Single Pedulum Comparison

Solving single pendulums by DifferentialEquations.jl

In this notebook, we shall solve the single pendulum equation:

\[\ddot q = -\sin q,\]

where $q$ means the angle.

Hamiltonian:

\[H(q,p) = \frac{1}{2}p^2 - \cos q + 1.\]

Canonical equation:

\[\dot q = p, \quad \dot p = - \sin q.\]

Initial condition:

\[q(0) = 0, \quad p(0) = 2k.\]

Exact solution:

\[q(t) = 2\arcsin(k\,\mathrm{sn}(t,k)).\]

Maximum of $q(t)$:

\[\sin(q_{\max}/2) = k, \quad q_{\max} = \max\{q(t)\}.\]

Define $y(t)$ by

\[y(t) = \sin(q(t)/2) = k\,\mathrm{sn}(t,k), \quad y_{\max} = k.\]

# Single pendulums shall be solved numerically.
#
using OrdinaryDiffEq, Elliptic, Printf, DiffEqPhysics, Statistics
using OrdinaryDiffEqSymplecticRK

sol2q(sol) = [sol.u[i][j] for i in 1:length(sol.u), j in 1:(length(sol.u[1]) ÷ 2)]
function sol2p(sol)
    [sol.u[i][j] for i in 1:length(sol.u), j in (length(sol.u[1]) ÷ 2 + 1):length(sol.u[1])]
end
sol2tqp(sol) = (sol.t, sol2q(sol), sol2p(sol))

# The exact solutions of single pendulums can be expressed by the Jacobian elliptic functions.
#
sn(u, k) = Jacobi.sn(u, k^2) # the Jacobian sn function

# Use Plot.
#
using Plots

# Define the color list
colorlist = [
    "#1f77b4", "#ff7f0e", "#2ca02c", "#d62728", "#9467bd",
    "#8c564b", "#e377c2", "#7f7f7f", "#bcbd22", "#17becf"
]
cc(k) = colorlist[mod1(k, length(colorlist))]

# Function to plot the solution of a Hamiltonian problem
function plotsol(sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local d = size(q)[2]
    p1 = plot(title = "Solution", xlabel = "t", grid = :on)
    for j in 1:d
        j_str = d > 1 ? "[$j]" : ""
        plot!(p1, t, q[:, j], color = cc(2j-1), label = "q$(j_str)", linewidth = 1)
        plot!(p1, t, p[:, j], color = cc(2j), label = "p$(j_str)",
            linewidth = 1, linestyle = :dash)
    end
    return p1
end

# Function to plot the solution on the 2D phase space
function plotsol2(sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local d = size(q)[2]
    p2 = plot(title = "Phase Space", xlabel = "q", ylabel = "p", grid = :on)
    for j in 1:d
        j_str = d > 1 ? "[$j]" : ""
        plot!(p2, q[:, j], p[:, j], color = cc(j),
            label = "(q$(j_str), p$(j_str))", linewidth = 1)
    end
    return p2
end

# Function to plot the energy of a Hamiltonian problem
function plotenergy(H, sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local energy = [H(q[i, :], p[i, :], nothing) for i in 1:size(q)[1]]
    p3 = plot(t, energy, label = "energy", color = "red", linewidth = 1,
        xlabel = "t", title = "Energy", grid = :on)
    stdenergy_str = @sprintf("%.3e", std(energy))
    title!("std(energy) = $stdenergy_str", fontsize = 10)
    return p3
end

# Function to compare the numerical and exact solutions of a single pendulum
function plotcomparison(k, sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local y = sin.(q / 2)
    local y_exact = k * sn.(t, k) # the exact solution

    p4 = plot(t, y, label = "numerical", linewidth = 1, grid = :on,
        xlabel = "t", ylabel = "y = sin(q(t)/2)", title = "Comparison")
    plot!(p4, t, y_exact, label = "exact", linewidth = 1, linestyle = :dash)
    error_str = @sprintf("%.3e", maximum(abs.(y - y_exact)))
    title!("maximum(abs(numerical - exact)) = $error_str", fontsize = 10)
    return p4
end

# Plot solution and energy
function plotsolenergy(H, integrator, Δt, sol::ODESolution)
    integrator_str = replace("$integrator", r"^[^.]*\\." => "")

    p1 = plotsol(sol)
    p2 = plotsol2(sol)
    p3 = plotenergy(H, sol)

    suptitle = "===== $integrator_str, Δt = $Δt ====="
    plot(p1, p2, p3, layout = (3, 1), title = suptitle)
end

# Solve a single pendulum
function singlependulum(k, integrator, Δt; t0 = 0.0, t1 = 100.0)
    H(p, q, params) = p[1]^2 / 2 - cos(q[1]) + 1
    q0 = [0.0]
    p0 = [2k]
    prob = HamiltonianProblem(H, p0, q0, (t0, t1))

    integrator_str = replace("$integrator", r"^[^.]*\\." => "")
    @printf("%-25s", "$integrator_str:")
    sol = solve(prob, integrator, dt = Δt)
    @time sol = solve(prob, integrator, dt = Δt)

    sleep(0.1)

    p1 = plotsol(sol)
    p2 = plotsol2(sol)
    p3 = plotenergy(H, sol)
    p4 = plotcomparison(k, sol)

    suptitle = "===== $integrator_str, Δt = $Δt ====="
    plot(p1, p2, p3, p4, layout = (2, 2), title = suptitle)
end

singlependulum (generic function with 1 method)

Tests

# Single pendulum
using Plots
using OrdinaryDiffEq

function singlependulum(k, integrator, Δt; t0 = 0.0, t1 = 100.0)
    local H(p, q, params) = p[1]^2 / 2 - cos(q[1]) + 1  # Hamiltonian for single pendulum
    local q0 = [0.0]  # Initial position
    local p0 = [2k]   # Initial momentum
    local prob = HamiltonianProblem(H, p0, q0, (t0, t1))

    local integrator_str = replace("$integrator", r"^[^.]*\." => "")
    @printf("%-25s", "$integrator_str:")
    sol = solve(prob, integrator, dt = Δt)
    @time local sol = solve(prob, integrator, dt = Δt)

    sleep(0.1)

    # Create plots using Plots.jl
    p1 = plot(sol.t, map(x -> x[1], sol.u), label = "Position", title = "Position vs Time")
    p2 = plot(sol.t, map(x -> x[2], sol.u), label = "Momentum", title = "Momentum vs Time")
    p3 = plot(sol.t, map(p -> H(p[1], p[2], nothing), sol.u), label = "Energy", title = "Energy vs Time")
    p4 = plot(sol.t, map(x -> x[1] - k, sol.u), label = "Comparison", title = "Comparison Plot")

    # Combine all plots in a layout
    layout = @layout [a b; c d]
    plot(p1, p2, p3, p4, layout = layout, size = (1000, 800),
        title = "===== $integrator_str, Δt = $Δt =====")
end

k = rand()
integrator = VelocityVerlet()
Δt = 0.1
singlependulum(k, integrator, Δt, t0 = -20.0, t1 = 20.0)

VelocityVerlet():          0.001171 seconds (7.71 k allocations: 261.898 Ki
B)

# Two single pendulums

using Plots
using OrdinaryDiffEq

function plotsolenergy(H, integrator, Δt, sol::ODESolution)
    local integrator_str = replace("$integrator", r"^[^.]*\." => "")

    # Create plots using Plots.jl
    p1 = plot(sol, label = "Position", title = "Position vs Time")
    p2 = plot(sol, label = "Momentum", title = "Momentum vs Time")
    p3 = plot(sol.t, map(p -> H(p[2], p[3], nothing), sol.u), label = "Energy", title = "Energy vs Time")

    # Combine all plots in a layout
    layout = @layout [a b; c]
    plot(p1, p2, p3, layout = layout, size = (1000, 800),
        title = "===== $integrator_str, Δt = $Δt =====")
end

H(q, p, param) = sum(p .^ 2 / 2 .- cos.(q) .+ 1)
q0 = pi * rand(2)
p0 = zeros(2)
t0, t1 = -20.0, 20.0
prob = HamiltonianProblem(H, q0, p0, (t0, t1))

integrator = McAte4()
Δt = 0.1
sol = solve(prob, integrator, dt = Δt)
@time sol = solve(prob, integrator, dt = Δt)

sleep(0.1)
plotsolenergy(H, integrator, Δt, sol)

0.004798 seconds (45.33 k allocations: 1.954 MiB)

Comparison of symplectic Integrators

SymplecticIntegrators = [
    SymplecticEuler(),
    VelocityVerlet(),
    VerletLeapfrog(),
    PseudoVerletLeapfrog(),
    McAte2(),
    Ruth3(),
    McAte3(),
    CandyRoz4(),
    McAte4(),
    CalvoSanz4(),
    McAte42(),
    McAte5(),
    Yoshida6(),
    KahanLi6(),
    McAte8(),
    KahanLi8(),
    SofSpa10()
]

k = 0.999
Δt = 0.1
for integrator in SymplecticIntegrators
    singlependulum(k, integrator, Δt)
end

SymplecticEuler():         0.002843 seconds (24.12 k allocations: 786.992 K
iB)
VelocityVerlet():          0.002089 seconds (19.11 k allocations: 646.273 K
iB)
VerletLeapfrog():          0.002823 seconds (24.11 k allocations: 786.898 K
iB)
PseudoVerletLeapfrog():    0.004254 seconds (34.11 k allocations: 1.043 MiB
)
McAte2():                  0.003206 seconds (39.11 k allocations: 1.181 MiB
)
Ruth3():                   0.006940 seconds (49.11 k allocations: 1.455 MiB
)
McAte3():                  0.006940 seconds (49.11 k allocations: 1.455 MiB
)
CandyRoz4():               0.009175 seconds (59.11 k allocations: 1.730 MiB
)
McAte4():                  0.008601 seconds (59.11 k allocations: 1.730 MiB
)
CalvoSanz4():              0.008832 seconds (64.11 k allocations: 1.868 MiB
)
McAte42():                 0.009306 seconds (69.11 k allocations: 2.005 MiB
)
McAte5():                  0.010488 seconds (79.11 k allocations: 2.280 MiB
)
Yoshida6():                0.011598 seconds (94.11 k allocations: 2.692 MiB
)
KahanLi6():                0.010998 seconds (114.11 k allocations: 3.242 Mi
B)
McAte8():                  0.017425 seconds (174.11 k allocations: 4.890 Mi
B)
KahanLi8():                0.019348 seconds (194.11 k allocations: 5.440 Mi
B)
SofSpa10():                0.036190 seconds (374.11 k allocations: 10.386 M
iB)

k = 0.999
Δt = 0.01
for integrator in SymplecticIntegrators[1:4]
    singlependulum(k, integrator, Δt)
end

SymplecticEuler():         0.014730 seconds (240.12 k allocations: 7.635 Mi
B)
VelocityVerlet():          0.009636 seconds (190.11 k allocations: 6.262 Mi
B)
VerletLeapfrog():          0.062434 seconds (240.11 k allocations: 7.635 Mi
B, 66.20% gc time)
PseudoVerletLeapfrog():    0.027005 seconds (340.11 k allocations: 10.382 M
iB)

k = 0.999
Δt = 0.001
singlependulum(k, SymplecticEuler(), Δt)

SymplecticEuler():         0.403384 seconds (2.40 M allocations: 76.300 MiB
, 34.58% gc time)

k = 0.999
Δt = 0.0001
singlependulum(k, SymplecticEuler(), Δt)

SymplecticEuler():         3.563311 seconds (24.00 M allocations: 762.945 M
iB, 16.29% gc time)

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/DynamicalODE","single_pendulums.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/DynamicalODE/Project.toml`
  [459566f4] DiffEqCallbacks v4.19.4
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  [b305315f] Elliptic v1.0.1
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [af6ede74] OrdinaryDiffEqRKN v2.2.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v2.2.2
  [9c7f1690] OrdinaryDiffEqTaylorSeries v2.3.0
  [65888b18] ParameterizedFunctions v5.27.0
  [91a5bcdd] Plots v1.41.7
  [31c91b34] SciMLBenchmarks v0.2.1
  [90137ffa] StaticArrays v1.9.20
  [10745b16] Statistics v1.11.5
  [92b13dbe] TaylorIntegration v0.18.16
  [37e2e46d] LinearAlgebra v1.11.0
  [de0858da] Printf v1.11.0

And the full manifest:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/DynamicalODE/Manifest.toml`
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  [64909d44] StateSelection v1.11.1
  [90137ffa] StaticArrays v1.9.20
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.5
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.13
  [69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
  [09ab397b] StructArrays v0.7.3
  [2efcf032] SymbolicIndexingInterface v0.3.55
  [19f23fe9] SymbolicLimits v1.2.1
  [d1185830] SymbolicUtils v4.46.6
  [0c5d862f] Symbolics v7.39.2
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.14.0
  [ed4db957] TaskLocalValues v0.1.3
  [b36ab563] TaylorDiff v0.3.5
  [92b13dbe] TaylorIntegration v0.18.16
  [6aa5eb33] TaylorSeries v0.22.4
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [a759f4b9] TimerOutputs v1.2.1
  [781d530d] TruncatedStacktraces v1.4.0
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [41fe7b60] Unzip v0.2.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.7+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
  [a3f928ae] Fontconfig_jll v2.17.1+0
  [d7e528f0] FreeType2_jll v2.14.3+1
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.5.1+0
  [d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [7746bdde] Glib_jll v2.88.3+0
  [3b182d85] Graphite2_jll v1.3.16+0
  [2e76f6c2] HarfBuzz_jll v100.14004.0+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.2.0+1
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.2.0+0
  [1d63c593] LLVMOpenMP_jll v23.1.1+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`