---
title: "Hamiltonian Phase Space & Integrators Interactive Lab | Sandbox Physics"
description: "When a computer simulates an orbit, can a small energy error hide a wrong trajectory? It can. A symplectic integrator, one built to respect the structure of Hamilton's equations, keeps the energy error small and bounded for a very long time. The computed body can still drift away from the true trajectory, running ahead of or behind it along the orbit. Energy behavior, accuracy, symplectic structure and time reversibility are separate properties; check each one."
canonical_url: "https://sandboxphysics.com/labs/hamiltonian-integrators/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hamiltonian-integrators/"
last_modified: "2026-10-22"
---

M084 · Geometry of numerical integrators

# Hamiltonian Phase Space & Integrators

**When a computer simulates an orbit, can a small energy error hide a wrong trajectory?** It can. A symplectic integrator, one built to respect the structure of Hamilton's equations, keeps the energy error small and bounded for a very long time. The computed body can still drift away from the true trajectory, running ahead of or behind it along the orbit. Energy behavior, accuracy, symplectic structure and time reversibility are separate properties; check each one.

Run Euler, RK4, symplectic Euler and Verlet side by side. Compare how closely each follows the true phase or orbit over long times, how its energy drifts, and whether its step map keeps the local canonical geometry and runs backward correctly. Then halve the step and take on a Kepler periapsis where the motion changes fast.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The three systems are conservative, dimensionless and of unit mass: a harmonic oscillator, an ideal pendulum and a point mass on a planar Kepler orbit. All four methods use the same constant step. Nominal cycles count the harmonic or Kepler reference period, not the period of the pendulum at finite amplitude. The oscillator and Kepler references are analytic solutions; the pendulum reference uses RK4 at two finer steps. The difference between those two refinements is an estimate, not a rigorous error bound. There is no adaptive step size, damping or physical collision model.

Physics tutorial

## A numerical map has geometry as well as local order

**Background** A numerical method replaces continuous Hamiltonian motion with a discrete map that carries the canonical coordinates forward one step at a time.

**Why it matters** Geometric numerical integration asks which structures of the true motion a discrete method keeps, and what that means over long runs.

Start with the essentials

- **Focus question**: When a computer simulates an orbit, can a small energy error hide a wrong trajectory?
- **One-sentence intuition**: Accuracy, energy behavior, symplecticity and reversibility are related, but each one measures something different.

### Core mathematical model

#### Hamilton's equations for the three systems

$$
\dot{\mathbf q}=\mathbf p,\qquad\dot{\mathbf p}=-\nabla V,\qquad H=\frac{|\mathbf p|^2}{2}+V
$$

Unit mass and scaled units let the three conservative systems be compared directly. The harmonic frequency and the Kepler gravity parameter are both one.

#### A kick followed by a drift

$$
\mathbf p_{j+1}=\mathbf p_j-h\nabla V(\mathbf q_j),\qquad\mathbf q_{j+1}=\mathbf q_j+h\mathbf p_{j+1}
$$

Symplectic Euler is first order. It differs from explicit Euler because the drift uses the momentum that was just updated.

#### Symmetric Verlet steps

$$
\mathbf p_{j+1/2}=\mathbf p_j-\frac h2\nabla V(\mathbf q_j),\quad\mathbf q_{j+1}=\mathbf q_j+h\mathbf p_{j+1/2},\quad\mathbf p_{j+1}=\mathbf p_{j+1/2}-\frac h2\nabla V(\mathbf q_{j+1})
$$

Splitting the kick into two halves gives a second-order, time-reversible map. That does not conserve the true Hamiltonian exactly, and it does not make a periapsis taken in too few steps accurate.

#### Checking that the step map is symplectic

$$
D\Phi_h^{\mathsf T}J D\Phi_h=J,\qquad J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}
$$

The readout gives the largest deviation of any matrix entry, estimated with central differences. In four dimensions, a determinant of one is a weaker test on its own. The Kepler patch is a two-dimensional projection and never measures the full phase-space volume.

#### Energy error and state error are measured separately

$$
\epsilon_H=\frac{H_j-H_0}{\max(|H_0|,10^{-6})},\qquad\epsilon_z=\|\mathbf z_j-\mathbf z_{\rm ref}(t_j)\|_2
$$

The maximum energy error uses every step, but the plots keep at most about 600 samples per method. A steady energy curve can hide a phase error that keeps growing. A record that stops is not extended with a prediction.

#### How the logarithmic axes work

$$
y_H=\operatorname{sgn}(\epsilon_H)\log_{10}\left(1+\frac{|\epsilon_H|}{10^{-8}}\right),\qquad y_z=\log_{10}\left(1+\frac{\epsilon_z}{10^{-8}}\right)
$$

One shared logarithmic scale shows tiny and large errors together and keeps the sign of the energy drift. Nominal cycles use a reference period of two pi in scaled time. The pendulum reference compares runs with 32 and 64 RK4 substeps in each coarse step.

### Common difficulties

#### Good energy does not mean a good orbit

**Typical misconception** If the energy error is small, the orbit must be right.

**Better mental model** Phase error can build up while the energy stays nearly constant. Compare with the reference state at the same times.

#### Symplectic does not mean the energy is exact

**Typical misconception** A symplectic method conserves the exact Hamiltonian at every step.

**Better mental model** When the fixed step resolves the motion, the energy error usually oscillates within bounds, but the true energy is still not conserved exactly.

#### Periapsis still needs small steps

**Typical misconception** Preserving the geometry makes up for a large time step.

**Better mental model** The fast pass close to the centre still needs a short step. Halve the step and compare the orbit error, not just the determinant.

#### A projected area is not phase-space volume

**Typical misconception** If the projected Kepler patch keeps its area, the method is symplectic.

**Better mental model** Kepler phase space has four dimensions. The full matrix test gives the relevant local evidence.

### Run the experiment

1. 01

   #### Compare all methods

   Review a full harmonic record and switch the selected method.

   **What to observe:** Every color uses the same step, yet the energy behaves differently: it grows, drifts or oscillates within bounds.

2. 02

   #### Halve the step

   Use the long RK4 record, then halve the fixed step.

   **What to observe:** Both the state error and the energy drift change with the step size. A high local order is not the only thing that matters.

3. 03

   #### Leave the small-angle limit

   Compare the large-amplitude libration and rotating pendulum presets.

   **What to observe:** The reference here is a refined numerical solution, not a harmonic approximation.

4. 04

   #### Take on a Kepler periapsis

   Compare a resolved and an unresolved periapsis, check the full-map defect, then halve the step.

   **What to observe:** A symplectic map can keep its geometry and still give a poor trajectory.

Physical conclusion

Before you judge a method, compare its numerical diagnostics with the analytic or refined reference solution.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)