restrepo/Least Action

## Least Action
{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Least Action"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Load modules"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy.optimize \n",
      "global g  \n",
      "g=9.8"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Following:\n",
      "http://www.eftaylor.com/software/ActionApplets/LeastAction.html, we define the Action in one small segment of the free fall movement in one-dimension. \n",
      "    \n",
      "For one segment of the action between $(t_1,x_1)$, and $(t_2,x_2)$, with $\\Delta t$ sifficiently small such that $L$ can be considered constant, we have\n",
      "\\begin{eqnarray}\n",
      "S_1&=&\\int_{t_1}^{t_2} L dt \\\\\n",
      "&\\approx& \\left[\\frac12 m v^2-m g h \\right]\\Delta t\\\\\n",
      "&\\approx& \\left[\\frac12 m\\left(\\frac{x_2-x_1}{t_2-t_1}\\right)^2-m g \\frac{x_2+x_1}{2} \\right](t_2-t_1)\n",
      "\\end{eqnarray}\n",
      "that corresponds to Eq. (11) of Am. J. Phys, Vol. 72(2004)478: http://www.eftaylor.com/pub/Symmetries&ConsLaws.pdf"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import Image\n",
      "Image(filename='leastaction.png')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 2,
       "text": [
        "<IPython.core.display.Image at 0x7f2fcd92fd10>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Code implementation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We define the Action $S$ such of an object of mass $m$ throw vertically upward from $x_{\\hbox{ini}}$, such that $t_{\\hbox{end}}$ seconds later the object return to a height $x_{\\hbox{end}}$, as\n",
      "\\begin{eqnarray}\n",
      "S&=&\\int_{t_{\\hbox{ini}}}^{t_{\\hbox{end}}} L dt \\\\\n",
      "&=&\\sum_i S_i \\Delta t\n",
      "\\end{eqnarray}"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def S(x,t=3.,m=0.2,xini=0.,xend=0.):\n",
      "    t=float(t)\n",
      "    Dt=t/x[:-1].size\n",
      "    x=np.asarray(x)\n",
      "    #Fix initial and final point\n",
      "    x[0]=xini\n",
      "    x[-1]=xend\n",
      "    return ((0.5*m*(x[1:]-x[:-1])**2/Dt**2-0.5*m*g*(x[1:]+x[:-1]))*Dt).sum()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Function to find the least Action by using `scipy.optimize.fmin_powell`. It start from $\\mathbf{x}=(x_{\\hbox{ini}},0,0,\\ldots,x_{\\hbox{end}})$ and find the least action"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def xfit(n,t=3.,m=0.2,xini=0.,xend=0.,ftol=1E-8):\n",
      "    '''Find the array of n (odd) components that minimizes the action S(x)\n",
      "\n",
      "    :Parameters:\n",
      "\n",
      "    n: odd integer \n",
      "        dimension of the ndarray x that minimizes the action  S(x,t,m)\n",
      "    t,m: numbers\n",
      "       optional parameters for the action\n",
      "    ftol: number\n",
      "        acceptable relative error in S(x) for convergence.\n",
      "\n",
      "    :Returns: (x,xmax,Smin)\n",
      "    \n",
      "    x: ndarray\n",
      "        minimizer of the action S(x)\n",
      "        \n",
      "    xini:\n",
      "    \n",
      "    xend:\n",
      "\n",
      "    xmax: number\n",
      "        Maximum height for the object\n",
      "\n",
      "    Smin: number\n",
      "        value of function at minimum: Smin = S(x)\n",
      "    '''\n",
      "    t=float(t)\n",
      "    if n%2==0:\n",
      "        print 'x array must be odd'\n",
      "        sys.exit()\n",
      "  \n",
      "    x0=np.zeros(n)\n",
      "    a=scipy.optimize.fmin_powell(S,x0,args=(t,m,xini,xend),ftol=ftol,full_output=1)\n",
      "    x=a[0]\n",
      "    x[0]=xini;x[-1]=xend\n",
      "    xmax=np.sort(x)[-1]\n",
      "    Smin=a[1]\n",
      "    Dt=t/x[:-1].size\n",
      "    return x,xmax,Smin,Dt  \n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Least Action calculation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "_Problem_: Let an object of mass $m=0.2$ Kg throw vertically updward and returning back to the same hand after 3 s. Find the function of distance versus time of least Action. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us divide the intervals in 21 parts:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t=3.\n",
      "m=0.2\n",
      "y=xfit(21,t,m)\n",
      "x=y[0]\n",
      "Smin=y[2]\n",
      "Dt=t/x[:-1].size\n",
      "tx=np.arange(0,t+Dt,Dt)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: -21.554977\n",
        "         Iterations: 28\n",
        "         Function evaluations: 5784\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Plot"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(tx,x,label='$S_{\\mathrm{min}}=$%.2f' %Smin)\n",
      "plt.plot(tx,x,'ro')\n",
      "plt.ylabel('HEIGHT meters')\n",
      "plt.xlabel('TIME seconds')\n",
      "plt.title('Worldline of least action')\n",
      "plt.legend(loc='best')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "<matplotlib.legend.Legend at 0x7f2fcd92ff10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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QPXv2BCApKYn+ciY0cewYvPSS2s5akoBwJr6+sGoV/OMfkJqqdTSOp8REEB0d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\n6vfe0zqS0pESAXDqFLRpo05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AxcWFX3/9laNHj9KjRw/DCJcHDhxg//79VKxYkSZNmhAV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\ncpIIhBDCyUkiEEIIJyeJQAghnNz/Awr6aBJ1uQGmAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f2fcc063050>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Dynamics of the least action solution"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Velocity"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v=(x[1:]-x[:-1])/Dt\n",
      "Dt=t/v[:-1].size\n",
      "tx=np.arange(0,t+Dt,Dt)\n",
      "plt.plot(tx,v)\n",
      "plt.plot(tx,v,'ro')\n",
      "plt.xlabel('TIME seconds')\n",
      "plt.ylabel('VELOCITY meter/seconds')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "<matplotlib.text.Text at 0x7f2fc01bdf90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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yZViyBFJTYe5ciI+HTzdZ2JySgs+VKxQ0bMiQhATN7HYAFQoRcSuHDtmuLk4f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\nPKMtNg/kKQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f2fc0571150>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Aceleration"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dt=t/x[:-1].size\n",
      "a=(v[1:]-v[:-1])/Dt\n",
      "pa=plt.hist(a)\n",
      "plt.title('ACELERATION meter/second$^2$')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "<matplotlib.text.Text at 0x7f2fc03de5d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f2fcc043b10>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Energy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "T=0.5*m*v**2\n",
      "V=0.5*m*g*(x[1:]+x[:-1])\n",
      "E=T+V\n",
      "print np.round(E,2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 21.56  21.56  21.56  21.56  21.56  21.56  21.55  21.55  21.56  21.56\n",
        "  21.56  21.56  21.56  21.56  21.56  21.55  21.55  21.56  21.56  21.55]\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Action"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The Action is minimal in each interval!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "SS=(T-V)*Dt\n",
      "SS"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "array([ 2.61768077,  1.45086657,  0.41339493, -0.49446   , -1.2723905 ,\n",
        "       -1.92031546, -2.43943472, -2.8282667 , -3.0874324 , -3.21713113,\n",
        "       -3.21711687, -3.08747347, -2.82816487, -2.4388897 , -1.92039431,\n",
        "       -1.27224489, -0.49433123,  0.41338249,  1.45064066,  2.6171035 ])"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print 'S_MINIMUM=%g  Joules*second' %SS.sum()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "S_MINIMUM=-21.555  Joules*second\n"
       ]
      }
     ],
     "prompt_number": 11
    }
   ],
   "metadata": {}
  }
 ]
}