:orphan:

Examples
========

This gallery walks through every public feature of ``physicskit.classical``:
Newtonian, Lagrangian, and Hamiltonian formulations of the same mechanics,
coupled chains and their solitons, rigid-body rotations, the symplectic
integrators underneath all of it, and the interactive/animated visualizers
built on top.

Each script in this gallery is self-contained and can be run directly with
``python examples/classical/<section>/<script>.py``. Every script also
carries an RST module docstring as its title/description and uses ``# %%``
markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source of
truth for what you see, not a copy of it.

Sections
--------

- **newtonian** -- direct force-and-acceleration integration: the oblique
  cannonball problem, Kepler orbits with perihelion precession and a
  power-law perturbation, angular-momentum conservation as a diagnostic
  distinct from energy conservation, and Noether's theorem's manifest
  (rotational) versus hidden (Kepler's Laplace-Runge-Lenz) symmetries side
  by side.
- **lagrangian** -- systems built from a Lagrangian: the chaotic double
  pendulum, a bead on a rotating hoop (a pitchfork bifurcation), the normal
  modes of coupled oscillators, and how to write a custom system with
  ``LagrangianEngine`` from scratch.
- **hamiltonian** -- phase-space methods: Poincare sections and KAM torus
  breakdown in the Henon-Heiles system, Liouville's theorem illustrated with
  a swarm of pendulums, action-angle variables for the pendulum, and
  Hamilton's unified ``(q, p)`` canonical phase space shared by a librating
  and a rotating pendulum.
- **rotations** -- rigid-body dynamics: the intermediate axis theorem
  (Dzhanibekov effect), both as a stability argument and as a literal
  tumbling 3D rigid body, and the heavy symmetric top's precession and
  nutation.
- **chains** -- coupled degrees of freedom: the harmonic chain's exact normal
  modes, the Fermi-Pasta-Ulam-Tsingou recurrence, and Sine-Gordon kink
  propagation and kink-antikink breathers.
- **integrators** -- why the choice of integrator matters: symplectic
  (Yoshida4) versus RK4 long-horizon energy drift, and automatic timestep
  selection with ``estimate_dt``.
- **visualizers** -- live and interactive displays: ``SideBySideAnimator``
  pairing a physical-space animation with phase/energy diagnostics, and
  interactive Plotly views of the SO(3) momentum sphere and orbits.


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Coupled lattice chains
----------------------

One-dimensional lattices: the exactly-solvable harmonic chain, the
Fermi-Pasta-Ulam-Tsingou beta-lattice and its famous non-ergodic
recurrence, and the discrete sine-Gordon chain's topological solitons.


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    <div class="sphx-glr-thumbcontainer" tooltip="HarmonicChain models N identical point masses m, connected in a line by identical springs of constant k, with both ends clamped to fixed walls (q_0 = q_{N+1} = 0). Writing q_i and p_i for the displacement and momentum of mass i, the system is the separable Hamiltonian">

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  .. image:: /api/gallery/classical/chains/images/thumb/sphx_glr_plot_01_harmonic_chain_thumb.png
    :alt:

  :doc:`/api/gallery/classical/chains/plot_01_harmonic_chain`

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      <div class="sphx-glr-thumbnail-title">The harmonic chain: normal modes, exactly</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="FPUTChain is the classic Fermi-Pasta-Ulam-Tsingou \beta-lattice: N fixed-end masses m coupled by springs with an added quartic non-linearity, so the inter-site stretch r_i = q_{i+1} - q_i costs potential energy">

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  .. image:: /api/gallery/classical/chains/images/thumb/sphx_glr_plot_02_fput_recurrence_thumb.png
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  :doc:`/api/gallery/classical/chains/plot_02_fput_recurrence`

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      <div class="sphx-glr-thumbnail-title">The Fermi-Pasta-Ulam-Tsingou recurrence</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="SineGordonChain is a discretized chain of N coupled pendulums (the Frenkel-Kontorova model) with free (Neumann-like) ends, so a topological kink can propagate off either end. The potential energy is">

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  .. image:: /api/gallery/classical/chains/images/thumb/sphx_glr_plot_03_sine_gordon_thumb.png
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  :doc:`/api/gallery/classical/chains/plot_03_sine_gordon`

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      <div class="sphx-glr-thumbnail-title">Sine-Gordon solitons: kink propagation and a kink-antikink breather</div>
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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 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>
    </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
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  :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>
    </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
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  :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>
    </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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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 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
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  :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
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  :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>
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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 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
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  :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>
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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
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  :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
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  :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
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  :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>
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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
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  :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>
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Newtonian mechanics
-------------------

Vector dynamics under a force: free fall with an oblique launch
velocity (:class:`~physicskit.classical.systems.newtonian.ProjectileMotion`) and the
2-body Kepler problem (:class:`~physicskit.classical.systems.newtonian.KeplerSystem`),
including the post-Newtonian and power-law perturbations that make
orbits precess, plus a pendulum viewed from Earth's rotating frame
(:class:`~physicskit.classical.systems.newtonian.FoucaultPendulum`), whose
Coriolis-driven swing-plane precession demonstrated that rotation
directly.


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    <div class="sphx-glr-thumbcontainer" tooltip="ProjectileMotion is free fall under constant gravity g with an oblique initial velocity -- the historical &quot;cannonball&quot; problem. Without drag it is the separable Hamiltonian">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_01_cannonball_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_01_cannonball`

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      <div class="sphx-glr-thumbnail-title">Free fall with an oblique velocity: the cannonball problem</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="KeplerSystem is the planar, reduced 2-body problem in the center-of-mass frame, a separable Hamiltonian in Cartesian coordinates q=(x,y), p=(p_x,p_y),">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_02_kepler_precession_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_02_kepler_precession`

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      <div class="sphx-glr-thumbnail-title">Kepler orbits and perihelion precession</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="FoucaultPendulum models a small-oscillation pendulum&#x27;s horizontal displacement q=(x,y) (x = East, y = North) as seen from the rotating surface of the Earth. In a non-rotating (inertial) frame it would obey the isotropic harmonic equation \ddot q = -\omega_0^2 q, \omega_0 = \sqrt{g/\ell}; but Earth&#x27;s surface itself turns about the local vertical at rate \omega_z = \omega_\mathrm{earth}\sin(\text{latitude}), adding a velocity-dependent Coriolis term:">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_03_foucault_pendulum_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_03_foucault_pendulum`

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      <div class="sphx-glr-thumbnail-title">The Foucault pendulum and Earth's rotation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="plot_02_kepler_precession shows apsidal precession from the effective post-Newtonian c_\mathrm{pn}/r^3 correction. KeplerSystem has a second, independent perturbation family added to its radial potential:">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_03_kepler_perturbations_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_03_kepler_perturbations`

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      <div class="sphx-glr-thumbnail-title">KeplerSystem's other perturbation: the power-law term</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="physicskit.classical.utils.conservation has more than just energy-drift helpers. angular_momentum_2d and angular_momentum_drift track the planar angular momentum">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_04_conservation_diagnostics_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_04_conservation_diagnostics`

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      <div class="sphx-glr-thumbnail-title">Angular momentum: a different conservation law from energy</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Emmy Noether&#x27;s theorem ties each continuous symmetry of a system&#x27;s action to an exactly conserved quantity built from it: time-translation symmetry gives energy, rotational symmetry gives angular momentum. The Kepler problem carries a third, less obvious conserved quantity, the Laplace-Runge-Lenz vector">

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  .. image:: /api/gallery/classical/newtonian/images/thumb/sphx_glr_plot_05_noether_hidden_symmetry_thumb.png
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  :doc:`/api/gallery/classical/newtonian/plot_05_noether_hidden_symmetry`

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      <div class="sphx-glr-thumbnail-title">Noether's Theorem and the Kepler Problem's Hidden Symmetry</div>
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Rigid-body rotations
--------------------

Free and torqued rigid bodies: the intermediate axis theorem
(Dzhanibekov effect) for a torque-free top, rendered both abstractly
(the SO(3) momentum sphere) and literally (a tumbling 3D body); a heavy
symmetric top's precession and nutation under gravity; Euler's disk
settling into a finite-time singularity; and a rattleback's one-way
spin reversal.


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    <div class="sphx-glr-thumbcontainer" tooltip="EulerTop is a torque-free rigid body: Euler&#x27;s equations for the body-frame angular velocity \boldsymbol\omega = (\omega_1, \omega_2, \omega_3) about the three principal axes, with moments of inertia I_1, I_2, I_3,">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_01_dzhanibekov_thumb.png
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  :doc:`/api/gallery/classical/rotations/plot_01_dzhanibekov`

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      <div class="sphx-glr-thumbnail-title">The intermediate axis theorem (Dzhanibekov effect)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="HeavySymmetricTop is a symmetric top with one point fixed, spinning under gravity, with generalized coordinates q = (\phi, \theta, \psi) -- the standard z-x-z Euler angles (precession, nutation, spin) -- and Lagrangian">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_02_heavy_symmetric_top_thumb.png
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  :doc:`/api/gallery/classical/rotations/plot_02_heavy_symmetric_top`

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      <div class="sphx-glr-thumbnail-title">The heavy symmetric top: effective potential, precession and nutation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The same torque-free rigid body as plot_01_dzhanibekov -- EulerTop, governed by Euler&#x27;s equations I_i\dot\omega_i = (I_j - I_k)\,\omega_j\omega_k (cyclic in i,j,k) for the body-frame angular velocity, coupled to the orientation quaternion. The SO(3) momentum-sphere picture (plot_01_dzhanibekov) shows the abstract geometry behind the intermediate axis theorem. This instead draws the actual rigid body -- a flattened &quot;paddle&quot; (book/phone-like) box, elongated along its smallest-moment axis and flattened along its largest-moment axis, exactly like the real object in the famous ISS video -- and rotates it in 3D with animate_rigid_body_tumble, which builds the wireframe box from half_extents and rotates its vertices at each frame via rotation_matrix, built from the integrated orientation quaternion. Spin it about the intermediate axis and it periodically flips end over end; spin it about either other principal axis and it just spins.">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_03_rigid_body_3d_thumb.gif
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  :doc:`/api/gallery/classical/rotations/plot_03_rigid_body_3d`

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      <div class="sphx-glr-thumbnail-title">The Dzhanibekov effect, literally: a tumbling 3D rigid body</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Spin a disk and set it rolling flat on a table and it settles into a shuddering near-steady state described, at each instant, by just its inclination angle \theta (between the disk plane and the table) and precession angle \phi (the azimuthal angle of the rolling contact point). Its contact point precesses faster and faster -- the characteristic rattle rising in pitch -- right up until the disk suddenly stops: a finite-time singularity. Rather than committing to one first-principles dissipation mechanism (viscous air drag vs. rolling friction are both argued for in the literature; see the history page), EulersDisk uses a reduced phenomenological pair of ODEs,">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_04_eulers_disk_thumb.gif
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  :doc:`/api/gallery/classical/rotations/plot_04_eulers_disk`

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      <div class="sphx-glr-thumbnail-title">Euler's Disk: a finite-time singularity on a tabletop</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A rattleback (Celt stone) spins smoothly forever in one sense, but spun the other way it soon wobbles, stalls, and reverses into the stable sense of spin. Rattleback is a deliberately simplified toy model built directly on EulerTop&#x27;s free-rigid-body equations, for a reduced state (n_1, n_2, n_3) (n_3 playing the role of spin about the roughly-vertical axis, n_1, n_2 the two rocking/tipping-mode rates):">

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  .. image:: /api/gallery/classical/rotations/images/thumb/sphx_glr_plot_05_rattleback_thumb.gif
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  :doc:`/api/gallery/classical/rotations/plot_05_rattleback`

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      <div class="sphx-glr-thumbnail-title">The Rattleback: one-way spin reversal</div>
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Visualizers
-----------

Side-by-side physical-space/phase-space animation, and interactive
(draggable/zoomable) Plotly figures for the SO(3) momentum sphere and
a precessing orbit.


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    <div class="sphx-glr-thumbcontainer" tooltip="The three animation helpers in physicskit.classical.visualizers.animations and physicskit.classical.visualizers.modal_analysis, applied to three already-familiar conservative systems, each paired with a live diagnostic of its own governing physics: a double pendulum (~physicskit.classical.systems.lagrangian.DoublePendulum, governed by its Euler-Lagrange equations of motion) shown with its physical rods next to a growing (\theta_1, \dot\theta_1) phase trace; a precessing Kepler orbit (~physicskit.classical.systems.newtonian.KeplerSystem, whose post-Newtonian term c_\mathrm{pn}/r^3 in V(r) drives the precession) shown with its orbital path next to a growing energy trace; and a non-linear lattice chain (~physicskit.classical.systems.chains.FPUTChain) shown via its live modal-energy bar chart E_k(t).">

.. only:: html

  .. image:: /api/gallery/classical/visualizers/images/thumb/sphx_glr_plot_01_side_by_side_animation_thumb.gif
    :alt:

  :doc:`/api/gallery/classical/visualizers/plot_01_side_by_side_animation`

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      <div class="sphx-glr-thumbnail-title">SideBySideAnimator: physical space + phase/energy diagnostics, live</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Unlike the static Matplotlib plots elsewhere in physicskit.classical.visualizers, these are draggable/zoomable in a live Python session or a browser -- useful for directly inspecting how close a rigid-body trajectory passes to the unstable intermediate-axis pole, or for zooming into a precessing orbit&#x27;s perihelion. The two systems are governed by the equations already introduced elsewhere in the gallery: EulerTop&#x27;s torque-free Euler equations, I_i\dot\omega_i = (I_j - I_k)\,\omega_j\omega_k (cyclic in i,j,k) for the body-frame angular velocity (see /api/gallery/classical/rotations/plot_01_dzhanibekov), and KeplerSystem&#x27;s separable Hamiltonian H = |p|^2/(2\mu) + V(r) with the post-Newtonian perturbation c_\mathrm{pn}/r^3 in V(r) that drives the perihelion precession (see /api/gallery/classical/newtonian/plot_02_kepler_precession).">

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  .. image:: /api/gallery/classical/visualizers/images/thumb/sphx_glr_plot_02_interactive_plotly_thumb.png
    :alt:

  :doc:`/api/gallery/classical/visualizers/plot_02_interactive_plotly`

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      <div class="sphx-glr-thumbnail-title">Interactive Plotly visualizers: SO(3) momentum sphere and orbit</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/classical/chains/index.rst
   /api/gallery/classical/hamiltonian/index.rst
   /api/gallery/classical/integrators/index.rst
   /api/gallery/classical/lagrangian/index.rst
   /api/gallery/classical/newtonian/index.rst
   /api/gallery/classical/rotations/index.rst
   /api/gallery/classical/visualizers/index.rst



.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_