Lanchester's laws
Lanchester's laws are mathematical formulas for calculating the relative strengths of military forces.
[1][2] In 1915 and 1916 during World War I, M. Osipov[3]: vii–viii and Frederick Lanchester independently devised a series of differential equations to demonstrate the power relationships between opposing forces.
As of 2017 modified variations of the Lanchester equations continue to form the basis of analysis in many of the US Army’s combat simulations,[5] and in 2016 a RAND Corporation report examined by these laws the probable outcome in the event of a Russian invasion into the Baltic nations of Estonia, Latvia, and Lithuania.
The linear law also applies to unaimed fire into an enemy-occupied area.
In its basic form, the law is only useful to predict outcomes and casualties by attrition.
It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time.
For this reason, the law does not apply to machine guns, artillery with unguided munitions, or nuclear weapons.
The law requires an assumption that casualties accumulate over time: it does not work in situations in which opposing troops kill each other instantly, either by shooting simultaneously or by one side getting off the first shot and inflicting multiple casualties.
Each one has offensive firepower α, which is the number of enemy soldiers it can incapacitate (e.g., kill or injure) per unit time.
[7] Here, dA/dt represents the rate at which the number of Red soldiers is changing at a particular instant.
In a gun battle, bullets or shells are typically fired in large quantities.
Therefore, Lanchester's equations model gunfire as a stream of firepower that continuously weakens the enemy force over time.
By comparison, cruise missiles typically are fired in relatively small quantities.
Each one has a high probability of hitting its target, and carries a relatively powerful warhead.
Given their small size and large number, it is not practical to intercept bullets and shells in a gun battle.
Lanchester's laws have been used to model historical battles for research purposes.
[10] In modern warfare, to take into account that to some extent both linear and the square apply often, an exponent of 1.5 is used.
[13] The laws have also been applied to repeat battles with a range of inter-battle reinforcement strategies.
[14] Attempts have been made to apply Lanchester's laws to conflicts between animal groups.
[18] The Helmbold Parameters offer precise numerical indices, grounded in historical data, for quickly and accurately comparing battles in terms of bitterness and the degree of advantage held by each side.
While their definition is modeled after a solution of the Lanchester Square Law's differential equations, their numerical values are based entirely on the initial and final strengths of the opponents and in no way depend upon the validity of Lanchester's Square Law as a model of attrition during the course of a battle.
is small enough that the hyperbolic functions can, without any significant error, be replaced by their series expansion up to terms in the first power of
is a kind of "average" (specifically, the geometric mean) of the casualty fractions justifies using it as an index of the bitterness of the battle.
Statistical work prefers natural logarithms of the Helmbold Parameters.
See Helmbold (2021): Some observers have noticed a similar post-WWII decline in casualties at the level of wars instead of battles.
