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    "# Tutorial A12"
   ]
  },
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   "metadata": {},
   "source": [
    "__1__ How does logging behave when you rerun the script without restarting the kernel? What happens if you delete the log-file before you rerun?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__2__ Play around with the `datefmt` keyword when configuring the logger to omit displaying year and months."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__3__ Below you find a script that propagates a particle in a harmonic potential numerically (using the euler integrator). You do not need to know about details of the scientific background yet. Just note so much: The script monitors the total energy of the particle at every propagation step. Whenever the change in energy is larger than 0.3, the velocity should be rescaled (multiplied) by a factor of 0.9. This part is not yet included in the script below. If this happens a message should be written to a log file, stating the old and the new speed. Your task is it, to setup a logger and insert a logging statement at the right position in the code."
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    "<div class=\"alert alert-warning\">\n",
    "\n",
    "**Advanced:** If you are already a bit familar with the background of task 3. Try to implement the euler integrator yourself (and ignore the definition of the `propagate()` function below.\n",
    "    \n",
    "</div>"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "__4__ Plot the result of the simulation as velocity vs. position."
   ]
  },
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   "source": [
    "# Example code for task 3\n",
    "\n",
    "def force(x):\n",
    "    \"\"\"Compute the instantaneous force acting on the particle\n",
    "    \n",
    "    Args:\n",
    "        x: Current position\n",
    "        x0 (global): Position of the minimum of the potential\n",
    "        k (global): Force constant\n",
    "    \"\"\"\n",
    "    \n",
    "    return -k * (x - x0)\n",
    "\n",
    "\n",
    "def e_pot(x):\n",
    "    \"\"\"Compute the potential energy of the particle\n",
    "    \n",
    "    Args:\n",
    "        x: Current position\n",
    "        x0 (global): Position of the minimum of the potential\n",
    "        k (global): Force constant\n",
    "    \"\"\"\n",
    "\n",
    "    return 0.5 * k * (x - x0)**2\n",
    "\n",
    "\n",
    "def e_kin(v):\n",
    "    \"\"\"Compute the kinetic energy of the particle\n",
    "    \n",
    "    Args:\n",
    "        v: Particle velocity\n",
    "        m (global): Particle mass\n",
    "    \"\"\"\n",
    "\n",
    "    return 0.5 * m * v**2\n",
    "\n",
    "\n",
    "def propagate(x, v):\n",
    "    \"\"\"Compute the position of the particle one timestep further\n",
    "    \n",
    "    Args:\n",
    "        x: Current position\n",
    "        v: Current velocity\n",
    "        dt (global): Propagation timestep\n",
    "        m (global): Particle mass\n",
    "        \n",
    "    Returns:\n",
    "        New position and velocity\n",
    "    \"\"\"\n",
    "    \n",
    "    x_new = x + v*dt + force(x)/(2*m) * dt**2\n",
    "    v_new = v + force(x)/m * dt\n",
    "\n",
    "    return x_new, v_new\n",
    "\n",
    "\n",
    "# Preparing lists to store the computed trajectories\n",
    "x_t = []  # Position\n",
    "v_t = []  # Velocity\n",
    "e_t = []  # Energy\n",
    "\n",
    "# Particle/Simulation parameters\n",
    "x0 = 0  # Position of the potential minimum\n",
    "k = 1.  # Force constant\n",
    "m = 1.  # Particle mass\n",
    "\n",
    "dt = 1e-1  # Simulation timestep\n",
    "T = 10000  # Total number of steps\n",
    "\n",
    "x = 1  # starting position\n",
    "v = -2  # starting velocity\n",
    "\n",
    "# Put them in the output lists \n",
    "x_t.append(x)\n",
    "v_t.append(v)\n",
    "e_t.append(e_pot(x) + e_kin(v))\n",
    "\n",
    "# Simulate\n",
    "for i in range(T):\n",
    "    # Update position and velocity\n",
    "    x, v = propagate(x, v)\n",
    "    x_t.append(x)\n",
    "    v_t.append(v)\n",
    "    \n",
    "    # Update total energy\n",
    "    e_t.append(e_pot(x) + e_kin(v))    "
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