add the liouville transformation

[ChrisRackauckas]

@YingboMa this would be a nice case for a transformation to be able to set an initial condition, another indicator that we need some form of default values

[ChrisRackauckas]

@agerlach

[agerlach]

I’m not at my computer to look closer but I don’t think the implementation is correct. It should be dpdt = -p * tr(jac).

Also, why do you state the IC should be zero?

[ChrisRackauckas]

Oh yes, I found a good source:

https://abhishekhalder.bitbucket.io/F16ACC2013Final.pdf

Indeed I was missing a term, and that IC would come from the characteristic.

[ChrisRackauckas]

@YingboMa I think I'll go with this for now, though I don't quite like exposing a hardcoded symbol. I can't think of another good way to do it though.

[agerlach]

I don't quite like exposing a hardcoded symbol

Why not just have the user create and pass the symbol

function liouville_transform(sys, var)
    t = independent_variable(sys)
    D = ModelingToolkit.Differential(t)
    neweq = D(var) ~ var*-tr(calculate_jacobian(sys))
    neweqs = [equations(sys);neweq]
    vars = [states(sys);var]
    ODESystem(neweqs,t,vars,parameters(sys))
end

@variables q
liouville_transform(sys, q)

or wrap it in a type?

struct Liouville
    var
end

function transform(sys, alg::Liouville)
    var = alg.var
    t = independent_variable(sys)
    D = ModelingToolkit.Differential(t)
    neweq = D(var) ~ var*-tr(calculate_jacobian(sys))
    neweqs = [equations(sys);neweq]
    vars = [states(sys);var]
    ODESystem(neweqs,t,vars,parameters(sys))
end

@variables q
transform(sys,q)

[ChrisRackauckas]

Having the user pass a symbol might be a better option than having one hardcoded. @YingboMa what do you think?