IA idea · Differential equations & dynamics

Can mathematics predict a battle? Lanchester's laws and Iwo Jima

AI HLAA HL Ambitious Also in: Statistics

Research question

How well do Lanchester's square-law differential equations, with parameters estimated as in Engel's 1954 study, reproduce the troop numbers in the Battle of Iwo Jima, and what does the square law imply about concentration of force?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Lanchester's equations explain why concentration of force matters, and in 1954 J. H. Engel tested them against real battle data. Re-doing the analysis is a rare chance to verify a classic piece of applied mathematics.

The mathematics you'll need

Coupled linear differential equations
Deriving the square law invariant aM² − bN² = constant
Euler's method with reinforcements
Parameter estimation and goodness of fit

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use the daily troop and casualty figures reported in Engel's paper and later reanalyses; cite them carefully.

Engel (1954), A Verification of Lanchester's Law — The classic paper fitting Lanchester's combat equations to daily troop numbers from the Battle of Iwo Jima.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain the model and its assumptions.
Derive the square law from the equations.
Fit parameters and simulate the battle.
Compare with the recorded numbers.
Reflect on what the model ignores and on using maths to study war.

Pitfalls that cost marks

Treating a historical simulation as a precise prediction.
Copying Engel's results instead of reproducing them.
Ignoring the reinforcement term.

Showing personal engagement

Discuss the ethics of military modelling thoughtfully.
Test the linear law as an alternative and compare.
Apply the square law to a strategy game you play.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Fit both the square and linear laws by least squares and decide which fits better.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).