.. _sphx_glr_api_gallery_classical_integrators:

Integrators
-----------

Why ``physicskit.classical`` defaults every conservative system to a symplectic
integrator (RK4's energy drifts monotonically, Verlet/Yoshida4 stay
bounded), and automated timestep selection with
:func:`~physicskit.classical.utils.stepsize.estimate_dt`.


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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="All three integrators here advance the same eccentric two-body Kepler orbit (~physicskit.classical.systems.newtonian.KeplerSystem, semi-major axis a=1, eccentricity e=0.6), i.e. the same Hamiltonian H(q,p) and the same Hamilton&#x27;s equations \dot q = \partial H/\partial p,\ \dot p = -\partial H/\partial q; only the numerical scheme used to step them forward differs. Velocity Verlet updates the separable H = T(p) + V(q) by">

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  .. image:: /api/gallery/classical/integrators/images/thumb/sphx_glr_plot_01_symplectic_vs_rk4_thumb.png
    :alt:

  :doc:`/api/gallery/classical/integrators/plot_01_symplectic_vs_rk4`

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      <div class="sphx-glr-thumbnail-title">Why symplectic integration: RK4 leaks energy, Yoshida4 does not</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="None of physicskit.classical&#x27;s integrators are adaptive: each takes a fixed step \Delta t for the whole run, so choosing \Delta t is normally a manual &quot;integrate, check drift, halve or double, repeat&quot; search -- exactly what estimate_dt automates, by searching for the largest \Delta t whose relative energy drift |H(t) - H(0)| / |H(0)| stays below a given tolerance over the full run. This runs it on two very different conservative systems: a quasi-periodic Kepler orbit (~physicskit.classical.systems.newtonian.KeplerSystem, semi-major axis a=1, eccentricity e=0.5, governed by the inverse-square gravitational Hamiltonian), where a short probe already predicts the right \Delta t, and a chaotic double pendulum (~physicskit.classical.systems.lagrangian.DoublePendulum, governed by its coupled Euler-Lagrange equations), where estimate_dt&#x27;s own verification stage has to catch and shrink a fast-probe estimate that would otherwise fail once actually run for the full step count.">

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  .. image:: /api/gallery/classical/integrators/images/thumb/sphx_glr_plot_02_estimate_dt_thumb.png
    :alt:

  :doc:`/api/gallery/classical/integrators/plot_02_estimate_dt`

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      <div class="sphx-glr-thumbnail-title">Automated timestep selection with estimate_dt</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/classical/integrators/plot_01_symplectic_vs_rk4
   /api/gallery/classical/integrators/plot_02_estimate_dt