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Stability and equilibrium points

An equilibrium point has no net force. Its stability depends on what happens after a small displacement: a stable equilibrium tends to restore motion, while an unstable one amplifies the displacement.

Think like a programmer

Equilibrium is a fixed point of the update rule. Stability is a perturbation test: nudge the state, run the same rule, and compare whether the difference shrinks or grows.

Model checklist

Inputs
Position and a conservative potential model.
State
Position near an equilibrium.
Rule
Find places where force is zero or potential slope is zero.
Output
Stable or unstable response to a perturbation.
Check
A potential minimum is stable in the simple one-dimensional model.

Energy as an accounting check

Set mass and height. In this no-loss model, gravitational potential energy becomes kinetic energy at ground level.

Potential energy 98.07 J; predicted landing speed 9.90 m/s; kinetic energy 98.07 J.

Try this experiment

Prediction: A low potential-energy state has no lower nearby configuration in this model.

Change mass and height in the energy model. Then imagine a small displacement from a valley and from a hilltop; state which one restores motion.

Where this model breaks

Damping, driving, constraints, and multiple dimensions can change the stability picture. A static minimum alone does not describe transient behavior or how quickly a system returns.

Summary

Treat equilibrium as a fixed point and stability as a perturbation test. In a simple conservative model, minima are stable and maxima are unstable.

Glossary

Self-check

  1. What is a fixed point in code?
  2. Which potential shape is stable?
  3. What extra effect can change stability?

Sources

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 Stability and Equilibrium Points, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Potential Energy and Conservation Laws, 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 Stability and Equilibrium Points 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

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

Model review: turn Stability and Equilibrium Points into a test

Use potential-energy minima and maxima to distinguish stable, unstable, and neutral equilibrium in a simple model.

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