Lancaster Equation, an introduction to the Lanchester Equation

On the eve of World War I, the multi-talented Englishman Lanchester pioneered a semi-empirical method of combat simulation, establishing the classic Lanchester equation. Lanchester quantitatively explained the success of Nelson's various defeats at the Battle of Trafalgar (known as Nelson Touch) with the law of squares, and Engel used the linear law to accurately reproduce the situation of American forces on Iwo Jima in 54.

The classical Lanchester equation does not take into account morale, terrain, maneuver, reinforcements, and retreat, etc., but it is still instructive for the general laws of combat.

Lanchester reduced combat to two basic situations: long-range firefights and close-range concentrated fire. In a long-range firefight, the loss rate of one side is proportional to both the opponent's strength and the strength of one's own, expressed by differential equations.

dy/dt=-a*x*y

dx/dt=-b*x*y

where X and Y are the number of combat units of the Red Army and the Blue Army, respectively, and A and B are the average unit combat effectiveness of the Red Army and the Blue Army, respectively, so the conditions for the equality of the strength of the two sides are.

a*x=b*y

That is, the strength of either side is linearly related to the number of combat units of its own, also known as Lanchester's linear law. That is to say, if the average unit combat effectiveness of the Blue Army (including **, training and other factors) is four times that of the Red Army, the combat effectiveness of 100 Blues and 400 Reds is the same, and the result of the battle between 100 Blues and 400 Reds is the same end. Concentrating superior forces is just a matter of attrition, and it does not take advantage.

However, when killing with concentrated fire at close range, the loss rate of one side is only proportional to the number of combat units of the opponent, and has nothing to do with the number of combat units of the opponent, that is.

dy/dt=-a*x

dx/dt=-b*y

The condition for the equal strength of the two sides becomes.

a*x^2=b*y^2

That is, the strength of either side is directly proportional to the square of the number of combat units in itself, also known as Lanchester's law of squares. It is still assumed that the average unit combat effectiveness of the Blue Army is four times that of the Red Army, and after 100 Blue Army and 400 Red Army in close combat, when the Blue Army of 100 people is completely annihilated, the Red Army still has sqrt(400 2-4*100 2) = 346 people left (here sqrt is the square root, 2 is the square), that is, 54 people are lost. This.

It is the mathematical basis for concentrating forces to fight a war of annihilation, and the actual losses of the superior side are smaller than those of the inferior side.

The Lanchester equation, also known as the Lanchester battle theory or battle dynamics theory, is a branch of operations research that applies mathematical methods to study the process of annihilation of opposing sides in battle.

Lanchester EquationBritish engineer Lanchester proposed a system of differential equations that describe the relationship between the changes in the forces of the two sides of the war, which is known as the Lanchester equation.

The Lanchester equation is a system of differential equations that describe the relationship between the changes in the number of troops between opposing sides during an engagement. It includes the first linear law, the second linear law, and the square law. It is used to reveal the quantitative relationship between the changes in the battle results of the opposing sides under specific initial force and weapon conditions.

It is mainly used for operational research and analysis in operational command, military training, and equipment demonstration.

Lanchester's combat effectiveness equation is: combat effectiveness = total number of units participating in the battle unit combat efficiency.

It should be the derivative of BLU's strength = RED's strength multiplied by a

Derivative of Red's forces = Blue's forces multiplied by b

a, b are both numbers less than 0.