73 lines
1.9 KiB
Go
73 lines
1.9 KiB
Go
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// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
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// SPDX-License-Identifier: MIT
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// Command pendulum computes the exact period of a simple pendulum at
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// large amplitude through the complete elliptic integral of the first
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// kind, and shows how far the small-angle formula drifts once the
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// release angle stops being small. The period is
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//
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// T = 4·sqrt(L/g)·K(sin²(θ₀/2)),
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//
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// where K is EllipticK with the m = k² parameter convention.
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//
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// Usage: go run ./examples/pendulum
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package main
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import (
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"fmt"
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"log"
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"math"
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"sourcedock.dev/petrbalvin/tensor"
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)
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// kComplete evaluates EllipticK at a single parameter.
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func kComplete(m float64) float64 {
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arr, err := tensor.FromFloats([]float64{m}, 1)
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if err != nil {
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log.Fatal(err)
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}
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k, err := tensor.EllipticK(arr)
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if err != nil {
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log.Fatal(err)
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}
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v, _ := tensor.FloatAt(k, 0)
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return v
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}
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func main() {
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const (
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length = 1.0 // metres
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grav = 9.80665
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)
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small := 2 * math.Pi * math.Sqrt(length/grav)
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fmt.Println("release angle exact period small-angle period drift")
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for _, deg := range []float64{5, 15, 30, 45, 60, 90, 120, 170} {
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theta := deg * math.Pi / 180
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m := math.Sin(theta/2) * math.Sin(theta/2)
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period := 4 * math.Sqrt(length/grav) * kComplete(m)
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drift := (period/small - 1) * 100
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fmt.Printf("%10.0f° %12.6f s %14.6f s %+6.2f %%\n",
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deg, period, small, drift)
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}
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// The inverse problem: which release angle doubles the small-angle
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// period? Bisection on the angle, the period being monotone in it.
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target := 2 * small
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lo, hi := 0.0, math.Pi
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angle := 0.0
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for range 80 {
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mid := (lo + hi) / 2
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m := math.Sin(mid/2) * math.Sin(mid/2)
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if 4*math.Sqrt(length/grav)*kComplete(m) < target {
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lo = mid
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} else {
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hi = mid
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}
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angle = mid
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}
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fmt.Printf("\na release angle of %.2f° doubles the period (%.4f s)\n",
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angle*180/math.Pi, target)
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}
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