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Symplectic Integrators

Euler: H drift =
Leapfrog: H drift =
RK4: H drift =
Step: 0
Symplectic (Leapfrog/Störmer-Verlet) integrators preserve the symplectic 2-form. Energy drift is bounded for exponentially long times. Euler grows unboundedly; RK4 is accurate but slowly dissipates. Phase portrait shows the difference.