Examples

Examples are ordered from simple to advanced. Each covers one physical model and one objective, with parameters, problem construction, solving, and plotting in a single Python file.

Each page explains the situation and equations before showing the computed figures and videos. Dots are optimized node values; curves are interpolating polynomials. Black dashed curves show analytic solutions where available.

Basics: motion on a line and fastest descent

Small problems with few states and controls, suitable for learning the formulation and comparing with hand calculations.

Example 01Minimum-effort transferBalance shorter travel time against a smaller squared-control integral in a simple motion problem.

Example 02Accelerate and stopMove a cart 10 m from rest to rest in minimum time with bounded acceleration.

Example 03Fastest descentFind the frictionless track that takes a ball from rest to a vertical line 1 m away in minimum time.

Example 04Fastest descent with a depth limitFind a minimum-time track when a floor prevents descending below 0.3 m.

Example 05Fastest descent from Tokyo to OsakaDesign a 600 km gravity-powered tunnel and use scaling to solve a problem in physical units.

Example 06A simple two-dimensional rocketChoose the direction of a fixed-magnitude acceleration to reach a prescribed height and velocity in minimum time.

Rocket ascent and landing

Introduce changing mass, drag, phase splitting, and stage separation.

Example 07Goddard problem with one phaseMaximize the altitude of a vertical rocket with fixed propellant, accounting for drag and changing mass.

Example 08Goddard problem with a knotSplit the Goddard trajectory near burnout to represent a discontinuous thrust switch with separate polynomials.

Example 09Vertical soft landing on the MoonMinimize fuel for a lander descending from 100 m at 10 m/s, with zero velocity at touchdown.

Example 10Single-stage rocket ascentMaximize the altitude of a vertical rocket using physical units, atmospheric drag, variable gravity, and mass consumption.

Example 11Two-stage rocket ascentModel a mass jump at stage separation and maximize altitude with stage-specific propellant budgets.

Orbital insertion and transfer

Two-dimensional flight into and between Earth orbits, with scaling and more constraints.

Example 12Single-stage orbital insertion in polar coordinatesMaximize mass delivered to a 400 km circular orbit by optimizing thrust magnitude and direction.

Example 13Two-stage orbital insertion in polar coordinatesOptimize stage separation and stage-dependent thrust to maximize mass delivered to a 500 km circular orbit.

Example 14Two-stage orbital insertion from TaikiDeliver a 100 kg payload from Taiki, Hokkaido, to a 561 km circular orbit using tabulated atmosphere, Mach-dependent drag, and pressure-dependent engine performance.

Example 15Low-thrust orbit transferMinimize a componentwise thrust-acceleration integral while transferring from a unit circular orbit to radius 4 within a time limit.

Running the examples

Run from the repository root after installing the environment:

uv run python examples/07_goddard_0knot.py

Change the values in each script’s PARAMETERS dictionary or Rocket class to solve with different conditions. The scripts retain their original Japanese comments; these pages explain the models in English.