Constraints and manifolds

A constraint says where a system is allowed to be. A rail permits motion along one path but rejects position changes that would leave it; a surface does the same in one extra dimension.

Think like a programmer

Think of the allowed set as a data invariant. An update proposes new state, then a constraint projects or corrects it back to the valid shape. The correction belongs to the physical model, not merely input validation.

Model checklist

Inputs: A proposed position and a description of allowed positions.
State: Current position on the permitted path.
Rule: Correct proposals that leave the allowed set.
Output: Valid position and a constraint correction.
Check: The returned state always satisfies the path rule.

A constrained position

Ask for a position outside a line segment and compare it with the allowed state.

Requested position: 12.0 m

0 m10 m

Requested 12.0 m; constrained position 10.0 m; boundary correction -2.0 m.

function constrainToSegment(position: number, start: number, end: number) {
  return Math.max(start, Math.min(end, position));
}

Try this experiment

Prediction: A proposal beyond either stop is returned to the nearest valid endpoint.

Try values below 0 m and above 10 m. State the invariant, then explain what extra state a physically accurate bounce would need.

Where this model breaks

A clamp only projects position. Real constrained dynamics needs forces, velocity changes, time of contact, friction, and sometimes a solver that handles several constraints together.

Summary

Represent an allowed geometry explicitly, enforce it after updates, and test that every returned state remains on the allowed path.

Glossary

Manifold: an allowed shape or space that locally behaves like ordinary coordinates.
Projection: correction of a value back onto an allowed set.
Constraint force: physical force that prevents forbidden motion.

Self-check

What invariant does the segment enforce?
Why is a clamp not a full collision solver?
What would a surface constraint add?

Sources

D. E. Stewart, Rigid-Body Dynamics with Friction and Impact.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Constraints and Manifolds, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Motion in 2D and 3D, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Constraints and Manifolds as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Constraints and Manifolds into a test

Constrain a particle to an allowed path and distinguish the model's tangent motion from the boundary rule that enforces it.

Name the inputs and units that the motion in 2d and 3d model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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