{% import macros as m with context  %}

.. _harmonic_game:

Week 3: Harmonic Oscillator
=============================================================================================================

{{ m.embed_game('week3_harmonic') }}

.. topic:: Controls
    :class: margin

    :kbd:`p`
        Start the simulation
    :kbd:`r`
        Reset the game

    .. rubric:: Run locally

    :gitref:`../irlc/lectures/lec03/lecture_03_harmonic.py`


What you see
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The example show the Harmonic Oscillator (see (:cite:t:`herlau`, Subsection 11.1.2)) example. There is no control signal, :math:`u(t) = 0`, so the
system just oscillates forever.

How it works
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The bead has a position and a velocity which form the two coordinates of the state

.. math::

    \mathbf{x}(t) = \begin{bmatrix}\text{position} \\ \text{velocity} \end{bmatrix} = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}

The dynamics of the environment is then governed by the differential equation:

.. math::

    \dot{\mathbf x} = \begin{bmatrix}0 & 1 \\ -\frac{k}{m} & 0 \end{bmatrix}\mathbf{x}