Harmonic oscillator is a second order derivative equation
=========================================================

In classical mechanics a *simple harmonic oscillator* is a system that, after a small displacement from equilibrium, experiences a restoring force :math:`F` proportional to that displacement. Displacement is not only limited to physical motion, but should be interpreted in general terms. Examples include small-angle pendulums, mass–spring systems, acoustic resonators, and electrical RLC circuits.
If :math:`F` is the only force acting on the system, the system is called a simple harmonic oscillator.

**Conditions:**

#. There is no damping (i.e. friction) in the system.
#. The system experiences a single restoring force :math:`F` (for mechanical oscillators).

**Links:**

#. `Wikipedia – Simple harmonic oscillator <https://en.wikipedia.org/wiki/Harmonic_oscillator#Simple_harmonic_oscillator>`__

.. py:currentmodule:: symplyphysics.oscillations.natural_oscillations.harmonic_oscillator_is_second_derivative_equation

.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: displacement

    Displacement of oscillator from equilibrium as a function of time. See :attr:`~symplyphysics.symbols.basic.any_quantity`.

Symbol:
    :code:`x(t)`

Latex:
    :math:`x{\left(t \right)}`

Dimension:
    :code:`any_dimension`


.. py:data:: angular_frequency

    :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the oscillator.

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: law

    :code:`Derivative(x(t), (t, 2)) = -w^2 * x(t)`


    Latex:
        .. math::
            \frac{d^{2}}{d t^{2}} x{\left(t \right)} = - \omega^{2} x{\left(t \right)}