Symplectic Integrators & Energy Conservation

System

System: Step size dt: 0.20 Initial energy: 0.50 Steps to run: 2000

Leapfrog (symplectic):
p_{n+½} = pₙ − (dt/2)∇V
q_{n+1} = qₙ + dt·p_{n+½}
p_{n+1} = p_{n+½} − (dt/2)∇V

Euler (non-symplectic):
pₙ₊₁ = pₙ − dt·∇V(qₙ)
qₙ₊₁ = qₙ + dt·pₙ

Symplectic ↔ volume-preserving
area in phase space is conserved

Leapfrog ΔE/E₀: —

Euler ΔE/E₀: —

Symplectic integrators preserve the symplectic form dq∧dp, ensuring no secular energy drift. The Leapfrog/Störmer-Verlet method is 2nd-order and time-reversible. Euler's method is not symplectic — energy drifts systematically outward. Critical for N-body and MD simulations.