.. _sphx_glr_api_gallery_astro_nbody:

N-body dynamics
-----------------

Direct-summation gravity in :mod:`physicskit.astro.nbody`: the sensitive
dependence on initial conditions Poincare discovered in the three-body
problem, the figure-eight choreography, the virial theorem, and the
long-term stability that symplectic integration buys N-body simulation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Studying the three-body problem for King Oscar II&#x27;s 1889 prize, Poincare found that no general closed-form solution exists at all: two trajectories starting arbitrarily close together can separate at an exponential rate, the first rigorous demonstration of what would later be called deterministic chaos. Lagrange (1772) found one very special three-body configuration that is exactly solvable -- three masses at the vertices of an equilateral triangle, rigidly rotating about their common center of mass -- but Routh (1875) showed its linear stability depends on the mass ratios (stable only when one mass strongly dominates, as for the Sun-Jupiter-Trojan system). For masses like 3, 4, and 5 (in G=1 units), it is deep in the unstable regime, and provides a numerically clean, close-encounter-free setting to watch Poincare&#x27;s sensitive dependence directly with NBodySystem, without needing to resolve the extreme close encounters of a genuinely colliding configuration.">

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  .. image:: /api/gallery/astro/nbody/images/thumb/sphx_glr_plot_01_three_body_sensitivity_thumb.png
    :alt:

  :doc:`/api/gallery/astro/nbody/plot_01_three_body_sensitivity`

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      <div class="sphx-glr-thumbnail-title">Poincare's discovery: sensitive dependence in the three-body problem</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For two centuries after Newton, every known periodic three-body solution demanded either a special mass ratio (Lagrange&#x27;s equilateral triangle) or a rigidly symmetric configuration (Euler&#x27;s collinear solutions). Moore found something qualitatively different in 1993: three equal masses chasing one another endlessly around a single figure-eight curve, a genuine choreography with no distinguished body -- proven to exist by Chenciner and Montgomery in 2000. figure_eight_initial_conditions returns exactly this initial condition; this example integrates it with NBodySystem, checks that the configuration is genuinely periodic by finding the time at which body 1 first returns closest to its own starting point, and checks that energy and angular momentum stay conserved over one full period.">

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  .. image:: /api/gallery/astro/nbody/images/thumb/sphx_glr_plot_02_figure_eight_choreography_thumb.png
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  :doc:`/api/gallery/astro/nbody/plot_02_figure_eight_choreography`

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      <div class="sphx-glr-thumbnail-title">The figure-eight three-body choreography</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Clausius (1870) showed that for any bounded system interacting through conservative forces, the time-averaged kinetic energy and the time-averaged virial of the forces are related; for a system bound purely by inverse-square gravity, this reduces to">

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  .. image:: /api/gallery/astro/nbody/images/thumb/sphx_glr_plot_03_virial_theorem_thumb.png
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  :doc:`/api/gallery/astro/nbody/plot_03_virial_theorem`

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      <div class="sphx-glr-thumbnail-title">Clausius's virial theorem for a bound gravitational orbit</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Wisdom and Holman (1991) showed that symplectic integrators -- of which the kick-drift-kick leapfrog scheme is the simplest example -- keep a conservative system&#x27;s numerically computed energy oscillating within a small bound forever, rather than drifting secularly away as an ordinary (non-symplectic) integrator&#x27;s does. This is exactly what made simulations at the scale of the Millennium Simulation&#x27;s billions of particles, evolved over cosmological timescales, numerically trustworthy at all. This example integrates the same eccentric two-body orbit with NBodySystem&#x27;s symplectic leapfrog_step for a thousand orbital periods, alongside a hand-written non-symplectic explicit-Euler stepper using the same force law (~physicskit.astro.nbody.gravitational_acceleration), and compares their energy error growth directly.">

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  .. image:: /api/gallery/astro/nbody/images/thumb/sphx_glr_plot_04_symplectic_long_term_stability_thumb.png
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  :doc:`/api/gallery/astro/nbody/plot_04_symplectic_long_term_stability`

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      <div class="sphx-glr-thumbnail-title">Symplectic integration and long-term N-body stability</div>
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.. toctree::
   :hidden:

   /api/gallery/astro/nbody/plot_01_three_body_sensitivity
   /api/gallery/astro/nbody/plot_02_figure_eight_choreography
   /api/gallery/astro/nbody/plot_03_virial_theorem
   /api/gallery/astro/nbody/plot_04_symplectic_long_term_stability