Degenerate Lagrangian Systems
A Lagrangian $L(x,v)$ is said to be degenerate if the determinant of the Hessian with respect to the velocities vanishes, i.e.,
\[\left\vert \frac{\partial^2 L}{\partial v^i \, \partial v^j} \right\vert = 0 .\]
The Euler-Lagrange equations,
\[\frac{d}{dt} \frac{\partial L}{\partial v} - \frac{\partial L}{\partial x} = 0 ,\]
of such Lagrangians are a set of first-order ordinary differential equations and not second-order differential equations as for non-degenerate Lagrangians.
Of particular interest in various applications are degenerate Lagrangians of the form
\[L(x,v) = \vartheta(x) \cdot v - H (x) .\]
Their Euler-Lagrange equations take the form
\[\frac{d \vartheta}{dt} (x) = \nabla \vartheta(x) \cdot v - \nabla H (x) .\]
We exemplify this with the Lotka-Volterra problem in 2d.
Lotka-Volterra
Before any use, we need to load EulerLagrange:
using EulerLagrange
using LinearAlgebraNext, we generate symbolic variables for a two-dimensional Lagrangian system:
t, x, v = lagrangian_variables(2)(t, (x(t))[1:2], (v(t))[1:2])We define a named tuple with typical values for the parameters, e.g.,
params = (
a₁ = -1.0,
a₂ = -1.0,
b₁ = 1.0,
b₂ = 2.0,
)(a₁ = -1.0, a₂ = -1.0, b₁ = 1.0, b₂ = 2.0)We use the function symbolize to generate a symbolic version of the parameters:
sparams = symbolize(params)(a₁ = a₁ₚ, a₂ = a₂ₚ, b₁ = b₁ₚ, b₂ = b₂ₚ)Define the Hamiltonian function and the symplectic potential:
ϑ(x, params) = [log(x[2]) / x[1] / 2, - log(x[1]) / x[2] / 2]
H(x, params) = params.a₁ * x[1] + params.a₂ * x[2] + params.b₁ * log(x[1]) + params.b₂ * log(x[2])H (generic function with 1 method)The Hamiltonian and the symplectic potential, evaluated on and together with the symbolic variables and parameters are used to construct a DegenerateLagrangianSystem:
lag_sys = DegenerateLagrangianSystem(ϑ(x,sparams) ⋅ v, H(x,sparams), t, x, v, sparams)
Degenerate Lagrangian system with
L = (log((x(t))[2])*(v(t))[1]) / (2(x(t))[1]) + (-log((x(t))[1])*(v(t))[2]) / (2(x(t))[2]) - a₁ₚ*(x(t))[1] - a₂ₚ*(x(t))[2] - b₁ₚ*log((x(t))[1]) - b₂ₚ*log((x(t))[2])The constructor computes the Euler-Lagrange equations and generates the corresponding Julia code.
In the last step, we can now construct a LODEProblem from the LagrangianSystem and some appropriate initial conditions, a time span to integrate over and a time step:
tspan = (0.0, 10.0)
tstep = 0.01
q₀ = [2.0, 1.0]
p₀ = ϑ(q₀, params)
lprob = LODEProblem(lag_sys, tspan, tstep, q₀, p₀; parameters = params)Geometric Equation Problem for Lagrangian Ordinary Differential Equation (LODE)
with vector fields
ϑ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x18c1e037, 0x5de5636e, 0x0ad018fb, 0x1c084ae0, 0x7fd328f8), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = (log)((getindex)(X, 2))
var"##cse#2" = 2
var"##cse#3" = (*)(var"##cse#2", (getindex)(X, 1))
var"##cse#4" = (/)(var"##cse#1", var"##cse#3")
var"##cse#5" = -1
var"##cse#6" = (log)((getindex)(X, 1))
var"##cse#7" = (*)(var"##cse#5", var"##cse#6")
var"##cse#8" = (*)(var"##cse#2", (getindex)(X, 2))
var"##cse#9" = (/)(var"##cse#7", var"##cse#8")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#4"
ˍ₋out[2] = var"##cse#9"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end))
f = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0xbfa2810e, 0x8818c74e, 0x952cfba1, 0x92782c52, 0x71552a4e), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(V, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)(var"##cse#1", (getindex)(V, 2))
var"##cse#9" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#10" = (/)(var"##cse#8", var"##cse#9")
var"##cse#11" = (*)(var"##cse#1", params.a₁)
var"##cse#12" = (*)(var"##cse#1", params.b₁)
var"##cse#13" = (/)(var"##cse#12", (getindex)(X, 1))
var"##cse#14" = (+)((+)((+)(var"##cse#7", var"##cse#10"), var"##cse#11"), var"##cse#13")
var"##cse#15" = (*)(var"##cse#1", params.b₂)
var"##cse#16" = (/)(var"##cse#15", (getindex)(X, 2))
var"##cse#17" = (*)(var"##cse#1", params.a₂)
var"##cse#18" = (/)((getindex)(V, 1), var"##cse#9")
var"##cse#19" = (log)((getindex)(X, 1))
var"##cse#20" = (*)(var"##cse#19", (getindex)(V, 2))
var"##cse#21" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#22" = (*)(var"##cse#4", var"##cse#21")
var"##cse#23" = (/)(var"##cse#20", var"##cse#22")
var"##cse#24" = (+)((+)((+)(var"##cse#16", var"##cse#17"), var"##cse#18"), var"##cse#23")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#14"
ˍ₋out[2] = var"##cse#24"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end))
g = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x5d9f8016, 0x1ff07f29, 0x02c54600, 0x2fb5c5a1, 0x0addab8e), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(V, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)(var"##cse#1", (getindex)(V, 2))
var"##cse#9" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#10" = (/)(var"##cse#8", var"##cse#9")
var"##cse#11" = (+)(var"##cse#7", var"##cse#10")
var"##cse#12" = (/)((getindex)(V, 1), var"##cse#9")
var"##cse#13" = (log)((getindex)(X, 1))
var"##cse#14" = (*)(var"##cse#13", (getindex)(V, 2))
var"##cse#15" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#16" = (*)(var"##cse#4", var"##cse#15")
var"##cse#17" = (/)(var"##cse#14", var"##cse#16")
var"##cse#18" = (+)(var"##cse#12", var"##cse#17")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#11"
ˍ₋out[2] = var"##cse#18"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end))
Lagrangian: L = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x3afebc21, 0x8917d62a, 0xadcb4689, 0xef70a11b, 0x211da6a7), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (*)((*)(var"##cse#1", params.a₁), (getindex)(X, 1))
var"##cse#3" = (log)((getindex)(X, 2))
var"##cse#4" = (*)(var"##cse#3", (getindex)(V, 1))
var"##cse#5" = 2
var"##cse#6" = (*)(var"##cse#5", (getindex)(X, 1))
var"##cse#7" = (/)(var"##cse#4", var"##cse#6")
var"##cse#8" = (log)((getindex)(X, 1))
var"##cse#9" = (*)((*)(var"##cse#1", var"##cse#8"), params.b₁)
var"##cse#10" = (*)((*)(var"##cse#1", var"##cse#8"), (getindex)(V, 2))
var"##cse#11" = (*)(var"##cse#5", (getindex)(X, 2))
var"##cse#12" = (/)(var"##cse#10", var"##cse#11")
var"##cse#13" = (*)((*)(var"##cse#1", params.b₂), var"##cse#3")
var"##cse#14" = (*)((*)(var"##cse#1", params.a₂), (getindex)(X, 2))
var"##cse#15" = (+)((+)((+)((+)((+)(var"##cse#2", var"##cse#7"), var"##cse#9"), var"##cse#12"), var"##cse#13"), var"##cse#14")
var"##cse#15"
end
end
end))
Invariants:
(h = EulerLagrange.var"#111#112"{@NamedTuple{L::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x3afebc21, 0x8917d62a, 0xadcb4689, 0xef70a11b, 0x211da6a7), Expr}, H::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x24bc6488, 0x361cb419, 0xecd0a66e, 0x9276431f, 0x71b5c036), Expr}, EL::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0xf84b9375, 0xf534bf2b, 0x6669aecd, 0xc0a8f4e4, 0x8b7fb5a8), Expr}, ∇H::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x2da0027d, 0x59135b9a, 0x8b5ed0e8, 0x81d73b5f, 0x4839df03), Expr}, ẋ::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x26722f2f, 0x94f8a530, 0x85aa1d3b, 0xe6a8ecf6, 0x67449dae), Expr}, v::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :P, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x26722f2f, 0x94f8a530, 0x85aa1d3b, 0xe6a8ecf6, 0x67449dae), Expr}, f::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0xbfa2810e, 0x8818c74e, 0x952cfba1, 0x92782c52, 0x71552a4e), Expr}, u::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x4889eaba, 0x3711d03f, 0x67dd90d3, 0x9ce688d2, 0x5021471b), Expr}, g::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x5d9f8016, 0x1ff07f29, 0x02c54600, 0x2fb5c5a1, 0x0addab8e), Expr}, ū::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :P, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x4889eaba, 0x3711d03f, 0x67dd90d3, 0x9ce688d2, 0x5021471b), Expr}, ḡ::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :P, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x2c05248f, 0xc17dd5b8, 0x28ef20ff, 0x1462a36a, 0x2783998a), Expr}, p::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x850e1451, 0x8053caa2, 0x58eb0e18, 0xc232466b, 0x32e2eb1f), Expr}, ϑ::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x18c1e037, 0x5de5636e, 0x0ad018fb, 0x1c084ae0, 0x7fd328f8), Expr}, ω::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x3cf8cfed, 0xd6d7e22d, 0x4212de94, 0x94ed9183, 0xc1aa406b), Expr}, ϕ::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :P, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x12cf9256, 0x5bf0fed2, 0xe8d17234, 0x9ff12e24, 0xc754a431), Expr}, ψ::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :P, :Ẋ, :F, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x352f3b2e, 0xd73cbdaa, 0x3f6b2acd, 0x6f358750, 0x55ce0351), Expr}, P::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x9ed75676, 0x76967b6b, 0xfacf5a2d, 0x927f7000, 0xf2eb5ea1), Expr}}}((L = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x3afebc21, 0x8917d62a, 0xadcb4689, 0xef70a11b, 0x211da6a7), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (*)((*)(var"##cse#1", params.a₁), (getindex)(X, 1))
var"##cse#3" = (log)((getindex)(X, 2))
var"##cse#4" = (*)(var"##cse#3", (getindex)(V, 1))
var"##cse#5" = 2
var"##cse#6" = (*)(var"##cse#5", (getindex)(X, 1))
var"##cse#7" = (/)(var"##cse#4", var"##cse#6")
var"##cse#8" = (log)((getindex)(X, 1))
var"##cse#9" = (*)((*)(var"##cse#1", var"##cse#8"), params.b₁)
var"##cse#10" = (*)((*)(var"##cse#1", var"##cse#8"), (getindex)(V, 2))
var"##cse#11" = (*)(var"##cse#5", (getindex)(X, 2))
var"##cse#12" = (/)(var"##cse#10", var"##cse#11")
var"##cse#13" = (*)((*)(var"##cse#1", params.b₂), var"##cse#3")
var"##cse#14" = (*)((*)(var"##cse#1", params.a₂), (getindex)(X, 2))
var"##cse#15" = (+)((+)((+)((+)((+)(var"##cse#2", var"##cse#7"), var"##cse#9"), var"##cse#12"), var"##cse#13"), var"##cse#14")
var"##cse#15"
end
end
end)), H = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x24bc6488, 0x361cb419, 0xecd0a66e, 0x9276431f, 0x71b5c036), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:170 =#
begin
begin
var"##cse#1" = (*)(params.a₁, (getindex)(X, 1))
var"##cse#2" = (log)((getindex)(X, 1))
var"##cse#3" = (*)(var"##cse#2", params.b₁)
var"##cse#4" = (log)((getindex)(X, 2))
var"##cse#5" = (*)(params.b₂, var"##cse#4")
var"##cse#6" = (*)(params.a₂, (getindex)(X, 2))
var"##cse#7" = (+)((+)((+)(var"##cse#1", var"##cse#3"), var"##cse#5"), var"##cse#6")
var"##cse#7"
end
end
end)), EL = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0xf84b9375, 0xf534bf2b, 0x6669aecd, 0xc0a8f4e4, 0x8b7fb5a8), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (*)(var"##cse#1", params.a₁)
var"##cse#3" = (*)(var"##cse#1", params.b₁)
var"##cse#4" = (/)(var"##cse#3", (getindex)(X, 1))
var"##cse#5" = (+)(var"##cse#2", var"##cse#4")
var"##cse#6" = (*)(var"##cse#1", params.b₂)
var"##cse#7" = (/)(var"##cse#6", (getindex)(X, 2))
var"##cse#8" = (*)(var"##cse#1", params.a₂)
var"##cse#9" = (+)(var"##cse#7", var"##cse#8")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#5"
ˍ₋out[2] = var"##cse#9"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ∇H = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x2da0027d, 0x59135b9a, 0x8b5ed0e8, 0x81d73b5f, 0x4839df03), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = (/)(params.b₁, (getindex)(X, 1))
var"##cse#2" = (+)(params.a₁, var"##cse#1")
var"##cse#3" = (/)(params.b₂, (getindex)(X, 2))
var"##cse#4" = (+)(params.a₂, var"##cse#3")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#2"
ˍ₋out[2] = var"##cse#4"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ẋ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x26722f2f, 0x94f8a530, 0x85aa1d3b, 0xe6a8ecf6, 0x67449dae), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = (/)(params.b₂, (getindex)(X, 2))
var"##cse#2" = (+)(params.a₂, var"##cse#1")
var"##cse#3" = (*)((*)((getindex)(X, 1), var"##cse#2"), (getindex)(X, 2))
var"##cse#4" = -1
var"##cse#5" = (/)(params.b₁, (getindex)(X, 1))
var"##cse#6" = (+)(params.a₁, var"##cse#5")
var"##cse#7" = (*)((*)((*)(var"##cse#4", (getindex)(X, 1)), var"##cse#6"), (getindex)(X, 2))
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#3"
ˍ₋out[2] = var"##cse#7"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), v = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :P, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x26722f2f, 0x94f8a530, 0x85aa1d3b, 0xe6a8ecf6, 0x67449dae), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = (/)(params.b₂, (getindex)(X, 2))
var"##cse#2" = (+)(params.a₂, var"##cse#1")
var"##cse#3" = (*)((*)((getindex)(X, 1), var"##cse#2"), (getindex)(X, 2))
var"##cse#4" = -1
var"##cse#5" = (/)(params.b₁, (getindex)(X, 1))
var"##cse#6" = (+)(params.a₁, var"##cse#5")
var"##cse#7" = (*)((*)((*)(var"##cse#4", (getindex)(X, 1)), var"##cse#6"), (getindex)(X, 2))
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#3"
ˍ₋out[2] = var"##cse#7"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), f = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0xbfa2810e, 0x8818c74e, 0x952cfba1, 0x92782c52, 0x71552a4e), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(V, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)(var"##cse#1", (getindex)(V, 2))
var"##cse#9" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#10" = (/)(var"##cse#8", var"##cse#9")
var"##cse#11" = (*)(var"##cse#1", params.a₁)
var"##cse#12" = (*)(var"##cse#1", params.b₁)
var"##cse#13" = (/)(var"##cse#12", (getindex)(X, 1))
var"##cse#14" = (+)((+)((+)(var"##cse#7", var"##cse#10"), var"##cse#11"), var"##cse#13")
var"##cse#15" = (*)(var"##cse#1", params.b₂)
var"##cse#16" = (/)(var"##cse#15", (getindex)(X, 2))
var"##cse#17" = (*)(var"##cse#1", params.a₂)
var"##cse#18" = (/)((getindex)(V, 1), var"##cse#9")
var"##cse#19" = (log)((getindex)(X, 1))
var"##cse#20" = (*)(var"##cse#19", (getindex)(V, 2))
var"##cse#21" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#22" = (*)(var"##cse#4", var"##cse#21")
var"##cse#23" = (/)(var"##cse#20", var"##cse#22")
var"##cse#24" = (+)((+)((+)(var"##cse#16", var"##cse#17"), var"##cse#18"), var"##cse#23")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#14"
ˍ₋out[2] = var"##cse#24"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), u = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x4889eaba, 0x3711d03f, 0x67dd90d3, 0x9ce688d2, 0x5021471b), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = (getindex)(V, 1)
ˍ₋out[2] = (getindex)(V, 2)
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end)), g = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x5d9f8016, 0x1ff07f29, 0x02c54600, 0x2fb5c5a1, 0x0addab8e), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(V, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)(var"##cse#1", (getindex)(V, 2))
var"##cse#9" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#10" = (/)(var"##cse#8", var"##cse#9")
var"##cse#11" = (+)(var"##cse#7", var"##cse#10")
var"##cse#12" = (/)((getindex)(V, 1), var"##cse#9")
var"##cse#13" = (log)((getindex)(X, 1))
var"##cse#14" = (*)(var"##cse#13", (getindex)(V, 2))
var"##cse#15" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#16" = (*)(var"##cse#4", var"##cse#15")
var"##cse#17" = (/)(var"##cse#14", var"##cse#16")
var"##cse#18" = (+)(var"##cse#12", var"##cse#17")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#11"
ˍ₋out[2] = var"##cse#18"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ū = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :P, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x4889eaba, 0x3711d03f, 0x67dd90d3, 0x9ce688d2, 0x5021471b), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = (getindex)(V, 1)
ˍ₋out[2] = (getindex)(V, 2)
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end)), ḡ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :Λ, :P, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x2c05248f, 0xc17dd5b8, 0x28ef20ff, 0x1462a36a, 0x2783998a), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(V, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#9" = (/)((getindex)(V, 2), var"##cse#8")
var"##cse#10" = (+)(var"##cse#7", var"##cse#9")
var"##cse#11" = (*)(var"##cse#1", (getindex)(V, 1))
var"##cse#12" = (/)(var"##cse#11", var"##cse#8")
var"##cse#13" = (log)((getindex)(X, 1))
var"##cse#14" = (*)(var"##cse#13", (getindex)(V, 2))
var"##cse#15" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#16" = (*)(var"##cse#4", var"##cse#15")
var"##cse#17" = (/)(var"##cse#14", var"##cse#16")
var"##cse#18" = (+)(var"##cse#12", var"##cse#17")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#10"
ˍ₋out[2] = var"##cse#18"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), p = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x850e1451, 0x8053caa2, 0x58eb0e18, 0xc232466b, 0x32e2eb1f), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:408 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:408 =#
begin
begin
var"##cse#1" = (log)((getindex)(X, 2))
var"##cse#2" = 2
var"##cse#3" = (*)(var"##cse#2", (getindex)(X, 1))
var"##cse#4" = (/)(var"##cse#1", var"##cse#3")
var"##cse#5" = -1
var"##cse#6" = (log)((getindex)(X, 1))
var"##cse#7" = (*)(var"##cse#5", var"##cse#6")
var"##cse#8" = (*)(var"##cse#2", (getindex)(X, 2))
var"##cse#9" = (/)(var"##cse#7", var"##cse#8")
begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1329 =#
(SymbolicUtils.Code.create_array)(Array, nothing, Val{1}(), Val{(2,)}(), var"##cse#4", var"##cse#9")
end
end
end
end)), ϑ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x18c1e037, 0x5de5636e, 0x0ad018fb, 0x1c084ae0, 0x7fd328f8), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = (log)((getindex)(X, 2))
var"##cse#2" = 2
var"##cse#3" = (*)(var"##cse#2", (getindex)(X, 1))
var"##cse#4" = (/)(var"##cse#1", var"##cse#3")
var"##cse#5" = -1
var"##cse#6" = (log)((getindex)(X, 1))
var"##cse#7" = (*)(var"##cse#5", var"##cse#6")
var"##cse#8" = (*)(var"##cse#2", (getindex)(X, 2))
var"##cse#9" = (/)(var"##cse#7", var"##cse#8")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#4"
ˍ₋out[2] = var"##cse#9"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ω = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x3cf8cfed, 0xd6d7e22d, 0x4212de94, 0x94ed9183, 0xc1aa406b), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = 0
var"##cse#2" = 1
var"##cse#3" = (*)((getindex)(X, 1), (getindex)(X, 2))
var"##cse#4" = (/)(var"##cse#2", var"##cse#3")
var"##cse#5" = -1
var"##cse#6" = (/)(var"##cse#5", var"##cse#3")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#1"
ˍ₋out[2] = var"##cse#4"
ˍ₋out[3] = var"##cse#6"
ˍ₋out[4] = var"##cse#1"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ϕ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :P, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x12cf9256, 0x5bf0fed2, 0xe8d17234, 0x9ff12e24, 0xc754a431), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)(var"##cse#1", var"##cse#2")
var"##cse#4" = 2
var"##cse#5" = (*)(var"##cse#4", (getindex)(X, 1))
var"##cse#6" = (/)(var"##cse#3", var"##cse#5")
var"##cse#7" = (+)(var"##cse#6", (getindex)(P, 1))
var"##cse#8" = (log)((getindex)(X, 1))
var"##cse#9" = (*)(var"##cse#4", (getindex)(X, 2))
var"##cse#10" = (/)(var"##cse#8", var"##cse#9")
var"##cse#11" = (+)((getindex)(P, 2), var"##cse#10")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#7"
ˍ₋out[2] = var"##cse#11"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), ψ = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :P, :Ẋ, :F, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x352f3b2e, 0xd73cbdaa, 0x3f6b2acd, 0x6f358750, 0x55ce0351), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = -1
var"##cse#2" = (log)((getindex)(X, 2))
var"##cse#3" = (*)((*)(var"##cse#1", var"##cse#2"), (getindex)(Ẋ, 1))
var"##cse#4" = 2
var"##cse#5" = (^)((getindex)(X, 1), var"##cse#4")
var"##cse#6" = (*)(var"##cse#4", var"##cse#5")
var"##cse#7" = (/)(var"##cse#3", var"##cse#6")
var"##cse#8" = (*)(var"##cse#1", var"##cse#7")
var"##cse#9" = (*)((*)(var"##cse#4", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#10" = (/)((getindex)(Ẋ, 2), var"##cse#9")
var"##cse#11" = (*)(var"##cse#1", var"##cse#10")
var"##cse#12" = (+)((+)((getindex)(F, 1), var"##cse#8"), var"##cse#11")
var"##cse#13" = (log)((getindex)(X, 1))
var"##cse#14" = (*)(var"##cse#13", (getindex)(Ẋ, 2))
var"##cse#15" = (^)((getindex)(X, 2), var"##cse#4")
var"##cse#16" = (*)(var"##cse#4", var"##cse#15")
var"##cse#17" = (/)(var"##cse#14", var"##cse#16")
var"##cse#18" = (*)(var"##cse#1", var"##cse#17")
var"##cse#19" = (*)(var"##cse#1", (getindex)(Ẋ, 1))
var"##cse#20" = (/)(var"##cse#19", var"##cse#9")
var"##cse#21" = (*)(var"##cse#1", var"##cse#20")
var"##cse#22" = (+)((+)((getindex)(F, 2), var"##cse#18"), var"##cse#21")
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#12"
ˍ₋out[2] = var"##cse#22"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)), P = RuntimeGeneratedFunctions.RuntimeGeneratedFunction{(:ˍ₋out, :t, :X, :V, :params), EulerLagrange.var"#_RGF_ModTag", EulerLagrange.var"#_RGF_ModTag", (0x9ed75676, 0x76967b6b, 0xfacf5a2d, 0x927f7000, 0xf2eb5ea1), Expr}(:(#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =# @inbounds begin
#= /home/runner/.julia/packages/Symbolics/mLvup/src/build_function.jl:410 =#
begin
begin
var"##cse#1" = 0
var"##cse#2" = -1
var"##cse#3" = (*)((*)(var"##cse#2", (getindex)(X, 1)), (getindex)(X, 2))
var"##cse#4" = (*)((getindex)(X, 1), (getindex)(X, 2))
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1264 =# @inbounds begin
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1260 =#
ˍ₋out[1] = var"##cse#1"
ˍ₋out[2] = var"##cse#3"
ˍ₋out[3] = var"##cse#4"
ˍ₋out[4] = var"##cse#1"
#= /home/runner/.julia/packages/SymbolicUtils/c9cTZ/src/code.jl:1262 =#
ˍ₋out
end
end
end
end)))),)
Timespan: (0.0, 10.0)
Timestep: 0.01
Initial conditions:
(t = fill(0.0), q = [2.0, 1.0], p = [0.0, -0.34657359027997264], v = [2.0, 1.0])
Parameters:
(a₁ = -1.0, a₂ = -1.0, b₁ = 1.0, b₂ = 2.0)We can integrate this system using GeometricIntegrators:
using GeometricIntegrators
sol = integrate(lprob, Gauss(1))
using CairoMakie
fig = lines(parent(sol.q[:,1]), parent(sol.q[:,2]);
axis = (; xlabel = "x₁", ylabel = "x₂", title = "Lotka-Volterra system in 2d"),
figure = (; size = (800,600), fontsize = 22))