Hamiltonian problem types about diffeqphysics.jl

[Datseris]

Yeap okay, I will do that after I learn how all these new stuff works.

I will also make a method
HamiltonianProblem(dHdq, dHdq, tspan) in case the user can do the differentiation by themselves. I know its is identical to the ODEProblem but having the Hamiltonian name will be not only nice but intuitive as well.

[ChrisRackauckas]

Some more details. Hamiltonians can be written as H(p,q). This then gives equations of motion

dp/dt = - dH/dq
dq/dt = dH/dp

Those equations of motion define a PartitionedODEProblem, where the partition has a set of ODEs for p and a set of ODEs for q. OrdinaryDiffEq.jl can natively handle this kind of PartitionedODEProblem and use things like Verlets, or it can (soon) transform it into a standard ODE for all of the other algorithms. So the actual solving can be handled by that.

What needs to be done is that the equations of motion need to be generated using differentiation tools. I.e., use something like ForwardDiff to define the equations of motion using a user defined H(p,q) function (for vectors p and q).

[Datseris]

I would like to try and solve this issue.
How do you envision the interface?
Hamiltonian(p,q) -> PartitionedODEProblem etc?

I have had some experience with ForwardDiff for the DynamicalSystems package, so probably I can handle this... (don't be so sure though)

[ChrisRackauckas]

How do you envision the interface?
ODEHamiltonian(p,q) -> PartitionedODEProblem etc?

Yes.

HamiltonianProblem(H,q0,p0,tspan)

Essentially what it will do is use ForwardDiff to make a function for dHdq and a function for dHdp, and build the ODEProblem((dHdq,dHdp),(q0,p0),tspan). This output can then be directly solved by the symplectic solvers.

So the only "hard part" is the differentiation, which should just be differentiating a closure ForwardDiff.derivative!((p)->H(p,q),...). And making tests of course. It actually should be pretty easy, but will be a nice convenience.