Calculus_of_variations

# Calculus of variations ## Microsim (three.js) <div class="microsim-player">  <p class="wt-pending"><strong>Microsim staged, not yet on the CDN.</strong> <code>Calculus_of_variations.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> *In the ALGORITHM hub this is where continuous optimization begins — the "find the best route" instinct behind [[Optimal_control]] and [[Dynamic_programming]], and the smooth-curve cousin of [[Combinatorial_optimization]], searching a whole space of functions instead of a finite list of choices.* > The calculus of variations looks for the *function* — not the number — that makes a total quantity as small or as large as possible. Here that quantity is the length of a path joining two fixed points, and one knob deforms the candidate path while a readout reports its length. Slide the knob to zero and the path snaps to a straight line, the shortest route: an infinite-dimensional search reduced to a single-slider experiment whose minimum you find by eye. ## About this microsim The canvas fixes two endpoints and draws a candidate path whose shape is set entirely by one control, **Bulge strength c**, running from **−1.5 to 1.5**. At $c=0$ the path is the straight segment; positive $c$ bows it one way, negative the other, with $\pm1.5$ giving the most pronounced bulge. A live arc-length readout updates as you drag. Because each $c$ names exactly one competing curve, sweeping the slider marches along a one-parameter family of paths, and the readout traces a function that visibly bottoms out at $c=0$. That show-before-tell moment — the minimum sitting precisely at the straight line — *is* the theorem, seen before any equation is written. ## Related microsims - [[Optimal_control]] — extends "best path" from fixed curves to steered trajectories; Pontryagin's principle generalizes Euler–Lagrange. - [[Dynamic_programming]] — Bellman's discrete route to shortest paths, working backward from the goal. - Extreme value theorem — the finite-dimensional cousin: why a continuous quantity on a closed interval attains its minimum. - [[Combinatorial_optimization]] — shortest paths when the choices are graph edges rather than smooth curves. - [[Model_predictive_control]] — re-solves an optimal-path problem at every time step. - [[Reinforcement_learning]] — learns optimal long-run paths from reward, sharing the Bellman backbone. - [[Graph_of_a_function]] — related ALGORITHM microsim ## Links (Wikipedia order)  `Abel's_test` · `Abstract_Wiener_space` · `Action_(physics)` · `Adrien-Marie_Legendre` · 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`Variational_analysis` · `Variational_bicomplex` · `Variational_method_(quantum_mechanics)` · `Variational_methods_in_general_relativity` · `Variational_principle` · `Variational_vector_field` · `Vector_calculus` · `Vector_calculus_identities` · `Vector_measure` · `Vincenzo_Brunacci` · `Volume_integral` · [[Wave_equation]] · `Weak_duality` · `Weakly_measurable_function` · `YouTube` · `Young_measure` · `Zonotope` ## Overview The calculus of variations finds extrema of *functionals*: rules that map an entire function to a single number, usually an integral of a function and its derivatives. The unknown is a whole curve or trajectory, not one variable. Isoperimetric questions reach back to antiquity, but the modern subject opened with Johann Bernoulli's 1696 *brachistochrone* — the curve of fastest descent. Euler and Lagrange forged the general method in the mid-18th century; Euler named the field, and Lagrange added the variational $\delta$-notation still in use. Its central result, the Euler–Lagrange equation, drives the principle of least action, Fermat's principle, geodesics, and the continuous-time root of [[Optimal_control]]. ## The mathematics Write a functional with fixed endpoints, $J[y] = \int_a^b F(x, y, y')\,dx .$ A necessary condition for $y$ to be an extremum is the **Euler–Lagrange equation** $\frac{\partial F}{\partial y} - \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right) = 0 .$ For shortest path the quantity is arc length, so $F=\sqrt{1+(y')^2}$ depends on $y'$ alone. Then $\partial F/\partial y = 0$ and the equation forces $y'/\sqrt{1+(y')^2}$ to be constant — hence $y'$ constant, a **straight line**. The microsim exposes the idea one step earlier. Replacing $y$ by a family $y_c=\bar y + c\,\eta$, with $\bar y$ the straight baseline and $\eta$ vanishing at both ends, collapses the functional into an ordinary function $J(c)$ — and bulge strength $c$ *is* that number. For a horizontal baseline, $J'(0)=0,\qquad J''(0)=\int_a^b\big(\eta'(x)\big)^2\,dx>0,$ so near the bottom $J(c)\approx J(0)+\tfrac12 J''(0)\,c^2$, an upward parabola. That is the U-shaped length curve the slider traces, with $J'(0)=0$ the vanishing first variation. ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | Bulge strength c | Amplitude of the perturbation $\eta$ added to the straight baseline — the variation parameter in $y_c=\bar y+c\,\eta$ | $-1.5$ to $1.5$ | How far, and in which direction, the candidate curve bows from the straight path; $c=0$ is the length-minimizing geodesic, $\pm1.5$ the largest deformations | ## Learning objective After playing, a learner can explain why the straight line minimizes length and predict that any nonzero bulge lengthens the path, linking the U-shaped readout to the vanishing first variation $J'(0)=0$. ## Limits and connections The sim searches only one hand-picked bulge family, so it *illustrates* the minimum rather than proving it over all curves — that is the Euler–Lagrange equation's job. The same reduction from a functional to a one-parameter function underlies gradient methods across [[Optimal_control]] and machine learning. ## Poster & source <div class="microsim-fallback">  <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Calculus_of_variations.html">open full</a> · source: Microsims for Dissemination/ALGORITHM_microsims/Calculus_of_variations.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/Calculus_of_variations) : [Wikitube](https://en.wikitube.io/wiki/Calculus_of_variations) ## Previous hub tags Tree parent: [[Decision_theory]]. Legacy hubs: `ALGORITHM`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*