What does a symplectic integrator preserve, and what does it not promise?
14-week course · 3 credits · Week 3
Motivation
What does a symplectic integrator preserve, and what does it not promise?
For a dominant central mass and weak perturbations, Wisdom-Holman splitting separates Keplerian drift from perturbing kicks. WHFast is fixed-step and symplectic; IAS15 is adaptive and high-order. Their errors have different structure and should be selected by the problem, not by a single accuracy slogan.
Today’s result must be interpretable as a physical statement with declared data, algorithm, uncertainty, and a check that could fail.
Learning objectives
After this lecture, you can
What does a symplectic integrator preserve, and what does it not promise?
Initialize units before particles
An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Core language
Terms for This Lecture
Symplectic integrator
Definition: A time-stepping map that preserves Hamiltonian phase-space geometry.
Why it matters here: It helps answer What does a symplectic integrator preserve, and what does it not promise? within the numerical model described here.
Relationships: Connect it to Kepler drift, the governing expression \(H=\sum_i\left(\frac{p_i^2}{2m_i}-\frac{GMm_i}{r_i}\right)+H_{\rm interaction}\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Kepler drift
Definition: Exact or specialized propagation under the dominant two-body Hamiltonian during a split step.
Why it matters here: It helps answer What does a symplectic integrator preserve, and what does it not promise? within the numerical model described here.
Relationships: Connect it to Kick step, the governing expression \(H=\sum_i\left(\frac{p_i^2}{2m_i}-\frac{GMm_i}{r_i}\right)+H_{\rm interaction}\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Kick step
Definition: Momentum update produced by the perturbing interaction part of a split Hamiltonian.
Why it matters here: It helps answer What does a symplectic integrator preserve, and what does it not promise? within the numerical model described here.
Relationships: Connect it to Timestep convergence, the governing expression \(H=\sum_i\left(\frac{p_i^2}{2m_i}-\frac{GMm_i}{r_i}\right)+H_{\rm interaction}\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Timestep convergence
Definition: Evidence obtained by comparing results from systematically refined step sizes.
Why it matters here: It helps answer What does a symplectic integrator preserve, and what does it not promise? within the numerical model described here.
Relationships: Connect it to Ephemeris, the governing expression \(H=\sum_i\left(\frac{p_i^2}{2m_i}-\frac{GMm_i}{r_i}\right)+H_{\rm interaction}\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Ephemeris
Definition: A time-tagged prediction of an object's position and velocity in a specified frame and dynamical model.
Why it matters here: It helps answer What does a symplectic integrator preserve, and what does it not promise? within the numerical model described here.
Relationships: Connect it to Symplectic integrator, the governing expression \(H=\sum_i\left(\frac{p_i^2}{2m_i}-\frac{GMm_i}{r_i}\right)+H_{\rm interaction}\), and the computed diagnostic developed later in the lecture.
Boundary or common confusion: An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Physical and mathematical model
State what is being evolved
For a dominant central mass and weak perturbations, Wisdom-Holman splitting separates Keplerian drift from perturbing kicks. WHFast is fixed-step and symplectic; IAS15 is adaptive and high-order. Their errors have different structure and should be selected by the problem, not by a single accuracy slogan.
Define the state, coordinates, units, initial/boundary conditions, and domain before interpreting solver output.
Numerical method
Algorithm choice is a modeling decision
Initialize units before particles; validate the two-body orbit; sweep fixed timesteps; record energy and angular momentum; compare a one-day Sun-Earth propagation with the JPL state in a consistent heliocentric frame.
Write the model and its assumptions.
Choose a solver and identify its stability, accuracy, and conservation properties.
Record configuration, tolerances or timestep, and output sampling.
Compare against an analytic limit, independent method, or controlled refinement.
Visual reasoning
Heliocentric Earth orbit and JPL checkpoints
An explicitly schematic orbital ellipse plus the exact two JPL epoch points; the points are close because they are one day apart.
Computed example
One result, with a check
Using Astropy GM_sun, the circular 1-AU Kepler period is 365.25690 d. After one day, REBOUND and independent SciPy two-body runs differ from the DE441 Earth-center vector by 126.0 km and 126.1 km respectively; the residual is model/frame scope, not an accuracy claim.
Inputs and method are generated by src/astr474_computations.py; the machine-readable values are in data/generated/computed_results.json.
An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Verification
What would falsify this result?
Repeat with a smaller timestep, tighter tolerance, or finer spatial grid.
Track conservation or residual structure in addition to visual agreement.
Check units, coordinate frame, boundary conditions, and invariants independently.
Change a random seed when the method is stochastic; report the resulting sampling uncertainty.
Do not report a smoother curve as proof of convergence.
Failure modes
Numerical agreement has a domain
An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.
Solver error, discretization error, stochastic error, parameter uncertainty, and model discrepancy are different quantities. Diagnose and report them separately.
Active reasoning
Make a prediction before rerunning
What does a symplectic integrator preserve, and what does it not promise?
State a measurable outcome, a competing explanation, and the smallest computation that distinguishes them. Include an expected direction or scaling, then test it.
Workflow
From parameters to auditable output
Initialize units before particles; validate the two-body orbit; sweep fixed timesteps; record energy and angular momentum; compare a one-day Sun-Earth propagation with the JPL state in a consistent heliocentric frame.
Record the exact command, package versions, data path, random seed if applicable, and output checksum. Keep raw inputs unchanged and generated files separate.
Transfer
Lab and problem-set connection
Lab 02 uses the level-2 JPL vector record and distinguishes integrator error from model discrepancy.
Submit plots with labeled axes/units, a concise uncertainty argument, and enough code or intermediate values for a peer to reproduce the claim.
Synthesis
Three durable checks
Does the implemented equation match the intended physical model?
Does the numerical result converge under a controlled method change?
Does the interpretation stay inside the validity and provenance limits?
An ideal two-body propagation is not an ephemeris. Earth-center state, Sun motion, Moon, other planets, and relativistic terms matter when comparing against DE441.