EPAL-v1/ex.I.4.20.iii.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "07364731-a05b-4a27-b653-c7578a8132cd",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "<table style=\"width:100%\">\n",
    "    <tr>\n",
    "        <td>\n",
    "            <a href=\"http://v-v-kisil.scienceontheweb.net/MoebInv-notebooks/EPAL-v1/ex.I.4.20.iii.html\">\n",
    "  <img src=\"../svg/view-html.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "<a href=\"https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.4.20.iii.ipynb\">\n",
    "  <img src=\"../svg/open-github.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "            <a href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/ex.I.4.20.iii.ipynb\">\n",
    "  <img src=\"../svg/exec-colab.svg\" alt=\"Open In Colab\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "        <td>\n",
    "            <a href=\"https://codeocean.com/capsule/7952650/tree\">\n",
    "  <img src=\"../svg/code-ocean.svg\" alt=\"Code Ocean\"/>\n",
    "            </a>\n",
    "        </td>\n",
    "    </tr>\n",
    " </table>\n",
    "\n",
    "<a  align=\"center\" href=\"https://colab.research.google.com/github/vvkisil/MoebInv-notebooks/blob/master/Introduction/Software_installation_GUI_integration.ipynb\">Google CoLab software installation</a>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41a7f43b-1dd6-4eca-a0cf-14c2a17e21ab",
   "metadata": {
    "collapsed": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# Transformations of the symplectic form under matrix multiplication\n",
    "\n",
    "**Exercise I.4.20.iii** [ [1] ](#[1])\n",
    "$\n",
    "\\newcommand{\\Space}[3][]{\\mathbb{#2}^{#3}_{#1}}\n",
    "  \\newcommand{\\cycle}[3][]{{#1 C^{#2}_{#3}}}\n",
    "  \\newcommand{\\realline}[3][]{#1 R^{#2}_{#3}}\n",
    "  \\newcommand{\\bs}{\\breve{\\sigma}}\n",
    "  \\newcommand{\\zcycle}[3][]{#1 Z^{#2}_{#3}}\n",
    "  \\newcommand{\\SL}[1][2]{\\mathrm{SL}_{#1}(\\Space{R}{})}\n",
    "  \\newcommand{\\rs}{\\sigma_r}\n",
    "  \\newcommand{\\lvec}[1]{\\overrightarrow{#1}}\n",
    "$    The symplectic form on\n",
    "    $\\Space[\\sigma]{R}{2}$ is invariant if vectors are multiplied\n",
    "    by a $2\\times 2$ real matrix with the unit determinant."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ef9d8ab-264b-47aa-a9ae-e8b277be866a",
   "metadata": {
    "collapsed": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "**Solution.** Define the symplectic form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3e139350-19d5-48ed-b0c5-eb739cc5c533",
   "metadata": {
    "autoscroll": false,
    "collapsed": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "from init_cycle import *\n",
    "def sf(P,P1):\n",
    "    return P.op(0)*P1.op(1)-P.op(1)*P1.op(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b43da07-b8a2-4b8e-82ab-3f6047969d57",
   "metadata": {
    "collapsed": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Two generic initial vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "be9e2f91-a906-4b3e-b940-11e24f4e9630",
   "metadata": {
    "autoscroll": false,
    "collapsed": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "Z=matrix([[u],[v]])\n",
    "Z1=matrix([[u1],[v1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f13fa65-76d6-4055-8641-f2e71b595796",
   "metadata": {
    "collapsed": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Now we define the generic matrix and transformed vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "427aad34-1ce9-48f2-a575-5935058e1614",
   "metadata": {
    "autoscroll": false,
    "collapsed": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "M=matrix([[a,b],[c,d]])\n",
    "Y=(M*Z).evalm()\n",
    "Y1=(M*Z1).evalm()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f2e7fa0-0483-407a-9809-6d67d9ec7012",
   "metadata": {
    "collapsed": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "The result check"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2d135182-361d-499e-8823-8f256f69260f",
   "metadata": {
    "autoscroll": false,
    "collapsed": false,
    "ein.hycell": false,
    "ein.tags": "worksheet-0",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "\"Symplectic form is invariant up to the factor det(M): %s\" %\\\n",
    "(sf(Y,Y1)-M.determinant()*sf(Z,Z1)).normal().is_zero()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1282ef20-1618-491b-8b67-6ca3bbba75f1",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "\n",
    "This notebook is a part of the *MoebInv notebooks* project [ [2] ](#[2]).\n",
    "\n",
    "### References\n",
    "<a id=\"references\"></a>\n",
    "<a id=\"[1]\"></a>\n",
    "1. Vladimir V. Kisil. *Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\\mathbb{R})$*. [Imperial College Press](https://www.worldscientific.com/worldscibooks/10.1142/p835), London, 2012. Includes a live DVD.\n",
    "\n",
    "   <a id=\"[2]\"></a>\n",
    "2.  Vladimir V. Kisil, [*MoebInv notebooks*](https://github.com/vvkisil/MoebInv-notebooks), 2019.\n",
    "\n",
    "[Back to Folder](https://github.com/vvkisil/MoebInv-notebooks/blob/master/EPAL-v1/)\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "argv": [
    "/usr/bin/python3",
    "-m",
    "ipykernel_launcher",
    "-f",
    "{connection_file}"
   ],
   "display_name": "Python 3 (ipykernel)",
   "env": null,
   "interrupt_mode": "signal",
   "language": "python",
   "metadata": {
    "debugger": true
   },
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  },
  "name": "ex.I.4.20.iii.ipynb"
 },
 "nbformat": 4,
 "nbformat_minor": 5
}