IntroductionWhen

researching the use of mathematics in various recognized approaches, I sought

to incorporate said mathematics to my interest in military history and tactics.

Thus, I derived to my topic which is the use of Lanchester’s Law. The aim of

this project is to study the ways Lanchester’s combat model works as well as use

differential equations to determine the victor in an engagement. The combat

model takes into account the size of an armed force, as well as, the

neutralization potential of the opposing armies. Lanchester’s Law has changed

over time to acclimatize to the dynamic nature of the battlefield. Modern

versions of Lanchester’s Law are currently in use in today’s militaries and is

used to determine the outcome of a battle. Essentially, predicting, to an

extent, which side will achieve victory.

Reason

for interestI

have always had in interest in military tactics and weaponry and try to

incorporate this hobby in other unorthodox subjects. Thus, I came to choose

this IA topic. project aims at studying the basic Lanchester’s combat models

using the knowledge of differential equations and hyperbolic sine-cosine

functions learnt in this course. It further analyses the model for its

application in a more complicated situation. The power of mathematics which can

predict the course as well as the result of a battle was a driving motivation

for this project. What is Lanchester’s law?Lanchester’s

linear and square law primary focus on wars of attrition, in addition, it is

calculated without the constrains of time. This model often includes a constant

to account for reinforcement rates, however, for the sake of simplicity this will

not be taken into account. The linear law focuses on primarily hand to hand

combat where usually the victor is determined by the size of the force rather

than the technological prowess of the force. This is generally expressed by the

equations; To

put this into context let’s say two opposing faction’s x and y are in combat.

The symbol x represents the quantity

of the armies’ forces for x and symbol

y represents the quantity for y. a and b are positive constants which depict the effectiveness of forces y

and x respectively. The larger the

values for a for the equivalent values of y would give a greater rate of loss

for the x forces. Henceforth, y is more potent for larger values of a. Lanchester’s

square Law is the equation that is used to predict the outcome of combat and is

commonly used, along with additional variables to determine the estimated

casualties and potential outcome of a war. This model, however, only works if

initial strength and combat effectiveness of both forces are known and remain

consistent throughout an engagement. Equation #2 is derived from the

integration of the equation below. This is also known as Lanchester’s square

law and will be the primary focus of this investigation. Lanchester’s

Law assumes that the loss rate of a force is directly proportional to an

enemy’s strength. This is what is outlined in Equation #1. The given this

constraint a and b are proportional constants Given this, we can create the

equation: To

more easily comprehend Equation #2 I will rewrite it in square root form.1. 2. 3.

Equation #3

If , x is a real number. In this case, shown

in figure one, when force y is destroyed, x survives. Thus, force x wins the

battle. The conditions of this are listed below.·

If , then force x wins battle·

If ,

then force y wins the battle·

If , then both forces die fighting a war of

attrition and the battle is a draw. To

exemplify this, the constrains for the graph are shown below. Figure 1: Graph of Square Law with a = 0.1,

b = 0.5, x0 = 40, y0 = 80Explanation

of this graph: Here we can see that force x has 40 men and force y has 80 men, one

might think that since force y has 2X

times larger army, they must surely win, right? Not quite, one must take into

account the lethality of each army as well as the force size, in this example,

we can see that army x may have 40

less soldiers, but they also have 5X the military capacity and killing

potential. This means that for every one soldier on team x is taken down, he or she

has defeated 5 of force y’s forces.

Following this trend force x will win

this battle. Meaning

of the graph: This graph can tell us which force will

succeeded in the battle by the way it is structured. For instance, if both

lines intersect, the battle will result in a draw as stated by this equation; . This is shown by Figure #2 below:If the graph concaves up and down, this

means that force y wins the battle. On the other hand, if the graph concaves

left and right, force x wins the battle.

Force

y wins as shown in the left hyperbola, and force

x wins in the right hyperbola. Principal

of ConcentrationIn the earliest forms of

combat, warfare was fundamentally a system of one to one duels. In turn, the potential

lethality relation during a given period of time did not depend on the warrior’s

force level. However, taking into concern modern conditions, effectiveness of a

firearm is more widely separated in firing location can be focused on objective

targets so that each armies’ loss rate is proportional to the number of enemy

soldiers. If one force can concentrate its total strength on a certain sector

or position of an opposing army and neutralize the threat, the principal of

concentration then states that the army may be able to attack the remaining

portion of the opposing army with its remaining forces still standing and

destroy it. Lanchester supposed that that air power would be the dominant force

because it can neutralize large amounts of belligerents in a short period of

time. An air support force can be concentrated in full force against select areas

of an opposing force. This principle also somewhat contradicts itself with the

idea of “divide and conquer.” Setting

up the problem1. In

the initial engagement army has a greater number of troops compared to

army , this can be written like so: 50,000 || a = 0.1 || b = 0.12.

Equation #3

The following equation in Lanchester’s law is used to

find the units of time and discover how long it takes for a force to be

annihilated;3. Plug

in the numbers into the variables =0Which

simplifies down to4. Solving

this gives us x(t) = 6.935. Now plugging

the units of time back into the equation we get (0.693)) This

gives the answer y = 40,000Thus,

there are 40,000 remaining soldiers on force y, which won the fight, but force x has 30,000 remaining. 6. 40,000 || a = 0.1 || b = 0.17. Even

without using the formula, we can see that force y will win. The battle if

both armies have the same lethality potential. What

does this mean?This means, that the Principal

of Concentration is superior to Divide and Conquer in this instance. In the real world there are many other factors

such as the element of surprise as well as technological advantages that

dictate the outcome. But in this instance, if an army were to separate and take

on an equality larger army they will lose because they lack the number of

troops. Homogenizing the forces resulted in a much better turn out for force y. ConclusionThrough the use of Lanchester

square law and Calculus, we have derived the