mwcraig/oscillator_sols.ipynb

## oscillator_sols.ipynb
{
 "metadata": {
  "name": "oscillator_sols"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# PUT YOUR NAME HERE (double click a text block to edit it)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from __future__ import division\n",
      "figsize(10, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Today's lab\n",
      "\n",
      "Your *eventual* goal in today's lab is to explore three different numerical techniques for integrating the equation of motion for a simple harmonic oscillator (specifically, a pendulum):\n",
      "\n",
      "+ The Euler method, which you have used in every other python lab this semester.\n",
      "+ The Euler-Cromer method, a variant of the Euler method which is much better than Euler for an oscillating system.\n",
      "+ The Runge-Kutta method, which is more accurate than either of the above, but also a bit more ecomplicated."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### You will turn in, as a single zip file:\n",
      "\n",
      "+ *This notebook*, modified as indicated in the notebook.\n",
      "+ One python file, called ``integrators.py``; it should include the code from previous weeks with the addition of new functions you write for this week's lab."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### You will need:\n",
      "\n",
      "+ This note book\n",
      "+ The file `integrators.py` included in the zip file with this notebook."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Part 1: Oscillating pendulum: the physics\n",
      "\n",
      "## Equation of motion \n",
      "\n",
      "The equation of motion for the pendulum shown on page 49 of  the book is\n",
      "\n",
      "$$\\frac{\\text{d}^2\\theta}{\\text{d}t^2} = - \\frac{g}{\\ell} \\sin\\theta,$$\n",
      "\n",
      "where $g$ is the acceleration due to gravity and $\\ell$ is the length of the pendulum.\n",
      "\n",
      "If the angle $\\theta$ is small then $\\sin\\theta \\approx \\theta$ and the equation of motion becomes\n",
      "\n",
      "$$\\frac{\\text{d}^2\\theta}{\\text{d}t^2} \\approx  - \\frac{g}{\\ell} \\theta$$\n",
      "\n",
      "Like we did in the projectiile motion lab it is convenient to break this second order differential equation into two spearate first order differential equations, one for the \"position\" $\\theta$ and one for the angular velocity $\\omega$:\n",
      "\n",
      "$$\\dfrac{\\text{d}\\theta}{\\text{d}t} = \\omega$$\n",
      "\n",
      "$$\\dfrac{\\text{d}\\omega}{\\text{d}t} = - \\frac{g}{\\ell} \\theta$$\n",
      "\n",
      "\n",
      "Throughout this lab we will work with a specific pendulum, with mass 1 kg and length $\\ell=1.0$ meters, and will assume we are on Earth.\n",
      "\n",
      "Rather than setting those numbers in all of the equations that follow, set the appropriate value of each variable below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mass = 1.0  # kg, replace with desired value\n",
      "length = 1.0  # meters, replace with desired value\n",
      "g = 9.8  # m/s/s, replace with desired value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Frequency and period\n",
      "\n",
      "Recall that for a simple harmonic oscillator, of which this is an example if the amplitude is small, the frequency of oscillation is given by\n",
      "\n",
      "$$\\Omega = \\sqrt{\\frac{g}{\\ell}}$$\n",
      "\n",
      "and the period is \n",
      "\n",
      "$$T = \\frac{2\\pi}{\\Omega}$$.\n",
      "\n",
      "We will need both of these later, so define the appropriate variables in the cell below.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Omega = sqrt(g/length)   ## REPLACE WITH CORRECT FORMULA\n",
      "period = 2*pi/Omega  ## REPLACE WITH CORRECT FORMULA"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Energy\n",
      "\n",
      "The total energy of the pendulum is, in general,\n",
      "\n",
      "$$E = \\frac{1}{2} m \\ell^2 \\omega^2 + mg\\ell (1-\\cos\\theta);$$\n",
      "\n",
      "in the limit of small angle $\\cos\\theta \\approx 1 - \\theta^2/2$, so the energy, in this limit, is\n",
      "\n",
      "$$E \\approx \\frac{1}{2} m \\ell^2 \\left(\\omega^2 + \\frac{g}{\\ell} \\theta^2\\right)$$\n",
      "\n",
      "As with any closed system the energy of this oscillator *should be conserved*. One way we will evaluate how accurate the numerical methods you will explore today will work is by looking at how well each conserves energy.\n",
      "\n",
      "The cell below defines a function that calculates the energy of the oscillator, in the case when the angle is small."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def pendulum_energy(theta, omega, m=mass, g=g, l=length):\n",
      "    return .5 * m * l**2 * (omega**2 + (g/l) * theta**2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "For the specific pendulum we are considering ($m=1.0$ kg, $\\ell=1.0$ m, $g=9.8\\text{m/s}^2$), what should the energy be if $\\omega=0$ and $\\theta=0.1$? Write your answer in the appropriate place in the cell below, and check that the function we defined above calculates the energy correctly."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expected_value = 0.0  # REPLACE WITH CORRECT VALUE\n",
      "computed_value = pendulum_energy(0.1, 0)\n",
      "print computed_value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.049\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### BRIEFLY DISCUSS HERE (double click to edit)\n",
      "\n",
      "Do the expected and computed values agree? "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Analytic solution for small angles\n",
      "\n",
      "In the limit of small angles there is an analytic solution for the angle as a function of time,\n",
      "\n",
      "$$\\theta(t) = \\theta(0) \\cos(\\Omega t) + \\frac{\\omega(0)}{\\Omega} \\sin(\\Omega t),$$\n",
      "\n",
      "where $\\theta(0)$ is the initial angle and $\\omega(0)$ is the initial angular velocity."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "Define, in the cell below, a function for computing the analytic solution for the pendulum. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def analytic_pendulum(time, init_theta, init_omega, Omega=Omega):\n",
      "    return init_theta*cos(Omega*time) + init_omega/Omega*sin(Omega*time) ## REPLACE THIS With The CORRECT EXPRESSION"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### DO THIS TOO:\n",
      "\n",
      "Check your anayltic solution by plotting it for the initial conditions $\\theta(0)=0.1$ and $\\omega(0)=0$ from $t=0$ to $t=5T$, where $T$ is the period. In other words, graph five full periods of the motion. The array of times to use for this part is already defined for you."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "end_time = 5*period\n",
      "time = linspace(0, end_time, num=100)\n",
      "plot(time, analytic_pendulum(time, 0.1, 0.0))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "[<matplotlib.lines.Line2D at 0x330a070>]"
       ]
      },
      {
       "output_type": "display_data",
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GEeSOSc+iRayIRRG0GgaIheNNTawUhHXkCFBbG+x2/poaoKKCFx1RBG1NAmxPRuW1Jbl1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\ngg==\n"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Part 2: Oscillating pendulum: numerical solution, Euler method\n",
      "\n",
      "## Review: Euler method\n",
      "\n",
      "Your first numerical method to look at today is the Euler method, which you have used in every python-based lab this semester.\n",
      "\n",
      "As a reminder, the Euler method finds the new value of a variable at time $t_{i+1}$ from the value of that variable and its derivative at time $t_{i}$ in  the simplest way possible. If we call the variable $y$ and its derivative $v$ then\n",
      "\n",
      "$$ y[i+1] = y[i] + v[i]\\Delta t$$\n",
      "\n",
      "In other words, the Euler method finds the new value of $y$ by assuming the function $y$ is linear from time $t_i$ to $t_{i+1}$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Euler method, applied to the pendulum\n",
      "\n",
      "There is a function called `pendulum_linear_euler` in the file `integrators.py` that calculates the angle as a function of time using the Euler method. You do **not** need to modify this function in any way today.\n",
      "\n",
      "The first thing we will do today is plot the angle as a function of time using the Euler method. As usual, we need to do some setup first."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Setup"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Load `integrators.py`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import integrators\n",
      "reload(integrators)  # using reload ensures that you are using the most recent version of integrators"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "<module 'integrators' from 'integrators.pyc'>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Set up arrays for time, angle, angular velocity \n",
      "\n",
      "We are doing this in a slightly different way than in past weeks. In this lab it will be useful to always find the position and velocity of the  pendulum for a fixed range of time (say from $t=0$ to $t=5T$ as in the analytic example above), but to change the number of steps, which will also change the timestep.\n",
      "\n",
      "The `linspace` function from numpy does exactly what we want: specifiy a starting and ending time and a number of steps."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 500\n",
      "time, delta_t = linspace( 0., end_time, num=N_steps, retstep=True)\n",
      "theta = zeros_like(time)\n",
      "omega = zeros_like(time)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Set the initial conditions\n",
      "\n",
      "We will use the initial conditions $\\theta=0.1$, $\\omega=0$ as above...not for any deep reason, really. We just have to choose *something*."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta[0] = 0.1\n",
      "omega[0] = 0.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Use Euler method to find angle and angular velocity\n",
      "\n",
      "The cell below calculates the angle and velocity as a function of time using the Euler method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "theta_euler, omega_euler = integrators.pendulum_linear_euler(time, theta, omega, g=g, length=length)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Plot the results\n",
      "\n",
      "In the cell below, plot the angle as a function of time. Some plotting reminders:\n",
      "\n",
      "+ When plotting numerical results, *use points not lines*. \n",
      "\n",
      "    - Tell python to use markers by adding the `marker` argument to `plot` (e.g. `marker='d'`)\n",
      "    - Tell python not to draw a line by adding `linestyle='None'` as one of the arguments to `plot`.\n",
      "    - A list of all marker styles is at http://matplotlib.org/api/artist_api.html#matplotlib.lines.Line2D.set_marker\n",
      "\n",
      "+ Always include labels for the $x$- and $y$-axes.\n",
      "\n",
      "    - `xlabel('some appropriate label')` puts a label on the $x$-axis.\n",
      "    - `ylabel` does the same thing for the $y$-axis.\n",
      "\n",
      "+ Include either a plot title or a legend. \n",
      "\n",
      "    - Add a title to the plot with `title('My cool plot')`\n",
      "    - Add a label to an individual line or set of points by putting `label='some sensible name'` as one of the arguments for `plot`\n",
      "    - Add a legend with `legend()`"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "Modify the cell below so that the plot follows all of the rules listed above.\n",
      "\n",
      "**NOTE:** For the rest of the plots in this lab do yourself a favor and start by copying the cell below once you have it all fixed up right."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, theta_euler, linestyle='-', marker='None')\n",
      "ylabel('Somethign')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<matplotlib.text.Text at 0x371ed70>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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nj+pKembLurVTp2SkO2CA6kpi441CGdbCY1NnjWvW3NNjWPvKV76C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\nVyvl+TuMNAAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "Discuss the plot you just made: is it reasonable or not? If not, what is unreasonable about it? \n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Is the energy any better?\n",
      "\n",
      "The angle calculated using the Euler method is clearly off...one might ask whether the energy looks any more reasonable.\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "Modify the cell below so that the plot follows all of the rules listed above."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, pendulum_energy(theta_euler, omega_euler))\n",
      "ylabel('bad label')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "<matplotlib.text.Text at 0x38cf2d0>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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SYNAg+PRTePDBsJMo2bghQJKkOHMTgKqba84kSdqBbzcBXHedmwBUfSxnkiTtwMSJbgJQ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\nAIwEdAIcB55EOPYAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Can you fix the Euler method with smaller time steps?\n",
      "\n",
      "One way to make a numerical integrator perform better is to simply use a smaller step size $\\Delta t$ or, equivalently, a larger number of steps.\n",
      "\n",
      "The plot above was made with with 500 steps, or 100 steps per period (because we set up the time so that it went from 0 to 5 periods). Perhaps it would look better if we used more points...so, in the cells below, make new versions of these plots, using more points."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Redo angle plot with 1000 points\n",
      "\n",
      "As a first attempt to improve things, try doubling the number of points, to 1000. To do so, you will need to copy the contents of a few of the code cells above to the single code cell below and make the appropriate change in $N$. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 1000\n",
      "time, delta_t = linspace( 0., end_time, num=N_steps, retstep=True)\n",
      "theta = zeros_like(time)\n",
      "omega = zeros_like(time)\n",
      "theta[0] = 0.1\n",
      "omega[0] = 0.0\n",
      "theta_euler, omega_euler = integrators.pendulum_linear_euler(time, theta, omega, g=g, length=length)\n",
      "\n",
      "plot(time, theta_euler, linestyle='-', marker='None')\n",
      "ylabel('Somethign')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<matplotlib.text.Text at 0x3a76f90>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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Ly2OIA+SHuIQE8Zrj6sazzc1iYU1qqvx7u7pClSHObs6HuFBInEZw+LDqkegp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\nQ/sQ4gAACBBvQ1xzs1ReLo0YEfya4FyWhLjGxkZNmjRJsbGxmjx5ss6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      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Replot energy with 1,000 points \n",
      "\n",
      "In the cell below make a plot of energy vs time for this new case. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, pendulum_energy(theta_euler, omega_euler))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "[<matplotlib.lines.Line2D at 0x3c477f0>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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lN2e0JJXa44/DPffAzJnQqFHoNJKU/Ha7ojV79mwaNmxI/fr1GTJkyC6fc8st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\nSZJiYtGSJEmKiUVLkiQpJv8PFmb25DjMRrYAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: DISCUSS \n",
      "\n",
      "Discuss this question: Did doubling the number of points improve the solution? Explain\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Redo plot with 5,000 points\n",
      "\n",
      "Doubling the number of points didn't seem to help much, so let's try something more extreme: increase the number of points by a factor of 10, from 500 to 5,000.\n",
      "\n",
      "You should be able to copythe contents of the cell above and paste it below, make one or two minor changes, and make the plot. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 5000\n",
      "time, delta_t = linspace( 0., end_time, num=N_steps, retstep=True)\n",
      "theta = zeros_like(time)\n",
      "omega = zeros_like(time)\n",
      "theta[0] = 0.1\n",
      "omega[0] = 0.0\n",
      "theta_euler, omega_euler = integrators.pendulum_linear_euler(time, theta, omega, g=g, length=length)\n",
      "\n",
      "plot(time, theta_euler, linestyle='-', marker='None')\n",
      "ylabel('Somethign')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<matplotlib.text.Text at 0x3f0dd30>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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VIc7F8MzhVLsxxA3D5flZKhY1AEBGhtg7ysVPyp2dQEuL/L2zALfnZ6maZJ+W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\nEQcAAGAgIg4AAMBA/wCsYec7m233+AAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Replot energy with 5,000 points \n",
      "\n",
      "In the cell below make a plot of energy vs time for this new case. Use the limits provided in the cell already."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, pendulum_energy(theta_euler, omega_euler))\n",
      "ylim(0.045, 0.06)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "(0.045, 0.06)"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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1TG5uLlOnTmXNmjVMmjSJrKwsADZs2MC6deuAqJgVFxdzxhlncMwxx/Dhhx+y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\nUOR2AAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: DISCUSS \n",
      "\n",
      "Discuss this question: Did increasing the number of points by a factor of 10 improve the solution? Explain\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Aside 1: DRY programming\n",
      "\n",
      "The acronym DRY stands for the programming idea \"Don't Repeat Yourself\". Whenever you find yourself copying and pasting large chunks of code to get something done you should ask if you can find a way to reduce the cutting and pasting, often by defining a function. \n",
      "\n",
      "There are a few problems with cutting and pasting:\n",
      "\n",
      "+ If you make a mistake in the first place you write the code, you copy the mistake everywhere else.\n",
      "+ Making the same change to all of the places you have pasted is tedious, and likely to be done incorrectly.\n",
      "+ It takes a fair bit of time.\n",
      "\n",
      "Admittedly, cutting and pasting can be a *fast* way to get things done...at least in the short term.\n",
      "\n",
      "## Make this notebook DRYer: define a function that generates initial arrays\n",
      "\n",
      "In the cell below we define a function that takes as its input the number of points desired for time, theta, and omega, and that returns those arrays.\n",
      "\n",
      "With this function it is easier to change the number of points in each array."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def init_arrays(n_points, start_time, end_time):\n",
      "    \"\"\"\n",
      "    Generate arrays for integration\n",
      "\n",
      "    Returns time, delta_t, theta, omega\n",
      "    \"\"\"\n",
      "    time, delta_t = linspace(start_time, end_time, num=n_points, retstep=True)\n",
      "    theta = zeros_like(time)\n",
      "    omega = zeros_like(time)\n",
      "    return time, delta_t, theta, omega"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Using this function \n",
      "\n",
      "If you want a list of, say, 32,000 positions and velocities, with time from 0 to 5 times the period, you would get those like this:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 32000  # technically, you don't need this, but it makes it easier to use the number of steps later\n",
      "time, delta_t, theta, omega = init_arrays(N_steps, 0., 5*period)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## We could be even DRYer...\n",
      "\n",
      "Since we will want to find the motion of the pendulum using several different methods of integration and for several different numbers of steps, but always withthe same start and stop time and the same initial conditions $\\theta(0)=0.1$ and $\\omega(0)=0$, we could go a bit further and make a function that only takes the number of points, and also sets the initial conditions. \n",
      "\n",
      "That function is below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def init_conditions(n_points):\n",
      "    \"\"\"\n",
      "    Generate arrays, including initial conditions, for integration\n",
      "\n",
      "    Assumes that period has been set in a previous cell.\n",
      "\n",
      "    Returns time, delta_t, theta, omega\n",
      "    \"\"\"\n",
      "    time, delta_t, theta, omega = init_arrays(n_points, 0.0, 5*period)\n",
      "    theta[0] = 0.1\n",
      "    omega[0] = 0\n",
      "    return time, delta_t, theta, omega"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## How did this help?\n",
      "\n",
      "Suppose we want to try out the Euler method for the pendulum for one more case, with N=10,000. What took several lines of code above now takes just a few:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 10000\n",
      "time, delta_t, theta, omega = init_conditions(N_steps)\n",
      "theta_euler, omega_euler = integrators.pendulum_linear_euler(time, theta, omega, g=g, length=length)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Part 3: Oscillating Pendulum, Euler-Cromer method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Overview\n",
      "\n",
      "There is a very minor change to the Euler method, called the Euler-Cromer method, that works much better than the Euler method for numerical simulations of oscillators.\n",
      "\n",
      "In the file `integrators.py` is a function called `pendulum_linear_euler_cromer`. Compare it to the function `pendulum_linear_euler` and identify the difference; it will also be helpful to read the description of the Euler-Cromer method in Section 1 of Chapter 3 of Giordano."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "Explain, in words, what the difference is between the Euler and  the Euler-Cromer methods, based on the code in `integrators.py`.\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Angle and energy from the Euler-Cromer method\n",
      "\n",
      "The cell below finds the angle and angular velocity as a function of time using the Euler-Cromer method with only 500 steps."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 500\n",
      "time, delta_t, theta, omega = init_conditions(N_steps)\n",
      "theta_euler_cromer, omega_euler_cromer = integrators.pendulum_linear_euler_cromer(time, theta, omega, g=g, length=length)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS:\n",
      "\n",
      "In the cell below add the code to make a plot of $\\theta$ vs time. Follow the rules above about plots (and feel free to copy and paste one of your plot commands from above and modify it)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, theta_euler_cromer, linestyle='None', marker='d')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "[<matplotlib.lines.Line2D at 0x40d1730>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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ETLwXsC+HQ0ydSGMIQi74EPZdB3CbXBe2Me7pMYbPoiCImbgvYNeHg+sNO1Y+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\nxAAAABz5/wGHAXv22sv6aAAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Energy plot for Euler-Cromer\n",
      "\n",
      "In the cell below make aplot of energy vs time for the oscillator using the Euler-Cromer method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, pendulum_energy(theta_euler_cromer, omega_euler_cromer))\n",
      "ylim(0.045, 0.06)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "(0.045, 0.06)"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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Hsxfv1VfNMFJGRvvPX3+9mWNYXx9oWa68/HLH4VEyISeMQ6mOI730kjnotycW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\nR0eHuru7dffuXV27ds10rDGB9xVzjh49qokTJyo9PX3EfUd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      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Discuss\n",
      "\n",
      "Compare the angle vs time and energy vs time plots to the same plots you made earlier using the Euler method. \n",
      "\n",
      "+ Which method is more accurate? Explain.\n",
      "+ Neither method conserves energy perfectly. Describe the difference between them in terms of how the energy changes with time.\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Euler-Cromer with more points\n",
      "\n",
      "Repeat your plots above (both energy and time) but with more steps in the integration--use 1,000 steps this time."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: \n",
      "\n",
      "In the cell below make a plot of angle vs time for 1000 steps using Euler-Cromer."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 1000\n",
      "time, delta_t, theta, omega = init_conditions(N_steps)\n",
      "theta_euler_cromer, omega_euler_cromer = integrators.pendulum_linear_euler_cromer(time, theta, omega, g=g, length=length)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, theta_euler_cromer, linestyle='None', marker='d')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "[<matplotlib.lines.Line2D at 0x45fb070>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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ZKbJOB7P0oTTsIWYGcWSV8UKs+N1b73GjzFgoUfyWJquicR9ekigmy6JxH16S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\n"
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: \n",
      "\n",
      "In the cell below make a plot of energy vs time for 1000 steps using Euler-Cromer."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(time, pendulum_energy(theta_euler_cromer, omega_euler_cromer))\n",
      "ylim(0.045, 0.06)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "(0.045, 0.06)"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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ozJgxys3NVVJSUsPr5eXlKi0tVUVFhQoLCzVr1iwtXbpUkhQIBFRcXKzExMTjyn3//fdV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\n1TEPAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Discuss\n",
      "\n",
      "How did the Euler-Cromer solution change when you increased the number of steps? Did the angle seem more accurate? What about the energy?\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Aside 2: The DRYest version of all (and functions as arguments)\n",
      "\n",
      "You may have noticed by now (after all the copy-and-pasting you have done) that we never really graph `theta` and `omega`...we only care about the output of the integration routines.\n",
      "\n",
      "It turns out we can go one step further in making this notebook DRY, and at the same time introduce a new idea in python: using functions as arguments to other functions.\n",
      "\n",
      "Consider the basic pattern we've used up to this point:\n",
      "\n",
      "    N_steps = 500                                            # set number of steps\n",
      "    time, delta_t, theta, omega = init_conditions(N_steps)   # set up time and initial arrays of angle and velocity \n",
      "                                                             # in the next step, integrate using the desried method\n",
      "    theta_euler_cromer, omega_euler_cromer = integrators.pendulum_linear_euler_cromer(time, theta, omega, g=g, length=length)\n",
      "\n",
      "The simplification we could make is to combine the second two steps into one, using the function below."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def integrate_pendulum(n_steps, integration_method):\n",
      "    \"\"\"\n",
      "    Find angle and angular velocity given a number of steps.\n",
      "\n",
      "    integration_method should be the name of the function to be used to find the angle and velocity \n",
      "\n",
      "    Returns time, theta_out, omega_out\n",
      "    \"\"\"\n",
      "    time, delta_t, theta, omega = init_conditions(n_steps)\n",
      "    \n",
      "    theta_out, omega_out = integration_method(time, theta, omega, g=g, length=length)\n",
      "    \n",
      "    return time, theta_out, omega_out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that the second argument to `integrate_pendulum` is a *function*, not a number or array. One of the magical things about python is that it is easy to treat functinos just like any other object."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## How to use this new function \n",
      "\n",
      "Want to find angle and velocity with 1237 steps using Euler-Cromer? No problem:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 1237\n",
      "time, theta_ec_1237, omega_ec_1237 = integrate_pendulum(N_steps, integrators.pendulum_linear_euler_cromer)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that the very last argument is the *name* of the function we want to use to find `theta` and `omega`."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Part 4: Oscillating pendulum, Runge-Kutta 2nd Order\n",
      "\n",
      "The book describes, in the first couple of sections of Appendix A, a method called the *Runge-Kutta* method, and describes in detail how to integrate numerically using the 2nd order Runge-Kutta method.\n",
      "\n",
      "The essence of the method is to:\n",
      "\n",
      "+ estimate the position and velocity at the *middle* of the time step\n",
      "+ calculate an acceleration at the midpoint using the esimated position and velocity\n",
      "+ calculate the position at the end of the step using the estimated velocity \n",
      "+ use the estimated acceleration to calcualte the velocity at the end of the timestep"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Angle and energy with Runge-Kutta, 500 steps\n",
      "\n",
      "Modify the cell below so that it calculates the angle and angular velocity for the pendulum using the Runge-Kutta method. \n",
      "\n",
      "**Note:** You will need to look in `integrators.py` for the name of the pendulum integrator that uses the Runge-Kutta method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 500\n",
      "time, theta_rk, omega_rk = integrate_pendulum(N_steps, integrators.pendulum_linear_runge_kutta2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Plot angle and energy vs time\n",
      "\n",
      "You should make **two** separate plots, one in each of the cells below. The first should be angle vs time. The second should be energy vs time."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# MAKE YOUR ANGLE-TIME PLOT IN THIS CELL\n",
      "plot(time, theta_rk, linestyle='None', marker='d')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1e4ead0>]"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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10t4/CmV2Sd/DQQDOJunjmtoQQW1JI82gR03PhvvnnuCDGOk/u6SBgIOeXdr3cBCA00n7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\nAABgCUEMAADAEoIYAACAJQQxAAAAS/4feKG2dlFqef4AAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# MAKE YOUR ENERGY-TIME PLOT IN THIS CELL....\n",
      "plot(time, pendulum_energy(theta_rk, omega_rk))\n",
      "ylim(0.045, 0.06) # Use these y limits to make it easier to compare plots"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "(0.045, 0.06)"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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YAEABIgwAoAARBgBQgAgDAChAhAEAFCDCAAAKEGEAAAWIMACAAkQYAEABIgwA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\nrkJggg==\n"
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### YOU DO THIS: Discuss\n",
      "\n",
      "Compare the angle vs time and energy vs time Runge-Kutta plots to the same plots you made earlier using the Euler and Euler-Cromer method. \n",
      "\n",
      "+ Which method is more accurate? Explain.\n",
      "+ None of the methods conserves energy perfectly. Describe the difference between them in terms of how the energy changes with time.\n",
      "\n",
      "**DOUBLE CLICK TO EDIT THIS CELL AND PUT YOUR ANSWER HERE**"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Summary: Comparison of methods\n",
      "\n",
      "### YOU DO THIS: \n",
      "\n",
      "Make a single plot that shows energy as a function of time for\n",
      "\n",
      "+ the Euler method\n",
      "+ the Euler-Cromer method\n",
      "+ the Runge-Kutta method\n",
      "+ the analytic solution \n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 500\n",
      "time, theta_rk, omega_rk = integrate_pendulum(N_steps, integrators.pendulum_linear_runge_kutta2)\n",
      "time, theta_euler, omega_euler = integrate_pendulum(N_steps, integrators.pendulum_linear_euler)\n",
      "time, theta_ec, omega_ec = integrate_pendulum(N_steps, integrators.pendulum_linear_euler_cromer)\n",
      "\n",
      "plot(time, pendulum_energy(theta_euler, omega_euler), linestyle='None', marker='d', label='Euler')\n",
      "plot(time, pendulum_energy(theta_ec, omega_ec), linestyle='None', marker='+', label='Euler-Cromer')\n",
      "plot(time, pendulum_energy(theta_rk, omega_rk), linestyle='None', marker='s', label='Runge-Kutta')\n",
      "plot(time, [pendulum_energy(0.1, 0)]*N_steps, label='Analytic')\n",
      "ylabel('Energy, Joules')\n",
      "xlabel('Time (sec)')\n",
      "ylim(0.04, 0.06)\n",
      "legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "<matplotlib.legend.Legend at 0x1ed5ff0>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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qVSM5jQoAAKpED5wLGIUKAACcRoBLEaNQAQCA0whwKWIeVAAA4DSm0rIA86AC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\nGIaAAwAAMAwBBwAAYJj/ACI/vqnTWf6vAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Epilogue: Why not always use Runge-Kutta?\n",
      "\n",
      "There are several answers to this. One is that the Euler method is simple to understand, and for many cases it works well enough. In most cases, though not for oscillations, you can make the Euler method work as well as you want by just increasing the number of steps.\n",
      "\n",
      "In addition, the Euler and Euler-Cromer methods are somewhat faster than the Runge-Kutta method. The cells below time how long it takes the computer to solve the pendulum problem using each one of the methods. Note that Runge-Kutta is somewhat longer than the others."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N_steps = 10000"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit \n",
      "time, theta_eu, omega_eu = integrate_pendulum(N_steps, integrators.pendulum_linear_euler)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1 loops, best of 3: 203 ms per loop\n"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit\n",
      "time, theta_ec, omega_ec = integrate_pendulum(N_steps, integrators.pendulum_linear_euler_cromer)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1 loops, best of 3: 230 ms per loop\n"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit\n",
      "time, theta_rk, omega_rk = integrate_pendulum(N_steps, integrators.pendulum_linear_runge_kutta2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1 loops, best of 3: 274 ms per loop\n"
       ]
      }
     ],
     "prompt_number": 37
    }
   ],
   "metadata": {}
  }
 ]
}