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Brachistochrone Gradient Flow

Watch a rough guessed path relax, step by step, into the true brachistochrone curve as gradient descent minimizes the descent-time functional directly -- the numerical method behind the calculus of variations, not a race between named curves.

Mathematics3DAdvanced60 FPS📱 Mobile-adapted⇄ 2D version

Details

CategoryMathematics

RenderingThree.js · WebGL

DifficultyAdvanced

Target FPS60

ControlsPick an initial guessed path (straight, bulged, or zigzag), start or single-step the gradient-descent relaxation, and drag sliders for the endpoint's drop and span, gravity, and descent rate while live readouts track iteration count, the functional's current descent time, and RMS convergence toward the analytic cycloid.

calculus of variationsbrachistochronegradient descentEuler-LagrangeRitz methodnumerical optimization

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Instead of racing pre-named curves against each other, this simulation solves the brachistochrone problem the way calculus of variations actually works: it discretizes the descent-time functional T[y] over a free path of movable points, computes its numerical gradient, and lets steepest descent relax an arbitrary guessed curve — straight, bulged, or zigzagged — step by step into the true minimum-time path. A dashed overlay of the closed-form cycloid solution, refit live as you move the endpoint, lets you watch the gap between guess and optimum close in real time, with the functional's value and an RMS convergence readout tracking the process exactly as an optimizer would.

⚙ Under the hood

Watch a roughly guessed path relax, step by step, into the true brachistochrone curve as numerical gradient descent minimizes the descent-time functional directly -- the method behind the calculus of variations, shown in action instead of a race between named curves.

calculus of variationsbrachistochronegradient descentEuler-LagrangeRitz methodnumerical optimization

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install