Building a Symplectic Integrator:
- It is usually possible to write where H1
and H2 are integrable.
- We can define a new Hamiltonian, , which:
- is a real Hamiltonian for a real dynamical system.
- is numerically integrable.
- Apply H1 for a timestep and then H2, then H1 again.
- is, under the right conditions (small timestep), always close to H.
- Small timestep depends on H2/H1.
- This is a symplectic integrator.
- Only first order.
- Can generalize to higher order by playing with the delta functions.
- Note: since is a function of
, we cannot change timestep during an integration.