### A Pluto.jl notebook ### # v0.19.11 using Markdown using InteractiveUtils # ╔═╡ 894c5827-242c-4746-a6f7-a7e6470f1ccd begin using Plots using LaTeXStrings end # ╔═╡ 1bbbd8fc-1ce3-11ed-2e9c-fddd6327f498 md" # Time evolution, radial potential ``H=\frac{1}{2} p_x^2 + \frac{1}{2} p_y^2 - \frac{\lambda}{\sqrt{x^2 + y^2}}`` Equations of motion: ``\dot{p}_x = -\frac{\lambda x}{(x^2+y^2)^{3/2}} \equiv K_x(x,y)`` ``\dot{p}_y = -\frac{\lambda y}{(x^2+y^2)^{3/2}} \equiv K_y(x,y)`` ``\dot{x} = p_x`` ``\dot{y} = p_y`` " # ╔═╡ f3e9e24d-fe00-49f6-96d4-3c246b769b3b begin λ = 1.0 V(x,y) = - λ / sqrt(x^2 + y^2) Kx(x,y) = - λ * x/ (x^2 + y^2)^(3/2) Ky(x,y) = - λ * y/ (x^2 + y^2)^(3/2) x0 = 0.5 y0 = 0.0 px0 = 0.0 py0 = 1.0 function setInitialConditions!(x,y,px,py) px[1] = px0 py[1] = py0 x[1] = x0 y[1] = y0 end tMax = 10.0 #Conserved quantities H(x,y,px,py) = 0.5 * px^2 + 0.5 * py^2 + V(x,y) L(x,y,px,py) = x * py - y * px end # ╔═╡ 433152ad-b183-4326-b28f-38dd21cd190d begin #Analytical solution (https://en.wikipedia.org/wiki/Kepler_problem) En0 = H(x0,y0,px0,py0) L0 = x0 * py0 - y0 * px0 ellipse(θ, e) = 1.0 / (1.0 + e * cos(θ)) ecc(L, E) = sqrt(1.0 + 2.0 * E * L^2) rθ(θ,L,E) = -L^2 * ellipse(θ, ecc(L,E)) ang = collect(0:0.01:2*pi) rAn = rθ.(ang, L0, En0) xAn = rAn.*cos.(ang) yAn = rAn.*sin.(ang) end # ╔═╡ 556820a6-d583-4097-b95a-c900e31aff5e md" ## Foward Euler" # ╔═╡ f586da02-75c7-4d80-8364-05def1b1d96d function forwardEuler!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 px[i+1] = px[i] + dt * Kx(x[i],y[i]) py[i+1] = py[i] + dt * Ky(x[i],y[i]) x[i+1] = x[i] + dt * px[i] y[i+1] = y[i] + dt * py[i] end end # ╔═╡ c84a933b-0b5f-4db8-af09-77909b0ae1ce begin dtFE = 0.003 nStepsFE = convert(Int64, round(tMax / dtFE)) xFE, yFE, pxFE, pyFE = zeros(nStepsFE), zeros(nStepsFE), zeros(nStepsFE), zeros(nStepsFE) setInitialConditions!(xFE,yFE,pxFE,pyFE) forwardEuler!(xFE,yFE,pxFE,pyFE,dtFE) tsFE = collect(1:nStepsFE) .* dtFE; plot(xFE,yFE,title="dt=$dtFE", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash, label="analytical") end # ╔═╡ 5e50cfde-943a-43f5-ba6c-cb33013fb717 begin enFE = H.(xFE,yFE,pxFE,pyFE) plot(tsFE, enFE, title="dt=$dtFE", label=:none, xaxis="t", yaxis="H") end # ╔═╡ 3b5c8f78-aeb8-4875-9cf8-e470a983c43c plot(xFE,pxFE,title="dt=$dtFE", label=:none, xaxis="x", yaxis=L"$p_x$") # ╔═╡ 7ca5a84b-b726-4fdf-af2f-edf4a4e0eda1 md"## Semi-implicit Euler" # ╔═╡ 5d01deb5-838a-488c-9e85-fccb4394c530 function semiImplicitEuler!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 px[i+1] = px[i] + dt * Kx(x[i],y[i]) py[i+1] = py[i] + dt * Ky(x[i],y[i]) x[i+1] = x[i] + dt * px[i+1] y[i+1] = y[i] + dt * py[i+1] end end # ╔═╡ 174c8be1-b443-41e0-a97d-3f964002732f begin dtSE = 0.003 nStepsSE = convert(Int64, round(tMax / dtSE)) xSE, ySE, pxSE, pySE = zeros(nStepsSE), zeros(nStepsSE), zeros(nStepsSE), zeros(nStepsSE) setInitialConditions!(xSE,ySE,pxSE,pySE) semiImplicitEuler!(xSE,ySE,pxSE,pySE,dtSE) tsSE = collect(1:nStepsSE) .* dtSE; plot(xSE,ySE,title="dt=$dtSE", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash) end # ╔═╡ a1dcd474-4029-400c-837d-28f264a71475 plot(tsFE,L.(xFE, yFE, pxFE, pyFE),title="dt=$dtSE", label=:none, xaxis="t", yaxis="L") # ╔═╡ 4a5d3c11-e84b-4de4-b378-36ffb7624681 begin enSE = H.(xSE,ySE,pxSE,pySE) plot(tsSE, enSE,title="dt=$dtSE", label=:none, xaxis="t", yaxis="H") end # ╔═╡ 99d2e083-4e02-41a5-9044-ad66c8576197 begin LSE=L.(xSE, ySE, pxSE, pySE) plot(tsSE,LSE,title="dt=$dtSE", label=:none, xaxis="t", yaxis="L") ylims!(0.4,0.6) end # ╔═╡ 9c1ba2ae-8aec-4f07-b20a-44d3b2e6e061 plot(tsSE,(LSE.-LSE[1])./LSE[1],title="dt=$dtSE", label=:none, xaxis="x", yaxis=L"$\Delta L$") # ╔═╡ ea1a96e5-2d79-47fb-8026-d897c3917cfa plot(ySE,pySE,title="dt=$dtSE", label=:none, xaxis="x", yaxis=L"$p_x$") # ╔═╡ 325f9889-77d0-456b-b890-eebba176696f begin xSE2, ySE2, pxSE2, pySE2 = zeros(2*nStepsSE), zeros(2*nStepsSE), zeros(2*nStepsSE), zeros(2*nStepsSE) setInitialConditions!(xSE2,ySE2,pxSE2,pySE2) semiImplicitEuler!(xSE2,ySE2,pxSE2,pySE2,dtSE/2.0) tsSE2 = collect(1:2*nStepsSE) .* dtSE/2.0; enSE2 = H.(xSE2,ySE2,pxSE2,pySE2) plot(tsSE, enSE,title="dt=$dtSE", label="dt=$dtSE", xaxis="t", yaxis="H") plot!(tsSE2, enSE2,label="dt=$(dtSE/2)") end # ╔═╡ 7850a4a1-bae2-4baf-812d-2f04343d377e let plot(tsSE, abs.((enSE .- enSE[1]) ./ enSE[1]),title="dt=$dtSE", label="dt=$dtSE", xaxis="t", yaxis=L"$\Delta E$") plot!(tsSE2, abs.((enSE2 .- enSE2[1]) ./ enSE2[1]),label="dt=$(dtSE/2)") end # ╔═╡ 78493adb-245a-469c-8f81-8b178566ba03 begin dt = 0.01 nSample = 10 enCFE, dts = zeros(nSample), zeros(nSample) for n in 1:nSample x, y, px, py = zeros(n*nStepsSE), zeros(n*nStepsSE), zeros(n*nStepsSE), zeros(n*nStepsSE) setInitialConditions!(x,y,px,py) semiImplicitEuler!(x,y,px,py,dt/n) dts[n] = dt / n en = H.(x,y,px,py) enCFE[n] = maximum(abs.((en .- en[1]) ./ en[1])) end scatter(dts, enCFE,title="Energy conservation", label=:none, xaxis="dt", yaxis=L"$\max(\Delta E)$") end # ╔═╡ b5eab516-ed4a-495f-af9b-1ac873c8e4b0 md"## Verlet" # ╔═╡ e7e935a1-5145-47e8-a8af-b097d0efe22d begin function velocityVerlet!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 px[i+1] = px[i] + dt / 2 * Kx(x[i],y[i]) py[i+1] = py[i] + dt / 2 * Ky(x[i],y[i]) x[i+1] = x[i] + dt * px[i+1] y[i+1] = y[i] + dt * py[i+1] px[i+1] = px[i+1] + dt / 2 * Kx(x[i+1],y[i+1]) py[i+1] = py[i+1] + dt / 2 * Ky(x[i+1],y[i+1]) end end function positionVerlet!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 x[i+1] = x[i] + dt / 2 * px[i] y[i+1] = y[i] + dt / 2 * py[i] px[i+1] = px[i] + dt * Kx(x[i+1],y[i+1]) py[i+1] = py[i] + dt * Ky(x[i+1],y[i+1]) x[i+1] = x[i+1] + dt / 2 * px[i+1] y[i+1] = y[i+1] + dt / 2 * py[i+1] end end end # ╔═╡ 7f160ec1-23cb-4d95-ac5c-bff66feb3d1d begin dtVV = 0.003 nStepsVV = convert(Int64, round(tMax / dtVV)) xVV, yVV, pxVV, pyVV = zeros(nStepsVV), zeros(nStepsVV), zeros(nStepsVV), zeros(nStepsVV) setInitialConditions!(xVV,yVV,pxVV,pyVV) positionVerlet!(xVV,yVV,pxVV,pyVV,dtVV) tsVV = collect(1:nStepsVV) .* dtVV; plot(xVV,yVV,title="dt=$dtVV", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash) end # ╔═╡ c7fd93a5-55e5-454d-b01d-c17c8943be50 let dt = 0.01 nSample = 10 enCVV, enCPV, dts = zeros(nSample), zeros(nSample), zeros(nSample) for n in 1:nSample x1, y1, px1, py1 = zeros(n*nStepsVV), zeros(n*nStepsVV), zeros(n*nStepsVV), zeros(n*nStepsVV) setInitialConditions!(x1,y1,px1,py1) velocityVerlet!(x1,y1,px1,py1,dt/n) en = H.(x1,y1,px1,py1) enCVV[n] = maximum(abs.((en .- en[1]) ./ en[1])) x2, y2, px2, py2 = zeros(n*nStepsVV), zeros(n*nStepsVV), zeros(n*nStepsVV), zeros(n*nStepsVV) setInitialConditions!(x2,y2,px2,py2) positionVerlet!(x2,y2,px2,py2,dt/n) en = H.(x2,y2,px2,py2) enCPV[n] = maximum(abs.((en .- en[1]) ./ en[1])) dts[n] = dt / n end scatter(dts, enCVV, label="Velocity verlet", xaxis="dt", yaxis=L"$\max(\Delta E)$",yscale=:log10,xscale=:log10) scatter!(dts, enCPV, label="Position verlet", xaxis="dt", yaxis=L"$\max(\Delta E)$",yscale=:log10,xscale=:log10) plot!(dts, 72.0.*dts.^2,title="Energy conservation", label=L"$\propto \mathrm{d}t^2$", xaxis="dt", yaxis=L"$\max(\Delta E)$",yscale=:log10,xscale=:log10) plot!(dts, 19.0.*dts.^2,title="Energy conservation", label=L"$\propto \mathrm{d}t^2$", xaxis="dt", yaxis=L"$\max(\Delta E)$",yscale=:log10,xscale=:log10) scatter!(dts, enCFE,title="Energy conservation", label="SI Euler", xaxis="dt", yaxis=L"$\max(\Delta E)$") plot!(dts, 7.0.*dts,title="Energy conservation", label=L"$\propto \mathrm{d}t$", xaxis="dt", yaxis=L"$\max(\Delta E)$",yscale=:log10,xscale=:log10) end # ╔═╡ 227a7cc3-7651-4116-a1b0-e0a526dcb1c3 md"## RK2" # ╔═╡ 9d1b57fb-cd21-4eef-ba09-48a9bbf620a0 function RK2!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 kx1 = Kx(x[i] , y[i]) ky1 = Ky(x[i] , y[i]) px2 = px[i] + dt * kx1 py2 = py[i] + dt * ky1 kx2 = Kx(x[i] + dt * px[i] , y[i] + dt * py[i]) ky2 = Ky(x[i] + dt * px[i] , y[i] + dt * py[i]) x[i+1] = x[i] + dt / 2.0 * (px[i] + px2) y[i+1] = y[i] + dt / 2.0 * (py[i] + py2) px[i+1] = px[i] + dt / 2.0 * (kx1 + kx2) py[i+1] = py[i] + dt / 2.0 * (ky1 + ky2) end end # ╔═╡ 51474c96-20c5-42a6-a3d5-3ea612f0d286 begin dtRK2 = 0.003 nStepsRK2 = convert(Int64, round(tMax / dtRK2)) xRK2, yRK2, pxRK2, pyRK2 = zeros(nStepsRK2), zeros(nStepsRK2), zeros(nStepsRK2), zeros(nStepsRK2) setInitialConditions!(xRK2,yRK2,pxRK2,pyRK2) RK2!(xRK2,yRK2,pxRK2,pyRK2,dtRK2) tsRK2 = collect(1:nStepsRK2) .* dtRK2; plot(xRK2,yRK2,title="dt=$dtRK2", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash) end # ╔═╡ 02b9abc1-2c8c-4b96-9d86-7b997cb5396c begin LRK2=L.(xRK2, yRK2, pxRK2, pyRK2) plot(tsRK2,(LRK2.-LRK2[1])./LRK2[1],title="dt=$dtRK2", label=:none, xaxis="x", yaxis=L"$\Delta L$") end # ╔═╡ 8223455c-142e-47ba-a483-25421ebead36 begin enRK2= H.(xRK2,yRK2,pxRK2,pyRK2) plot(tsRK2, abs.((enRK2 .- enRK2[1]) ./ enRK2[1]),title="dt=$dtRK2", label="dt=$dtRK2", xaxis="t", yaxis=L"$\Delta E$") end # ╔═╡ 62cf52ab-b7a5-4592-8839-d7dea08fe39a md"## RK4" # ╔═╡ 59266541-f10a-414a-bca3-b86e493d475b function RK4!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 kx1 = Kx(x[i] , y[i]) ky1 = Ky(x[i] , y[i]) px2 = px[i] + dt / 2.0 * kx1 py2 = py[i] + dt / 2.0 * ky1 kx2 = Kx(x[i] + dt / 2.0 * px[i] , y[i] + dt / 2.0 * py[i]) ky2 = Ky(x[i] + dt / 2.0 * px[i] , y[i] + dt / 2.0 * py[i]) px3 = px[i] + dt / 2.0 * kx2 py3 = py[i] + dt / 2.0 * ky2 kx3 = Kx(x[i] + dt / 2.0 * px2 , y[i] + dt / 2.0 * py2) ky3 = Ky(x[i] + dt / 2.0 * px2 , y[i] + dt / 2.0 * py2) px4 = px[i] + dt * kx3 py4 = py[i] + dt * ky3 kx4 = Kx(x[i] + dt * px3 , y[i] + dt * py3) ky4 = Ky(x[i] + dt * px3 , y[i] + dt * py3) x[i+1] = x[i] + dt / 6.0 * (px[i] + 2.0 * px2 + 2.0 * px3 + px4) y[i+1] = y[i] + dt / 6.0 * (py[i] + 2.0 * py2 + 2.0 * py3 + py4) px[i+1] = px[i] + dt / 6.0 * (kx1 + 2.0 * kx2 + 2.0 * kx3 + kx4) py[i+1] = py[i] + dt / 6.0 * (ky1 + 2.0 * ky2 + 2.0 * ky3 + ky4) end end # ╔═╡ f831418b-3cf1-4887-80f8-ab35a5d5b309 begin dtRK4 = 0.003 nStepsRK4 = convert(Int64, round(tMax / dtRK4)) xRK4, yRK4, pxRK4, pyRK4 = zeros(nStepsRK4), zeros(nStepsRK4), zeros(nStepsRK4), zeros(nStepsRK4) setInitialConditions!(xRK4,yRK4,pxRK4,pyRK4) RK4!(xRK4,yRK4,pxRK4,pyRK4,dtRK4) tsRK4 = collect(1:nStepsRK4) .* dtRK4; plot(xRK4,yRK4,title="dt=$dtRK4", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash) end # ╔═╡ 49084e8d-7be6-4049-999f-f44ff88a9e18 begin LRK4=L.(xRK4, yRK4, pxRK4, pyRK4) plot(tsRK4,(LRK4.-LRK4[1])./LRK4[1],title="dt=$dtRK4", label=:none, xaxis="x", yaxis=L"$\Delta L$") end # ╔═╡ 5efcc7d1-6d82-4ae6-9091-1be9347a616b begin enRK4= H.(xRK4,yRK4,pxRK4,pyRK4) plot(tsRK4, abs.((enRK4 .- enRK4[1]) ./ enRK4[1]),title="dt=$dtRK4", label="dt=$dtRK4", xaxis="t", yaxis=L"$\Delta E$") end # ╔═╡ 17856811-445a-4f7f-876e-dc7bb2ef95bf md"## Mid-point" # ╔═╡ 85915e91-c3fd-4529-a4da-f22498130723 # ╠═╡ disabled = true #=╠═╡ function midPoint!(x,y,px,py, dt) nSteps = length(x) for i in 1:nSteps-1 x[i+1] = x[i] + dt * (px[i] + dt / 2 * Kx(x[i] , y[i])) y[i+1] = y[i] + dt * (py[i] + dt / 2 * Ky(x[i] , y[i])) xHalf = x[i] + dt / 2.0 * px[i] yHalf = y[i] + dt / 2.0 * py[i] px[i+1] = px[i] + dt * Kx(xHalf, yHalf) py[i+1] = py[i] + dt * Ky(xHalf, yHalf) end end ╠═╡ =# # ╔═╡ 0bea3f6b-93f0-44b5-9d76-c06af7358f2c #=╠═╡ begin dtMid = 0.003 nStepsMid = convert(Int64, round(tMax / dtMid)) xMid, yMid, pxMid, pyMid = zeros(nStepsMid), zeros(nStepsMid), zeros(nStepsMid), zeros(nStepsMid) setInitialConditions!(xMid,yMid,pxMid,pyMid) midPoint!(xMid,yMid,pxMid,pyMid,dtMid) tsMid = collect(1:nStepsMid) .* dtMid; plot(xMid,yMid,title="dt=$dtMid", label=:none, xaxis="x", yaxis="y") plot!(xAn, yAn, color=:black, ls=:dash) end ╠═╡ =# # ╔═╡ 306a07ab-24c5-41e4-bbc6-a7e2d76721bc #=╠═╡ begin LMid=L.(xMid, yMid, pxMid, pyMid) plot(tsMid,LMid,title="dt=$dtMid", label=:none, xaxis="t", yaxis="L") end ╠═╡ =# # ╔═╡ 764b2155-f107-4beb-b77b-2a48477327f0 #=╠═╡ begin enMid= H.(xMid,yMid,pxMid,pyMid) plot(tsMid, enMid,title="dt=$dtMid", label="dt=$dtMid", xaxis="t", yaxis="H") end ╠═╡ =# # ╔═╡ 14971364-ff53-4f01-8a95-c94a9a5b26f3 #=╠═╡ plot(tsMid, abs.((enMid .- enMid[1]) ./ enMid[1]),title="dt=$dtMid", label="dt=$dtMid", xaxis="t", yaxis=L"$\Delta E$") ╠═╡ =# # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" [compat] LaTeXStrings = "~1.3.0" Plots = "~1.31.7" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised julia_version = "1.8.0" manifest_format = "2.0" project_hash = "924bd2910dbb1fe9f152f9d43526b902c5aa722d" [[deps.Adapt]] deps = ["LinearAlgebra"] git-tree-sha1 = "195c5505521008abea5aee4f96930717958eac6f" uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" version = "3.4.0" [[deps.ArgTools]] uuid = 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"975044d2-76e6-5fbe-bf08-97ce7c6574c7" version = "0.4.0+1" [[deps.Xorg_xcb_util_renderutil_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" version = "0.3.9+1" [[deps.Xorg_xcb_util_wm_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" version = "0.4.1+1" [[deps.Xorg_xkbcomp_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" version = "1.4.2+4" [[deps.Xorg_xkeyboard_config_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" uuid = "33bec58e-1273-512f-9401-5d533626f822" version = "2.27.0+4" [[deps.Xorg_xtrans_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" version = "1.4.0+3" [[deps.Zlib_jll]] deps = ["Libdl"] uuid = "83775a58-1f1d-513f-b197-d71354ab007a" version = "1.2.12+3" [[deps.Zstd_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927" uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" version = "1.5.2+0" [[deps.libaom_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" version = "3.4.0+0" [[deps.libass_jll]] deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" version = "0.15.1+0" [[deps.libblastrampoline_jll]] deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" version = "5.1.1+0" [[deps.libfdk_aac_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" version = "2.0.2+0" [[deps.libpng_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" version = "1.6.38+0" [[deps.libvorbis_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" version = "1.3.7+1" [[deps.nghttp2_jll]] deps = ["Artifacts", "Libdl"] uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" version = "1.48.0+0" [[deps.p7zip_jll]] deps = ["Artifacts", "Libdl"] uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" version = "17.4.0+0" [[deps.x264_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" version = "2021.5.5+0" [[deps.x265_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" version = "3.5.0+0" [[deps.xkbcommon_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" version = "1.4.1+0" """ # ╔═╡ Cell order: # ╠═894c5827-242c-4746-a6f7-a7e6470f1ccd # ╟─1bbbd8fc-1ce3-11ed-2e9c-fddd6327f498 # ╠═f3e9e24d-fe00-49f6-96d4-3c246b769b3b # ╠═433152ad-b183-4326-b28f-38dd21cd190d # ╟─556820a6-d583-4097-b95a-c900e31aff5e # ╠═f586da02-75c7-4d80-8364-05def1b1d96d # ╠═c84a933b-0b5f-4db8-af09-77909b0ae1ce # ╠═5e50cfde-943a-43f5-ba6c-cb33013fb717 # ╠═3b5c8f78-aeb8-4875-9cf8-e470a983c43c # ╠═a1dcd474-4029-400c-837d-28f264a71475 # ╟─7ca5a84b-b726-4fdf-af2f-edf4a4e0eda1 # ╠═5d01deb5-838a-488c-9e85-fccb4394c530 # ╠═174c8be1-b443-41e0-a97d-3f964002732f # ╠═4a5d3c11-e84b-4de4-b378-36ffb7624681 # ╠═99d2e083-4e02-41a5-9044-ad66c8576197 # ╠═9c1ba2ae-8aec-4f07-b20a-44d3b2e6e061 # ╠═ea1a96e5-2d79-47fb-8026-d897c3917cfa # ╠═325f9889-77d0-456b-b890-eebba176696f # ╠═7850a4a1-bae2-4baf-812d-2f04343d377e # ╠═78493adb-245a-469c-8f81-8b178566ba03 # ╟─b5eab516-ed4a-495f-af9b-1ac873c8e4b0 # ╠═e7e935a1-5145-47e8-a8af-b097d0efe22d # ╠═7f160ec1-23cb-4d95-ac5c-bff66feb3d1d # ╠═c7fd93a5-55e5-454d-b01d-c17c8943be50 # ╟─227a7cc3-7651-4116-a1b0-e0a526dcb1c3 # ╠═9d1b57fb-cd21-4eef-ba09-48a9bbf620a0 # ╠═51474c96-20c5-42a6-a3d5-3ea612f0d286 # ╠═02b9abc1-2c8c-4b96-9d86-7b997cb5396c # ╠═8223455c-142e-47ba-a483-25421ebead36 # ╟─62cf52ab-b7a5-4592-8839-d7dea08fe39a # ╠═59266541-f10a-414a-bca3-b86e493d475b # ╠═f831418b-3cf1-4887-80f8-ab35a5d5b309 # ╠═49084e8d-7be6-4049-999f-f44ff88a9e18 # ╠═5efcc7d1-6d82-4ae6-9091-1be9347a616b # ╟─17856811-445a-4f7f-876e-dc7bb2ef95bf # ╠═85915e91-c3fd-4529-a4da-f22498130723 # ╠═0bea3f6b-93f0-44b5-9d76-c06af7358f2c # ╠═306a07ab-24c5-41e4-bbc6-a7e2d76721bc # ╠═764b2155-f107-4beb-b77b-2a48477327f0 # ╠═14971364-ff53-4f01-8a95-c94a9a5b26f3 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002