Symplectic Integrators
Time-stepping methods for Hamiltonian systems that preserve phase-space structure, keeping energy bounded over very long integrations.
Preserving the geometry of motion
Hamiltonian systems, such as charged particles orbiting in magnetic fields, evolve so that phase-space volume is conserved (Liouville's theorem) and the flow is symplectic. Standard integrators like Runge-Kutta do not respect this structure; over long runs they slowly drift in energy, either gaining or losing it artificially. Symplectic integrators are designed so that each step is itself a symplectic map, preserving the geometric structure exactly.
The payoff: bounded energy error
A symplectic method does not conserve the true energy exactly, but it exactly conserves a nearby shadow Hamiltonian. As a result the energy error stays bounded and oscillatory over arbitrarily long integrations rather than drifting without limit. For studying particle confinement over millions of orbits, this qualitative correctness matters far more than the local truncation error of a single step.
Common schemes
- The symplectic Euler method: implicit in one variable, explicit in the other
- The Stormer-Verlet (leapfrog) method: second-order, time-reversible, and the workhorse of molecular dynamics and particle-in-cell codes
- Higher-order composition methods built by composing leapfrog steps with carefully chosen substep sizes
- Implicit Runge-Kutta methods of Gauss type, which are symplectic to all orders
The Boris push
For charged particles in electromagnetic fields, the Boris algorithm is the standard particle mover in plasma particle-in-cell codes. It separates the electric acceleration from the magnetic rotation, handling the magnetic part as an exact rotation. It conserves phase-space volume and reproduces the gyration and drift motion faithfully over long runs, which is why it dominates kinetic plasma simulation. Symplectic and structure-preserving integrators like these are essential to trustworthy long-time particle-orbit modeling in fusion devices.