Symplectic Integrator

Author: Tianjiang Shuo
Site: https://cislunarspace.cn

Definition

A symplectic integrator is a class of numerical integration methods that preserve the symplectic geometric structure (the symplectic form) of Hamiltonian systems in phase space. For conservative systems in celestial mechanics (Hamiltonian systems), symplectic integrators maintain energy and other conserved quantities without systematic drift over long-term integration.

Principles

Hamiltonian Systems

The equations of motion for a Hamiltonian system are:

\dot{\mathbf{q}} = \frac{\partial H}{\partial \mathbf{p}}, \quad \dot{\mathbf{p}} = -\frac{\partial H}{\partial \mathbf{q}}

where $H(\mathbf{q}, \mathbf{p})$ is the Hamiltonian, $\mathbf{q}$ are generalized coordinates, and $\mathbf{p}$ are generalized momenta.

Symplectic Geometry

The phase space flow of a Hamiltonian system is symplectic, preserving the symplectic 2-form $d\mathbf{q} \wedge d\mathbf{p}$ . Standard numerical integrators (e.g., standard Runge-Kutta methods) do not preserve the symplectic structure, leading to systematic energy drift over long-term integration.

Störmer-Verlet Method

The second-order symplectic integrator, also known as the leapfrog method:

\mathbf{p}_{n+1/2} = \mathbf{p}_n + \frac{\Delta t}{2} \cdot \frac{\partial H}{\partial \mathbf{q}}(\mathbf{q}_n)

\mathbf{q}_{n+1} = \mathbf{q}_n + \frac{\Delta t}{2} \cdot \left( \frac{\partial H}{\partial \mathbf{p}}(\mathbf{q}_n) + \frac{\partial H}{\partial \mathbf{p}}(\mathbf{q}_{n+1}) \right)

\mathbf{p}_{n+1} = \mathbf{p}_{n+1/2} + \frac{\Delta t}{2} \cdot \frac{\partial H}{\partial \mathbf{q}}(\mathbf{q}_{n+1})

Splitting Methods

Decompose the Hamiltonian as $H = T(\mathbf{p}) + V(\mathbf{q})$ and integrate the kinetic and potential energy separately:

\mathbf{p} \leftarrow e^{\Delta t \cdot \nabla_{\mathbf{p}} T} \mathbf{p} \quad \text{(drift)}

\mathbf{q} \leftarrow e^{\Delta t \cdot \nabla_{\mathbf{q}} V} \mathbf{q} \quad \text{(kick)}

Applications in Cislunar Space

Long-term orbit evolution simulation: interplanetary trajectory prediction requiring $10^5$ – $10^8$ revolution integration: symplectic integrators ensure no energy drift and reliable results
Multi-body problem integration: long-term integration of the restricted three-body problem, where symplectic integrators outperform standard RK methods
Solar system nested three-body problems: long-term orbital evolution of Jupiter, Saturn, and other giant planets
Periodic orbit computation: symplectic integrators can be used to search for periodic orbits via phase space analysis

Comparison with Runge-Kutta Methods

Property	Symplectic Integrator	Standard Runge-Kutta
Energy conservation	Long-term preservation	Systematic drift
Phase space structure	Preserves symplectic form	Not preserved
Accuracy	Comparable at same order	Comparable at same order
Computational cost	Comparable	Comparable
Best for	Long-term integration, separable Hamiltonians	Short-term integration, non-conservative systems

Circular Restricted Three-Body Problem (CR3BP)
Ephemeris Model
Shooting Method

References

Hairer E, Lubich C, Wanner G. Geometric numerical integration[M]. Springer, 2006.
Sanz-Serna J M, Calvo M P. Numerical Hamiltonian problems[M]. Chapman & Hall, 1994.