Symplectic Orbit Propagation — Energy Conservation and Long-Term Stability

For long-term orbit propagation, preserving orbital structure can be more important than simply increasing solver order.

Euler spirals. RK improves accuracy. Symplectic methods preserve conservative-system structure.
Problem Setup

Orbit propagation as a physics-preservation test

This problem compares Euler, RK4, RK7-style, Symplectic Leapfrog, and Velocity Verlet for ideal two-body orbit propagation.

The goal is not only to see which curve looks smooth. The goal is to check whether the numerical method respects the physical structure of orbital motion.

Core question:
does the orbit stay physically bounded, or does the integrator create false drift?
Two-Body Orbit as a Conservative System

No drag, no thrust, no energy loss

In the ideal two-body problem, gravity is the only force. There is no aerodynamic drag, thrust, or perturbation.

\[ \ddot{\mathbf r} = -\frac{\mu}{\lVert \mathbf r \rVert^3}\mathbf r \]

Because the system is conservative, specific mechanical energy should remain constant:

\[ \epsilon = \frac{v^2}{2} - \frac{\mu}{r} \]
Diagnostic rule: If energy drifts strongly in this ideal problem, check the numerical method before blaming the physics.
Why Euler Spirals

Euler does not preserve orbital geometry

Euler integration is easy to understand, but it often injects numerical energy into orbital motion. This can make the orbit spiral outward even though the physical model contains no thrust.

Euler error can look like real orbital change.
But in an ideal two-body problem, that drift is numerical.
Why RK Is Accurate but Not Structure-Preserving

Higher order reduces error, but does not guarantee structure preservation

RK4 and RK7-style methods reduce local truncation error. They often produce much better trajectories than Euler for the same step size.

However, standard RK methods are not designed specifically to preserve the Hamiltonian structure of orbital motion over very long durations.

RK methods: strong local accuracy
Symplectic methods: stronger long-term structure preservation
Symplectic Leapfrog / Velocity Verlet

Updating velocity and position in a structure-aware way

Symplectic Leapfrog and Velocity Verlet split the update into velocity and position stages. This makes them especially useful for conservative systems such as orbital mechanics.

half velocity step → position step → half velocity step
\[ \mathbf v_{n+\frac{1}{2}} = \mathbf v_n + \frac{\Delta t}{2}\mathbf a(\mathbf r_n) \] \[ \mathbf r_{n+1} = \mathbf r_n + \Delta t\,\mathbf v_{n+\frac{1}{2}} \] \[ \mathbf v_{n+1} = \mathbf v_{n+\frac{1}{2}} + \frac{\Delta t}{2}\mathbf a(\mathbf r_{n+1}) \]
Energy Conservation Check

Interactive orbit propagation comparison

Adjust altitude, eccentricity, timestep, number of orbits, and method selector. The simulation compares orbit path, energy drift, radius drift, final position error, and step-size sensitivity.

Euler energy drift —
RK4 energy drift —
RK7-style drift —
Leapfrog drift —
Velocity Verlet drift —
Run the simulation to compare long-term orbit behaviour.
Long-Term Orbit Comparison

Watch small numerical differences grow

The plots above use the selected number of orbits. Increase the orbit count and timestep to see how long-term differences become more visible.

Key point: A method that looks acceptable after one orbit may become unreliable after many revolutions.
Step-Size Sensitivity

Generate a timestep sensitivity map

This section sweeps timestep values and compares final position error for all methods.

Engineering Interpretation

When structure matters more than solver order

Euler
Useful for teaching, but poor for long-term orbit propagation.
RK4 / RK7-style
Accurate and useful, but not automatically structure-preserving.
Leapfrog / Verlet
Strong for conservative orbital systems because they preserve long-term structure.
Energy check
A required diagnostic for ideal two-body propagation.
Takeaway

What this problem shows

For long-term orbit propagation, preserving orbital structure can be more important than simply increasing solver order.
Orbit path + energy drift + radius drift + final error
together reveal whether the numerical result is physically trustworthy.
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