\\partial x^i}\\Big|_{\\gamma(t)}.$$\n", "\n", "Since from $\\ \\ (Xf)(p)=X_p(f)\\ $ we obtain $X(x^i)(\\gamma(t))=X_{\\gamma(t)}(x^i)$, then by (8.5)\n", "the right hand side of (18.1) can be written as \n", "$$X_{\\gamma(t)} = \n", "X_{\\gamma(t)}(x^i)\\frac{∂}{∂ x^i}\\Big|_{\\gamma(t)}=\n", "X(x^i)(\\gamma(t))\n", "\\frac{∂}{∂ x^i}\\Big|_{\\gamma(t)}=\n", "(X^i\\circ\\gamma)(t) \\frac{∂}{∂ x^i}\\Big|_{\\gamma(t)},$$\n", "where $X^i=X(x^i).$
\n", "The last two equations imply, that the integral curves $\\gamma$ of $X$ defined by (18.1) **satisfy the system of ordinary differential equations**\n", "\\begin{equation}\n", "\\frac{d}{dt}(x^i\\circ\\gamma)=X^i\\circ\\gamma,\\quad i=1,\\ldots,n.\n", "\\tag{18.2}\n", "\\end{equation}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The right-hand side of (18.2) ca be written in the form\n", "$$\n", "(X^i ◦ \\gamma)(t) = (X^i ◦ φ^{−1} ) φ(\\gamma(t))\\\\\n", "= (X^i ◦ φ^{−1} ) (x^1 (\\gamma(t)), . . . , x^n (\\gamma(t)))\\\\\n", "= (X^i ◦ φ^{−1} ) ((x^1 ◦ \\gamma)(t),..., (x^n ◦ \\gamma)(t)),\n", "$$\n", "so (18.2) is equivalent to\n", "\\begin{equation}\n", "\\frac{d(x^i\\circ\\gamma)}{dt}= (X^i ◦ φ^{−1} )(x^1 ◦ \\gamma, . . . , x^n ◦ \\gamma).\n", "\\tag{18.3}\n", "\\end{equation}\n", "If we put $f^i = X^i ◦ φ^{−1} $ and replace $x^i ◦ \\gamma$ by $x^i$\n", "we obtain the simplified form of the system (18.3)\n", "\\begin{equation}\n", "\\frac{dx^i}{dt}=f^i(x^1,\\ldots,x^n),\\quad i=1,\\ldots,n.\n", "\\tag{18.4}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For a smooth vector field $X$, on a smooth manifold $M$, given $x\\in M$ there exist a unique integral curve $\\gamma$ of the vector field $X$ starting at $x$, defined in some interval $I\\subset R$. Define\n", "\n", "\\begin{equation}\n", "\\phi(x,t)=\\gamma(t).\n", "\\tag{18.5}\n", "\\end{equation}\n", "
\n", "We want to check that **if the integral curves of $X$ are defined for all $t\\in R$, then $\\phi$ is a one-parameter group of transformations**.\n", "\n", "Consider the curve $\\chi$ defined by \n", "$$\\chi(t)=\\gamma(t+s),\\quad s-\\mbox{fixed}.$$\n", "The curve $\\chi$ is an integral curve of $X$, since for $f\\in C^\\infty(M)$\n", "\n", "$$\\chi'_t(f)=\\frac{d(f\\circ\\chi)}{dt}\\Big|_t\n", "=\\lim_{h\\to 0}\\frac{f(\\chi(t+h))-f(\\chi(t))}{h}\\\\\n", "=\\lim_{h\\to 0}\\frac{f(\\gamma(t+h+s))-f(\\gamma(t+s))}{h}\n", "=\\gamma'_{t+s}(f)=X_{\\gamma(t+s)}(f)=X_{\\chi(t)}(f).\n", "$$
\n", "The curve $\\chi$ starts at $\\chi(0)=\\gamma(s)$ and the curve $\\phi(\\gamma(s),t)$ also starts at $\\gamma(s)$. By the uniqueness of the integral curves of smooth vector fields we have\n", "\n", "$$\\chi(t)=\\phi(\\gamma (s),t)=\\phi(\\phi(x,s),t).\n", "$$\n", "We have also \n", "$$\n", "\\chi(t)=\\gamma(t+s)=\\phi(x,t+s),\n", "$$\n", "so\n", "\\begin{equation}\n", "\\phi(\\phi(x,s),t)=\\phi(x,t+s).\n", "\\tag{18.6}\n", "\\end{equation}\n", "\n", "Thus $\\phi$ defines a one-parameter group of transformations." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.1** \n", "\n", "For $\\ X=y\\frac{\\partial}{\\partial x}+x\\frac{\\partial}{\\partial y}$, the system (18.4) takes the form\n", "$$\\frac{dx}{dt}=y,\\quad \\frac{dy}{dt}=x.$$\n", "\n", "Let us solve the system with Sympy. " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle x_{0} \\cosh{\\left(t \\right)} + y_{0} \\sinh{\\left(t \\right)}$" ], "text/plain": [ "x0*cosh(t) + y0*sinh(t)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import * # import Sympy\n", "init_printing() # use latex to output results\n", "t,x0,y0 = symbols('t,x0,y0') # Sympy variables\n", "x = Function('x') # Sympy function x(t)\n", "y = Function('y') # Sympy function y(t)\n", "ics = {x(0): x0,y(0):y0} # initial conditions\n", " # solve ODE system in Sympy:\n", "sol = dsolve([x(t).diff(t)-y(t), y(t).diff(t) -x(t)],ics=ics)\n", "import sympy # \"sympy\" not defined previously\n", "(sol[0].rhs).rewrite(sympy.sin).simplify() # use Sympy rewrite method\n", " # to simplify first component" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(first component of the solution)," ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle x_{0} \\sinh{\\left(t \\right)} + y_{0} \\cosh{\\left(t \\right)}$" ], "text/plain": [ "x0*sinh(t) + y0*cosh(t)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(sol[1].rhs).rewrite(sympy.sin).simplify() # do the same for second comp." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(second component)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.2**\n", "\n", "Check, that the integral curves from previous example define one-parameter group of transformations.\n", "\n", "
\n", "\n", "Let us define the components of the one-parameter group of transformations $\\phi$ corresponding to the vector field \n", "\n", "$$X=(𝑥_0\\cosh(𝑡)+𝑦_0\\sinh(𝑡))\\frac{\\partial}{\\partial x}\n", "+(𝑥_0\\sinh(𝑡)+𝑦_0\\cosh(𝑡))\\frac{\\partial}{\\partial y}\n", "$$\n", "\n", "from the previous example, using (18.5):" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "reset()\n", "var('t,s,x,y') # symbolic variables\n", " # \\phi[Tab] gives the greek letter phi\n", "ϕ1(x,y,t)=x*cosh(t) + y*sinh(t) # define ϕ1\n", "ϕ2(x,y,t)=x*sinh(t) + y*cosh(t) # define ϕ2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let us compute the difference between the right and left hand sides of the first components of (18.6):" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a1=ϕ1(x,y,t+s).trig_expand().expand() # right hand side of (18.6) for ϕ1\n", "b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (18.6) for ϕ1\n", "a1-b1 # rhs-lhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The same for the second components:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a2=ϕ2(x,y,t+s).trig_expand().expand() # right hand side of (18.6) for ϕ2\n", "b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s).expand() # left hand side of (18.6) for ϕ2\n", "a2-b2 # rhs-lhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We have checked that (18.6) is fulfilled." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.3**\n", "\n", "For the vector field $\\displaystyle X = x^2 \\frac{∂}{∂ x} + 2x y \\frac{∂}{∂ y}$ the system (18.4) takes the form\n", "$$\\frac{dx}{dt}=x^2,\\quad \\frac{dy}{dt}=2xy.$$\n", "
\n", "Solve it.\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "from sympy import * # import Sympy\n", "x0=symbols('x0') # initial value for x in Sympy\n", "y0=symbols('y0') # initial value for y in Sympy\n", "t = symbols('t') # symbolic variable t in Sympy\n", "x = Function('x') # function x(t) in Sympy\n", "y = Function('y') # function y(t) in Sympy\n", "ics = {x(0): x0,y(0):y0} # initial conditions\n", " # solve ODE system in Sympy:\n", "sol = dsolve([x(t).diff(t) - x(t)**2, \n", " y(t).diff(t) - 2*x(t)*y(t)],ics=ics)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system has the following solution:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ x{\\left(t \\right)} = - \\frac{x_{0}}{t x_{0} - 1}, \\ y{\\left(t \\right)} = \\frac{y_{0}}{\\left(t x_{0} - 1\\right)^{2}}\\right]$" ], "text/plain": [ "[Eq(x(t), -x0/(t*x0 - 1)), Eq(y(t), y0/(t*x0 - 1)**2)]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "init_printing()\n", "list(map(simplify,sol))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression is not defined for $t = 1/x_0$ , and therefore we are not dealing\n", "with a one-parameter group of transformations (by definition defined for $t\\in R$), despite the fact that $X$ is smooth." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.4**\n", "\n", "Show that the integral curve from the previous example defines a local version of \n", "one parameter group of transformations $\\phi$ with $t$ in a sufficiently small neighborhood of zero.\n", "\n", "
\n", "\n", "Define components of $\\phi$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "reset()\n", "var('t,s,x,y') # symbolic variables\n", "ϕ1(x,y,t)=-x/(t*x - 1) # define ϕ1\n", "ϕ2(x,y,t)=y/(t*x - 1)**2 # define ϕ2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "next, the right and left hand sides of (18.6):" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "a1=ϕ1(x,y,t+s).normalize() # right hand side of (18.6) for ϕ1\n", "b1=ϕ1(ϕ1(x,y,t),ϕ2(x,y,t),s).normalize() # lhs of (18.6) for ϕ1\n", "\n", "a2=ϕ2(x,y,t+s).normalize() # right hand side of (18.6) for ϕ2\n", "b2=ϕ2(ϕ1(x,y,t),ϕ2(x,y,t),s).normalize() # lhs of (18.6) for ϕ2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and the differences between the right and left hand sides" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0, 0)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a1-b1,a2-b2 # lhs - rhs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So (18.6) is fulfilled for sufficiently small $t$ and $s$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.5**\n", "\n", "For $\\displaystyle X = \\frac{1}{2} (x^2 − y^2 ) \\frac{∂}{∂ x} +\n", "x y \\frac{∂}{∂ y}$ on $M = \\{(x, y) ∈ R^2 : y > 0\\}$ we have\n", "$$\\frac{dx}{dt}=\\frac{1}{2}(x^2-y^2),\\quad \\frac{dy}{dt}=xy.$$\n", "Solve the system.

\n", "\n", "The system implies the single differential equation\n", "$$\\frac{dy}{dx} = 2x y/(x^2 − y^2 ).$$\n", "We can solve this equation with Sympy:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[(log(x**2/y(x) + y(x)),)]" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from sympy import * # import Sympy\n", "init_printing(False) # we want to copy-paste the result\n", "x=symbols('x') # Sympy variable x\n", "y = Function('y') # Sympy function y(x)\n", "sol = dsolve(y(x).diff(x) -2*x*y(x)/(x**2-y(x)**2))#,ics=ics\n", "var('C1') # C1 is the constant from solution\n", "solve(sol,C1) # solve the result with respect to C1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The result means that \n", "$$C1=\\log(x^2/y+y).$$\n", "If we replace $C1$ by $\\log C$, we obtain $$C=x^2/y + y.$$\n", "or\n", "$$x^2+y^2=Cy,$$\n", "so the integral curves are circles centered at $(0,C/2)$ with radius $C/2.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example 18.6**\n", "\n", "Show the integral curves from the previous example." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "reset()\n", "var('x y') # symbolic variables\n", "# show curves x^2/y+y=C for 30 values of C selected by the system\n", "p=contour_plot(log(x^2/y + y),(x,-8,8),(y,0,16),\n", "fill=false,contours=30,linewidths=2)\n", "p.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Remark.** In many cases exact solutions of the ODE systems defining the integral curves are not available. In that case one can use numerical tools." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "**Example 18.6**\n", "\n", "Using numerical tools find selected integral curve of the vector field from previous example." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 1 graphics primitive" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x,y,t=var('x y t')\n", "# find numerically the integral curve of X=(X1,X2), where\n", "# X1=1/2(x^2-y^2), X2=xy, passing through (0.1,0.01)\n", "P=desolve_system_rk4([0.5*(x^2-y^2),x*y],[x,y],ics=[0,0.1,0.01],ivar=t,\n", " end_points=60)\n", "Q=[ [j,k] for i,j,k in P] # list of points of the integral curve\n", "LP=list_plot(Q,plotjoined=True,color='grey') # line through points of Q\n", "plot(LP).show(aspect_ratio=1) # show with the same scale on both axes" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What's next?\n", "\n", "Take a look at the notebook [Lie derivative](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/19Manifold_Lie_Derivative.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 }