Symplectic Integrators

Building a Symplectic Integrator:

• It is usually possible to write where H₁ and H₂ are integrable.
  • More terms work too.
  
  
• We can define a new Hamiltonian, , which:
  1. is a real Hamiltonian for a real dynamical system.
  2. is numerically integrable.
     • Apply H₁ for a timestep and then H₂, then H₁ again.
  3. 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.