.. _sphx_glr_api_gallery_classical_hamiltonian:

Hamiltonian phase space
-----------------------

Phase-space structure: KAM torus breakdown in the non-integrable
Henon-Heiles system, Liouville's theorem via a sheared swarm of
pendulums, action-angle variables for the simple pendulum, and
Hamilton's unified ``(q, p)`` canonical phase space itself, shared by a
librating and a rotating pendulum.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="HenonHeilesSystem is the classic non-integrable 2-DOF Hamiltonian (originally a model of stellar orbits in an axisymmetric galactic potential),">

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  .. image:: /api/gallery/classical/hamiltonian/images/thumb/sphx_glr_plot_01_henon_heiles_thumb.png
    :alt:

  :doc:`/api/gallery/classical/hamiltonian/plot_01_henon_heiles`

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      <div class="sphx-glr-thumbnail-title">The Henon-Heiles system: Poincare sections and KAM torus breakdown</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="PendulumSwarm evolves N mutually independent, identical simple pendulums together as one separable Hamiltonian system, one pair (q_i, p_i) per pendulum,">

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  .. image:: /api/gallery/classical/hamiltonian/images/thumb/sphx_glr_plot_02_liouville_swarm_thumb.gif
    :alt:

  :doc:`/api/gallery/classical/hamiltonian/plot_02_liouville_swarm`

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      <div class="sphx-glr-thumbnail-title">Liouville's theorem: a swarm of pendulums</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For a 1-DOF system like the simple pendulum, H = p^2/2 - (g/l)\cos q, every librating (bounded, non-circulating) orbit lies on a closed invariant curve in phase space labeled by a single number: the action">

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  .. image:: /api/gallery/classical/hamiltonian/images/thumb/sphx_glr_plot_03_action_angle_thumb.png
    :alt:

  :doc:`/api/gallery/classical/hamiltonian/plot_03_action_angle`

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      <div class="sphx-glr-thumbnail-title">Action-angle variables for the pendulum</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Lagrange&#x27;s mechanics lives in a &quot;position and velocity&quot; space (q, qdot); Hamilton&#x27;s 1834 reformulation trades each qdot for its canonically conjugate momentum p = dL/dqdot via a Legendre transform, and rewrites the second-order Euler-Lagrange equation as twice as many first-order equations for (q, p) moving through a single unified phase space:">

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  .. image:: /api/gallery/classical/hamiltonian/images/thumb/sphx_glr_plot_04_canonical_phase_space_thumb.png
    :alt:

  :doc:`/api/gallery/classical/hamiltonian/plot_04_canonical_phase_space`

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      <div class="sphx-glr-thumbnail-title">Hamilton's canonical (q, p) phase space</div>
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.. toctree::
   :hidden:

   /api/gallery/classical/hamiltonian/plot_01_henon_heiles
   /api/gallery/classical/hamiltonian/plot_02_liouville_swarm
   /api/gallery/classical/hamiltonian/plot_03_action_angle
   /api/gallery/classical/hamiltonian/plot_04_canonical_phase_space