.. _sphx_glr_api_gallery_classical_lagrangian:

Lagrangian mechanics
--------------------

Systems whose equations of motion are derived symbolically from a
Lagrangian :math:`L(q, \dot q, t)` via
:class:`~physicskit.classical.utils.symbolic.LagrangianEngine` -- a double
pendulum, a bead on a rotating hoop, coupled oscillators, and an elastic
(spring) pendulum showing 1:2 autoparametric resonance -- including one
built entirely from scratch, outside :mod:`physicskit.classical.systems`,
showing the raw symbolic-derivation workflow directly.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="DoublePendulum is two point masses m_1, m_2 on massless rigid rods of length l_1, l_2, the second hanging from the first, with generalized coordinates q = (\theta_1, \theta_2) measured from the downward vertical:">

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  .. image:: /api/gallery/classical/lagrangian/images/thumb/sphx_glr_plot_01_double_pendulum_thumb.png
    :alt:

  :doc:`/api/gallery/classical/lagrangian/plot_01_double_pendulum`

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      <div class="sphx-glr-thumbnail-title">The double pendulum and deterministic chaos</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="BeadOnRotatingHoop is a bead sliding without friction on a circular hoop of radius R, forced to rotate at fixed angular speed \Omega about its vertical diameter. With generalized coordinate \theta (the bead&#x27;s polar angle on the hoop), the Lagrangian in the rotating frame is">

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  .. image:: /api/gallery/classical/lagrangian/images/thumb/sphx_glr_plot_02_bead_on_rotating_hoop_thumb.png
    :alt:

  :doc:`/api/gallery/classical/lagrangian/plot_02_bead_on_rotating_hoop`

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      <div class="sphx-glr-thumbnail-title">The bead on a rotating hoop: a pitchfork bifurcation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="CoupledOscillators is a chain of N masses m connected in a line by identical linear springs k, both ends fixed to immovable walls -- the small-N, exactly-linear prototype for the larger lattice chains in physicskit.classical.systems.chains. With generalized coordinates q_i (displacement of mass i from equilibrium) and q_0 = q_{N+1} = 0, the Lagrangian L = T - V built symbolically by LagrangianEngine is">

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  .. image:: /api/gallery/classical/lagrangian/images/thumb/sphx_glr_plot_03_coupled_oscillators_thumb.png
    :alt:

  :doc:`/api/gallery/classical/lagrangian/plot_03_coupled_oscillators`

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      <div class="sphx-glr-thumbnail-title">Coupled oscillators: normal modes of a small harmonic chain</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Every Lagrangian system built into physicskit.classical (DoublePendulum, BeadOnRotatingHoop, CoupledOscillators, HeavySymmetricTop) hides its use of LagrangianEngine inside a system class. This example shows the raw workflow directly, for a system that is not already built in: the elastic (spring) pendulum -- a mass m on a spring of natural length l_0 and stiffness k, free to swing as a pendulum too, so its radial &quot;stretch&quot; and angular &quot;swing&quot; motions are coupled. Using polar-style coordinates q = (r, \theta) for the pivot-to-mass distance and swing angle,">

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  .. image:: /api/gallery/classical/lagrangian/images/thumb/sphx_glr_plot_04_custom_lagrangian_engine_thumb.png
    :alt:

  :doc:`/api/gallery/classical/lagrangian/plot_04_custom_lagrangian_engine`

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      <div class="sphx-glr-thumbnail-title">Building your own system: LagrangianEngine from scratch</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="ElasticPendulum is a mass m on a Hookean spring of natural length L_0 and stiffness k, free to both stretch and swing. With generalized coordinates q = (s, \theta) -- s the spring&#x27;s stretch beyond L_0 (so the pivot-to-mass distance is r = L_0 + s) and \theta the swing angle from the downward vertical -- its Lagrangian is">

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  .. image:: /api/gallery/classical/lagrangian/images/thumb/sphx_glr_plot_05_elastic_pendulum_thumb.png
    :alt:

  :doc:`/api/gallery/classical/lagrangian/plot_05_elastic_pendulum`

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      <div class="sphx-glr-thumbnail-title">The Elastic Pendulum and 1:2 Autoparametric Resonance</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/classical/lagrangian/plot_01_double_pendulum
   /api/gallery/classical/lagrangian/plot_02_bead_on_rotating_hoop
   /api/gallery/classical/lagrangian/plot_03_coupled_oscillators
   /api/gallery/classical/lagrangian/plot_04_custom_lagrangian_engine
   /api/gallery/classical/lagrangian/plot_05_elastic_pendulum