Differential geometry

⋅(X::Function, f::Function) -> Function

Lie derivative of a scalar function along a function. In this case both functions will be considered autonomous and non-variable.

Example

julia> φ = x -> [x[2], -x[1]]
julia> f = x -> x[1]^2 + x[2]^2
julia> (φ⋅f)([1, 2])
0
julia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]
julia> f = (t, x, v) -> t + x[1]^2 + x[2]^2
julia> (φ⋅f)(1, [1, 2], [2, 1])
MethodError

⋅(
    X::VectorField{Autonomous, <:VariableDependence},
    f::Function
) -> CTBase.var"#90#92"{VectorField{Autonomous, var"#s206"}, <:Function} where var"#s206"<:VariableDependence

Lie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in autonomous case

Example

julia> φ = x -> [x[2], -x[1]]
julia> X = VectorField(φ)
julia> f = x -> x[1]^2 + x[2]^2
julia> (X⋅f)([1, 2])
0

⋅(
    X::VectorField{NonAutonomous, <:VariableDependence},
    f::Function
) -> CTBase.var"#94#96"{VectorField{NonAutonomous, var"#s206"}, <:Function} where var"#s206"<:VariableDependence

Lie derivative of a scalar function along a vector field : L_X(f) = X⋅f, in nonautonomous case

Example

julia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]
julia> X = VectorField(φ, NonAutonomous, NonFixed)
julia> f = (t, x, v) -> t + x[1]^2 + x[2]^2
julia> (X⋅f)(1, [1, 2], [2, 1])
10

Lie(
    X::Function,
    f::Function,
    dependences::DataType...
) -> Function

Lie derivative of a scalar function along a vector field or a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.

Example

julia> φ = x -> [x[2], -x[1]]
julia> f = x -> x[1]^2 + x[2]^2
julia> Lie(φ,f)([1, 2])
0
julia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]
julia> f = (t, x, v) -> t + x[1]^2 + x[2]^2
julia> Lie(φ, f, NonAutonomous, NonFixed)(1, [1, 2], [2, 1])
10

Lie(
    X::Function,
    f::Function;
    autonomous,
    variable
) -> Function

Lie derivative of a scalar function along a function. Dependencies are specified with boolean : autonomous and variable.

Example

julia> φ = x -> [x[2], -x[1]]
julia> f = x -> x[1]^2 + x[2]^2
julia> Lie(φ,f)([1, 2])
0
julia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]
julia> f = (t, x, v) -> t + x[1]^2 + x[2]^2
julia> Lie(φ, f, autonomous=false, variable=true)(1, [1, 2], [2, 1])
10

Lie(X::VectorField, f::Function) -> Function

Lie derivative of a scalar function along a vector field.

Example

julia> φ = x -> [x[2], -x[1]]
julia> X = VectorField(φ)
julia> f = x -> x[1]^2 + x[2]^2
julia> Lie(X,f)([1, 2])
0
julia> φ = (t, x, v) -> [t + x[2] + v[1], -x[1] + v[2]]
julia> X = VectorField(φ, NonAutonomous, NonFixed)
julia> f = (t, x, v) -> t + x[1]^2 + x[2]^2
julia> Lie(X, f)(1, [1, 2], [2, 1])
10

Lie(
    X::VectorField{Autonomous, V<:VariableDependence},
    Y::VectorField{Autonomous, V<:VariableDependence}
) -> VectorField

Lie bracket of two vector fields: [X, Y] = Lie(X, Y), autonomous case

Example

julia> f = x -> [x[2], 2x[1]]
julia> g = x -> [3x[2], -x[1]]
julia> X = VectorField(f)
julia> Y = VectorField(g)
julia> Lie(X, Y)([1, 2])
[7, -14]

Lie(
    X::VectorField{NonAutonomous, V<:VariableDependence},
    Y::VectorField{NonAutonomous, V<:VariableDependence}
) -> VectorField{NonAutonomous}

Lie bracket of two vector fields: [X, Y] = Lie(X, Y), nonautonomous case

Example

julia> f = (t, x, v) -> [t + x[2] + v, -2x[1] - v]
julia> g = (t, x, v) -> [t + 3x[2] + v, -x[1] - v]
julia> X = VectorField(f, NonAutonomous, NonFixed)
julia> Y = VectorField(g, NonAutonomous, NonFixed)
julia> Lie(X, Y)(1, [1, 2], 1)
[-7,12]

Lift(X::Function, dependences::DataType...) -> Function

Return the Lift of a function. Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.

Example

julia> H = Lift(x -> 2x)
julia> H(1, 1)
2
julia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)
julia> H(1, 1, 1, 1)
2

Lift(X::Function; autonomous, variable) -> Function

Return the Lift of a function. Dependencies are specified with boolean : autonomous and variable.

Example

julia> H = Lift(x -> 2x)
julia> H(1, 1)
2
julia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)
julia> H(1, 1, 1, 1)
2

Lift(X::VectorField) -> HamiltonianLift

Return the HamiltonianLift of a VectorField.

Example

julia> HL = Lift(VectorField(x -> [x[1]^2,x[2]^2], autonomous=true, variable=false))
julia> HL([1, 0], [0, 1])
0
julia> HL = Lift(VectorField((t, x, v) -> [t+x[1]^2,x[2]^2+v], autonomous=false, variable=true))
julia> HL(1, [1, 0], [0, 1], 1)
1
julia> H = Lift(x -> 2x)
julia> H(1, 1)
2
julia> H = Lift((t, x, v) -> 2x + t - v, autonomous=false, variable=true)
julia> H(1, 1, 1, 1)
2
julia> H = Lift((t, x, v) -> 2x + t - v, NonAutonomous, NonFixed)
julia> H(1, 1, 1, 1)
2

Poisson(
    f::Function,
    g::Function,
    dependences::DataType...
) -> Hamiltonian

Poisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with DataType : Autonomous, NonAutonomous and Fixed, NonFixed.

Example

julia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2
julia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]
julia> Poisson(f, g)([1, 2], [2, 1])
-20            
julia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]
julia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]
julia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])
-76

Poisson(
    f::Function,
    g::Function;
    autonomous,
    variable
) -> Hamiltonian

Poisson bracket of two functions : {f, g} = Poisson(f, g) Dependencies are specified with boolean : autonomous and variable.

Example

julia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2
julia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]
julia> Poisson(f, g)([1, 2], [2, 1])
-20            
julia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]
julia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]
julia> Poisson(f, g, autonomous=false, variable=true)(2, [1, 2], [2, 1], [4, 4])
-76

Poisson(
    f::AbstractHamiltonian{Autonomous, V<:VariableDependence},
    g::AbstractHamiltonian{Autonomous, V<:VariableDependence}
) -> HamiltonianLift

Poisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), autonomous case

Example

julia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2
julia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]
julia> F = Hamiltonian(f)
julia> G = Hamiltonian(g)
julia> Poisson(f, g)([1, 2], [2, 1])
-20            
julia> Poisson(f, G)([1, 2], [2, 1])
-20
julia> Poisson(F, g)([1, 2], [2, 1])
-20

Poisson(
    f::AbstractHamiltonian{NonAutonomous, V<:VariableDependence},
    g::AbstractHamiltonian{NonAutonomous, V<:VariableDependence}
) -> HamiltonianLift

Poisson bracket of two Hamiltonian functions (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g), non autonomous case

Example

julia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]
julia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]
julia> F = Hamiltonian(f, autonomous=false, variable=true)
julia> G = Hamiltonian(g, autonomous=false, variable=true)
julia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])
-76
julia> Poisson(f, g, NonAutonomous, NonFixed)(2, [1, 2], [2, 1], [4, 4])
-76

Poisson(
    f::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence},
    g::Function
) -> Hamiltonian

Poisson bracket of an Hamiltonian function (subtype of AbstractHamiltonian) and a function : {f, g} = Poisson(f, g), autonomous case

Example

julia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2
julia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]
julia> F = Hamiltonian(f)
julia> Poisson(F, g)([1, 2], [2, 1])
-20
julia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]
julia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]
julia> F = Hamiltonian(f, autonomous=false, variable=true)
julia> Poisson(F, g)(2, [1, 2], [2, 1], [4, 4])
-76

Poisson(
    f::Function,
    g::AbstractHamiltonian{T<:TimeDependence, V<:VariableDependence}
) -> Hamiltonian

Poisson bracket of a function and an Hamiltonian function (subtype of AbstractHamiltonian) : {f, g} = Poisson(f, g)

Example

julia> f = (x, p) -> x[2]^2 + 2x[1]^2 + p[1]^2
julia> g = (x, p) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1]
julia> G = Hamiltonian(g)          
julia> Poisson(f, G)([1, 2], [2, 1])
-20
julia> f = (t, x, p, v) -> t*v[1]*x[2]^2 + 2x[1]^2 + p[1]^2 + v[2]
julia> g = (t, x, p, v) -> 3x[2]^2 + -x[1]^2 + p[2]^2 + p[1] + t - v[2]
julia> G = Hamiltonian(g, autonomous=false, variable=true)
julia> Poisson(f, G)(2, [1, 2], [2, 1], [4, 4])
-76

Poisson(
    f::HamiltonianLift{T<:TimeDependence, V<:VariableDependence},
    g::HamiltonianLift{T<:TimeDependence, V<:VariableDependence}
) -> HamiltonianLift

Poisson bracket of two HamiltonianLift functions : {f, g} = Poisson(f, g)

Example

julia> f = x -> [x[1]^2+x[2]^2, 2x[1]^2]
julia> g = x -> [3x[2]^2, x[2]-x[1]^2]
julia> F = Lift(f)
julia> G = Lift(g)
julia> Poisson(F, G)([1, 2], [2, 1])
-64
julia> f = (t, x, v) -> [t*v[1]*x[2]^2, 2x[1]^2 + + v[2]]
julia> g = (t, x, v) -> [3x[2]^2 + -x[1]^2, t - v[2]]
julia> F = Lift(f, NonAutonomous, NonFixed)
julia> G = Lift(g, NonAutonomous, NonFixed)
julia> Poisson(F, G)(2, [1, 2], [2, 1], [4, 4])
100

∂ₜ(f) -> CTBase.var"#99#101"

Partial derivative wrt time of a function.

Example

julia> ∂ₜ((t,x) -> t*x)(0,8)
8

Macros for Poisson brackets

Example

julia> H0 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)
julia> H1 = (x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)
julia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7]) autonomous=true variable=false
#
julia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)
julia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)
julia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false variable=false
#
julia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)
julia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)
julia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) autonomous=true variable=true
#
julia> H0 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)
julia> H1 = (t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)
julia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2) autonomous=false variable=true

Macros for Lie and Poisson brackets

Example

julia> H0 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2)
julia> H1 = (t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2)
julia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7]) autonomous=false
#
julia> H0 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v)
julia> H1 = (x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v)
julia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2) variable=true
#

Macros for Lie and Poisson brackets

Example

julia> F0 = VectorField(x -> [x[1], x[2], (1-x[3])])
julia> F1 = VectorField(x -> [0, -x[3], x[2]])
julia> @Lie [F0, F1]([1, 2, 3])
[0, 5, 4]
#
julia> F0 = VectorField((t, x) -> [t+x[1], x[2], (1-x[3])], autonomous=false)
julia> F1 = VectorField((t, x) -> [t, -x[3], x[2]], autonomous=false)
julia> @Lie [F0, F1](1, [1, 2, 3])
#
julia> F0 = VectorField((x, v) -> [x[1]+v, x[2], (1-x[3])], variable=true)
julia> F1 = VectorField((x, v) -> [0, -x[3]-v, x[2]], variable=true)
julia> @Lie [F0, F1]([1, 2, 3], 2)
#
julia> F0 = VectorField((t, x, v) -> [t+x[1]+v, x[2], (1-x[3])], autonomous=false, variable=true)
julia> F1 = VectorField((t, x, v) -> [t, -x[3]-v, x[2]], autonomous=false, variable=true)
julia> @Lie [F0, F1](1, [1, 2, 3], 2)
#
julia> H0 = Hamiltonian((x, p) -> 0.5*(2x[1]^2+x[2]^2+p[1]^2))
julia> H1 = Hamiltonian((x, p) -> 0.5*(3x[1]^2+x[2]^2+p[2]^2))
julia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7])
3.0
#
julia> H0 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[1]^2), autonomous=false)
julia> H1 = Hamiltonian((t, x, p) -> 0.5*(x[1]^2+x[2]^2+p[2]^2), autonomous=false)
julia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7])
#
julia> H0 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), variable=true)
julia> H1 = Hamiltonian((x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), variable=true)
julia> @Lie {H0, H1}([1, 2, 3], [1, 0, 7], 2)
#
julia> H0 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[1]^2+v), autonomous=false, variable=true)
julia> H1 = Hamiltonian((t, x, p, v) -> 0.5*(x[1]^2+x[2]^2+p[2]^2+v), autonomous=false, variable=true)
julia> @Lie {H0, H1}(1, [1, 2, 3], [1, 0, 7], 2)
#