# Lanchester Law

The law of Lanchester (Lanchester's Law) is a deterministic model from the military tactics of the losses in combat situations under certain conditions. It was formulated by Frederick W. Lanchester during the First World War .

## Simplest form

In its simplest form, the conditions under which the law applies include:

• Two opposing armies face each other with a known number of units.
• Both armies begin firing at the same time.
• Every unit has a defined and constant firepower , this firepower is the same on both sides in the simplest case of the Lanchester Law.

These conditions can be approximated , for example, in an artillery battle.

Under these conditions, the outcome of the battle is understandably dependent solely on the number of units involved. In such a way that the difference in squares between the opposing units remains the same until all units on one side are eliminated.

Example: Army Green (G) has 3 units (square = 9), Army Red (R) has 5 units (Q = 25, the difference is 16) then Army Green will be eliminated after the fight, while Army Red still has has 4 units.

From and follows after multiplying both equations and thus ${\ displaystyle {\ frac {dR} {dt}} = - G}$${\ displaystyle {\ frac {dG} {dt}} = - R}$${\ displaystyle \ int RdR = \ int GdG}$

${\ displaystyle R ^ {2} -G ^ {2} = \ mathrm {constant}}$

## Taking into account the firepower

The model described above can be extended by taking into account the firepower of the units on both sides (g for the firepower of the green units, r for that of the red units):

           ${\displaystyle rR^{2}-gG^{2}=\mathrm {konstant} }$


According to this, the firepower only has a linear influence and is hardly significant compared to the absolute number of opposing units. Firepower only becomes a decisive factor when the number of units is approximately the same.

In Lancester's Law, the old strategic wisdom shows impressively that the bundling of forces - or, in contrast, the division of the enemy - is decisive for the success of a battle.

## Practical applicability

### In military tactics

In real battles, the conditions of the Lanchester Act in a broad context, for example for the description of a battle, are inadequate, since here the units involved differ greatly in the type of their armament. Furthermore, the units are mobile and units are added / withdrawn. However, Lanchester's Law is applicable if one delimits the consideration of real battles in terms of time and space. The simulations used as an alternative to Lanchester's Law, on the other hand, are usually very complex and hardly comprehensible. In this respect, the law formulated by Lanchester impresses with its simplicity.

### Use in the non-military area

Lanchester's Law is also used in biology, such as immunology. There are also comparable conditions in economic competition.

## literature

• Frederick W. Lanchester: Aircraft in Warfare - The Dawn of the Fourth Arm . Constable and Company Limited, London 1916 online from Internet Archive (English)