[Physics] A partial differential equation for kinetic energy

classical-mechanicsenergy

The kinetic energy of a point particle of mass $m$ and speed $v$ is $K = \frac{1}{2}mv^2$. An elementary mathematics textbook I saw asked one to show that

$$ \frac{\partial K}{\partial m}\frac{\partial^2 K}{\partial v^2} = K.$$

While this is a straightforward exercise in partial differentiation, is there supposed to be any physical meaning behind this formula? For example, is there a significance to quantities that satisfy this nonlinear PDE?

Best Answer

I can't see where this has any utility at all. The point of having any equation, differential or algebraic, is to put constraints on a system. We then solve these equations to obtain an unknown subject to the constraint. However in this case, we already know the answer and the equation gives us no new constraints and hence no new info information.

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