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Lanchester's Square Law states that given two armies, $x$ and $y$, with the army units' relative effectiveness $\alpha$ and $\beta$, respectively, this can be written as two differential equations for the sizes of the armies as a function of time: $$\dot{x}=-\beta y,$$ $$\dot{y}=-\alpha x.$$

My question is as follows: Are $\alpha$ and $\beta$ independent of time and constant throughout the battle such that: $$\alpha x^2-\beta y^2= c$$ If so, how can this be shown?

user975
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1 Answers1

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That $\alpha$ and $\beta$ are constants is an assumption, not a conclusion.

From $$\frac{dx}{dt} = - \beta y$$ and $$\frac{dy}{dt} = - \alpha x$$ we have, by the chain rule, $$\frac{dx}{dy} = \frac{\beta}{\alpha} \frac{y}{x}$$ so $$\alpha x \; dx = \beta y \;dy$$ Integrating, $$\frac{1}{2} \alpha x^2 = \frac{1}{2} \beta y^2 + C_1$$ so $$\alpha x^2 - \beta y^2 = C_2$$

awkward
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