# IntroductionWhen is Lanchester’s law?Lanchester’s linear and square law primary

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

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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