Configuration_space_(physics)

# Configuration space (physics) ## Microsim (three.js) <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Configuration_space_%28physics%29.html</code> is built and deploy-ready in <code>Microsims for Dissemination/</code>, but the Netlify project still serves the geometry+spintronics set only. The player is disabled until the deploy lands; the explanatory text below is unchanged.</p>   </div> *Where the FRACTALS hub meets mechanics: the double pendulum's two joint angles trace a path on a doughnut-shaped space, a geometric picture that ties the Curvature of a torus to the Rotation angles of its links and the step-by-step [[Numerical_integration]] that produces its famously chaotic motion.* > The **configuration space** of a mechanical system is the set of all positions it can occupy, with one dimension per independent coordinate (degree of freedom). A double pendulum has two swinging links, so two angles $(\theta_1,\theta_2)$ fix its state completely — and because each angle wraps around, its configuration space is a **torus**. This microsim swings a double pendulum and simultaneously traces the point $(\theta_1,\theta_2)$ so you can watch the abstract space fill in. Small changes in the start send the trace onto wildly different paths: chaos, seen geometrically. ## About this microsim **Speed** (0.1–2.5) sets how fast physical time advances, letting you slow the motion to follow each swing or speed through long runs. **Trail length** (200–6000) controls how many past points of the tip (and of the configuration-space trace) remain drawn, so you can see either a clean instantaneous state or a dense, chaos-filled history. **Pause** freezes the integration to inspect the current configuration, and **Reset** returns the pendulum to its release position so you can compare trajectories from nearly identical starts. ## Related microsims - Curvature — the configuration space here is a torus, whose Gaussian curvature that sim maps out. - Rotation — each pendulum link's state is a planar rotation angle. - [[Numerical_integration]] — the chaotic path is produced by numerically integrating the equations of motion. - Fourier series — a regular (non-chaotic) oscillation would decompose into the harmonics that sim builds. ## Links (Wikipedia order)  `Bloch_sphere` · `C-Space,_Beijing` · `Classical_mechanics` · `Configuration_space_(mathematics)` · `Cotangent_bundle` · `Cotangent_space` · `Degrees_of_freedom_(mechanics)` · `Euclidean_space` · `Forward_kinematics` · `Free_University_of_Berlin` · `Generalized_coordinates` · `Hamiltonian_mechanics` · `Holonomy` · `Inverse_kinematics` · `Lagrangian_mechanics` · `Manifold` · `Map_(mathematics)` · `Motion_planning` · `Mott_problem` · `Parameter_space` · [[Phase_space]] · [[Physical_system]] · `Projective_Hilbert_space` · [[Quantum_mechanics]] · `Quantum_state_space` · `Quaternion` · `Robert_Ghrist` · `Tangent_space` · `Tautological_one-form` · `Wave_function` · `Wigner's_theorem` ## Overview Configuration space is a central idea of Lagrangian and Hamiltonian mechanics. A system of $N$ degrees of freedom has an $N$-dimensional configuration space; its full dynamical state also needs the velocities, doubling the count in the $2N$-dimensional **phase space**. The double pendulum is the textbook example of a simple system with only two degrees of freedom that is nonetheless chaotic — exquisitely sensitive to initial conditions — which is why it is a favourite demonstration of deterministic unpredictability. ## The mathematics The generalized coordinates are the two angles $q=(\theta_1,\theta_2)$. Since each is periodic, $\theta_i \in [0,2\pi)$ with $0$ and $2\pi$ identified, the configuration space is the 2-torus $T^2 = S^1\times S^1$. The motion follows the Euler–Lagrange equations from the Lagrangian $L=T-V$, $\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i}=0,$ a pair of coupled nonlinear second-order ODEs. They have no closed-form solution, so the sim advances them numerically. Sensitive dependence means two traces starting a hair apart diverge exponentially, at a rate set by the system's positive Lyapunov exponent. ## Controls → what each maps to | Control | Maps to (symbol) | Range / values | Meaning | |---|---|---|---| | Speed | time rate $dt$ | 0.1–2.5 | how fast simulated time advances | | Trail length | history depth | 200–6000 | number of past points retained in the trace | | Pause | integrator halt | button | freeze the current configuration | | Reset | initial state | button | return to the release configuration | ## Learning objective After playing, a learner can count a system's degrees of freedom, identify its configuration space (here a torus), and explain why two degrees of freedom already permit chaos. ## Limits and connections The model omits friction and air drag, so energy is conserved and the motion never settles; a real pendulum's trace would slowly spiral inward. The torus geometry links directly to Curvature, and each link's orientation is a plane Rotation driven forward by [[Numerical_integration]]. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Configuration_space_%28physics%29.html">open full</a> · source: Microsims for Dissemination/FRACTALS_ThreeJS_Microsims/Configuration_space_(physics).html</em></p> </div>  *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Configuration_space_%28physics%29) : [Wikitube](https://en.wikitube.io/wiki/Configuration_space_%28physics%29) ## Previous hub tags Tree parents: [[Dynamical_system]] · [[Phase_space]]. Legacy hubs: `FRACTALS`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*