{% 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/lec04/lecture_04_harmonic.py`


What you see
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The example show the Harmonic Oscillator example simulated using RK4. 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)`. The position
vector obey a linear differential equation, shown in the slide, and I then simply simulate that differential equation
using RK4 as implemented in the exercises.