On the eve of World War I, Lanchester, a versatile Englishman, created a semi-empirical combat simulation method with mathematics and established the classic Lanchester equation. Lanchester quantitatively explained Nelson's success in the Battle of Trafalgar (called Nelson Touch) with the square law, and Engel accurately reproduced the casualties of American troops in Iwo Jima in 1954 with the linear law. The classical Lanchester equation does not consider morale, terrain, maneuver, reinforcement and retreat, but it still has guiding significance to the general law of combat.
Lanchester simplified the battle into two basic situations: long-range crossfire killing and close-range concentrated fire killing. In a long-distance firefight, the loss rate of one side is proportional to the strength of the other side as well as its own strength. In other words, the more enemies, the greater the loss; On the other hand, the more we own, the bigger our goals and the greater our losses. This situation is expressed by differential equations.
dy/dt=-axy
dx/dt=-bxy
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, then the conditions for equal strength of the two sides are as follows.
ax=by
That is, there is a linear relationship between the strength of either side and the number of its own combat units, also known as Lanchester's linear law. That is to say, if the average unit combat effectiveness of the Blue Army (including weapons, training and other factors) is four times that of the Red Army, then the combat effectiveness of 100 Blue Army and 400 Red Army is the same, and the result of 100 Blue Army and 400 Red Army fighting is mutually assured destruction. Concentrating superior forces is only a struggle, not an advantage.
However, in close-range concentrated fire killing, the loss rate of one side is only proportional to the number of combat units of the other side, and has nothing to do with the number of combat units of its own side. In other words, the more enemies there are, the greater their losses will still be; However, in hand-to-hand combat, there is no problem with the size of the target on your own. So the differential equation becomes:
dy/dt=-ax
dx/dt=-by
The situation of equal strength between the two sides becomes
ax^2=by^2
That is, the strength of either side is directly proportional to the square of the number of its own combat units, also known as Lanchester's square law. Or suppose that the average unit combat effectiveness of the Blue Army is four times that of the Red Army. 100 After the melee between the Blue Army and the 400 Red Army, when 100 the Blue Army was wiped out, the Red Army remained √([400]2-4×[ 100]2)= 346 people, that is, 54 people were lost. This is the mathematical basis for concentrating forces to fight annihilation and avoiding refueling tactics.
Consider another situation: 200 Blue Army and 400 Red Army are in close combat, and the strength of both sides is equal (√ ([400] 2-4× [200] 2) = 0). If the Red Army divides the Blue Army into two halves, each with 100 men, but does not support each other through tactical actions or strategies, the Red Army can first annihilate the first 100 men of the Blue Army at the cost of 54 men, and then annihilate the second 100 men of the Blue Army with the remaining troops at the cost of 64 men. The total cost of the Red Army is 1 18 people. Such a total result is the mathematical explanation of the principle of "divide and conquer", and it is also the mathematical explanation of a mountain of defeat, because the typical feature of defeat is that they fight independently, regardless of the end, even if the problem of the disintegration of fighting will is not considered, it objectively strengthens the chance of being divided.
Consider another situation. We still consider the Blue Army 100 and the Red Army's 400. The combat effectiveness gap between the two sides is 4: 1, but the two sides are far apart. If the Red Army advances to close range at half the cost, according to the linear law of 4: 1, there are still 200 people left in the Red Army and 50 people left in the Blue Army. But then the Red Army can give full play to its melee advantage and destroy the second 50 people of the Blue Army at the cost of 27 people. This is a mathematical explanation for bravely breaking through and destroying the enemy in close combat to overcome the enemy's long-range firepower advantage.
Lanchester's square law and linear law can also have special cases. For example, in guerrilla warfare, guerrillas are in the dark, and it is easy to take the initiative to concentrate their forces and annihilate the enemy at close range; The anti-guerrilla side is in the open, so it needs to be divided and passive. In this way, the guerrillas obey the square law, the anti-guerrillas obey the linear law, and the guerrillas have the advantage. Air strikes and counter-air strikes are similar.
Lanchester equation, of course, simplifies and idealizes the battlefield situation, and later generations have greatly expanded it to study more realistic battlefield reality. But Lanchester equation is deterministic in nature, and random factors are not considered. For example, hostile troops of the same size have a greater chance of winning when they fight against the rabble, but accidental factors do not rule out that the rabble wins. The size of this odds is a category of probability and randomness.
Before Lanchester, the German General Staff took the lead in using sand table exercises in the world. Sand table exercise not only builds a model to simulate the battlefield, but also considers the mobility, weather and terrain conditions of troops when deploying troops. At the end of the battle, the roll of the dice determines the outcome of the battle. According to the difference of combat effectiveness, the strong side can throw more times and the weak side can throw less times, but the final result is still random. The scientific truth here is probability and stochastic process. The German army under the command of old Mao Qi relied on this scientific command system, and fought as accurately as a machine in the Franco-Prussian War, defeating the French army that once dominated the European continent. In the Ardennes counterattack at the end of World War II, the Germans lost the initiative and were countered by the US military. Marshal Mo Deer ordered the General Staff of A Army and all frontline officers who have not yet entered the battle to continue the combat simulation at the headquarters, taking the current situation as input data. The later war was really like a battle simulation.