## Tuesday, May 17, 2016

### 150th Motor Rifle Division And Much Much More Military Power Related-III

So, the question is: do Osipov-Lanchester Equations work in real life? Not quite. This, inevitably, creates another question: why bother then with studying these equations? The answer is simple--to understand highly non-linear dynamics of a war and to get both tactical and operational (and, in the end, strategic) sense of opposing forces' ratios which become an absolute must when planning and fighting a battle. These equations give us an entrance into the world of both quantitative and qualitative considerations for opposing forces, which is later described by Salvo Model, which can sometimes get fairly complicated, not too bad, though.

So, let's conduct some mental experiment--let's model a simplest battle from the point of view of non-military and non-mathematical average Joe, or Ivan, or Hans or what have you. Say we have two opposing forces A and B. Both forces are an exact match in terms of their weapons and skills, except for their numbers. Let's assume that force A has 1000 riflemen while force B has 750. These forces begin to shoot at each other and the intuitive, and very civilian, conclusion would be that by the time these forces A and B stop shooting at each-other, force A will have 1000-750=250 riflemen left after completely annihilating force B. After all, force A is simply larger (more numerous) than force B. Well, this is very wrong. This is not how it will happen. A way more realistic calculation will be done with the system of differential equations which you already saw in the previous post and we will simplify them even more by getting rid off those fancy Alpha and Betta which are merely numerical coefficients we will talk about later:

As you can see, we simplified these equations to merely reflect the changes in the numbers of both sides over time. Term dt merely stands for change in time: say our imaginary hostile forces started to shoot at 13:00 and stopped at 14:00, so dt=14:00-13:00=1 hour. dA and dB merely mean change (in our case this will be decrease in numbers) in respective forces. Thus, dA/dt and dB/dt are merely changes in respective forces over time or rates of losses. Now comes this tricky (not really, but still) moment--how are we going to calculate what's left of more numerical force A after it annihilates force B, and here our civilian non-mathematical intuition does not fail us--under all other conditions being equal (memorize this statement really well!!!), force A will eliminate force B, because it has numerical superiority. Here is how--all this is basic math: let us bring both equations in system to a single "floor":

As you may have already noticed, we are solving these equations in general, without plugging in our numbers yet. We also understand our initial conditions (and constrains) that once B is annihilated A stops sustaining casualties. So, let us simplify these equations even more, for people who are scared of math, by rewriting what these fancy dA and dB really are. They are nothing more than the difference between the numbers of respective forces before (start) and after (end) the battle, in our particular problem. In reality, we are the ones who choose the time period in which to see what those differences are. Thus:

By now some of you may have guessed already that our dB will be 750 (or -750, depending on how you view your equations), which means only one thing--everyone in force B will be either killed or taken out of action. As you can see, I am being very deliberate and tedious in explaining these transformations and this is for only one reason--we have to integrate both sides of both equations. This is how simplest differential equations with separable variables are solved. Some of you probably will recall now what anti-derivative is. So, let me make things somewhat easier here. We already know from the bottom equation in the system that in the end B force will reduce (or, rather, will be reduced by A) itself from 750 riflemen to 0. But laws of mathematics do not allow us to simply write this number in the left side of this equation, it still has to look like this. I deliberately eliminated any mention of limits of integration to make it look simpler:

I know, I know, believe me I am trying desperately to make this as simple as possible for those who are intimidated a bit by all this math mambo-jumbo. But since we are dealing here with integrals and some abstract variables we have to remember that integration is a finding of anti-derivative and anti-derivative for a simple variable X (or A, or B or whatever letter you want to use) is always:

So, the solution to our system of equations will look like this and we will call it Equation - I :

But, since we are back into simple math in Equation-I you may easily see that after all simplifications our equation becomes this, Equation-II:

This equation is in the foundation of what became known as Quadratic Law and we are about to demonstrate it. So, let's start plugging in our available data into this equation. We know that our B start=750, we also know that our B end=0, our A start is 1000 and A end is unknown and we will call it X. Look now at what our equation has become--yes, simplest quadratic equation:

From here you can easily establish that the value of X, that is the number of remaining riflemen in the force A after they annihilate whole force B will be a square root of 1,000,000-562,500=437,500 which is approximately 661 riflemen. That is 2.5 times more than linear approach would suggest. I hope those lively colors in equations helped all of you to keep focus on what matters here. So, the implications of this simple quadratic law are immense. In fact,  all those Alphas and Bettas which are, in this particular case, coefficients of combat efficiency enter the fray. You may get a somewhat expanded form of these equations which introduce combat efficiency here

Consider this simple problem: we know that combat efficiency of the machine gunner equals combat efficiency of 36 riflemen. How many machine gunners will we need to completely substitute 1000 riflemen. No, it is not 1000 divided by 36, it is 1000 divided by the square root of 36 which is 6. 1000/6 gives us about 167 machine gunners. That means that combat strength of a fighting force is calculated by multiplication of combat efficiency of a single unit (rifleman, squad, platoon etc.) by the square of numerical strength. In layman's lingo it means one very important thing: the more numbers you have (let alone when you have numbers more effective than that of your enemy), the more disproportionate will be the distribution of losses in your favor. Indeed, recalculate this same problem but now 2,000 against 750. You will lose roughly 146 of your riflemen, that is 1854 of your troops will survive the battle. These simple calculations lead us to a very fundamental conclusion which is one of the main principles of war:

PRINCIPLE OF CONCENTRATION OF FORCE

This principle also has a very famous application in everyday life in famous formula from physics: P=F/A. Pressure equals applied force divided by area. The larger is a force and the smaller is an area--the larger is a pressure. This is what went into the foundation of the Blitzkrieg and its famous schwerpunkt (focal point) principle, where a massive force, locally more numerous than that of the enemy, was concentrated on a small segment of the front thus applying and immense pressure on the enemy defense, eventually breaking the front and going on the exploitation which was a death knell to a broken enemy. For anyone who ever even remotely dealt with the issues of military technology, combat training or combat, let alone studied military history not from propaganda outlets, the arguments which Shlykov used in his article do not sound convincing at all. Basic operations research can give some fairly accurate answers.

To Be Continued......