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Is there any general significance to calculated quantities that depend purely on general powers of the velocity of a particle/system/etc? The first power being momentum and the second being kinetic energy.

I know that in relativistic mechanics the momentum and energy become quantities that must actually be expressed in infinite orders of velocity since energy and momentum are functions that can be expressed as power series of velocity. So, if there is any significance to the momentum and energy being to first and second order in Newtonian mechanics respectively, why do they go to functions of infinite order when special relativistic effects become a concern.

Or am I making connections that lead nowhere?

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You are probably thinking about the moments of a velocity distribution function.

Suppose you had some probability distribution that was a function of space, time, and velocity or $f$ = $f\left( \mathbf{x}, \mathbf{v}, t \right)$, where $f$ has the units of # per unit volume per cubic velocity. From this we can define things like the number density: $$ n\left( \mathbf{x}, t \right) = \int_{V} \ d^{3}v \ f\left( \mathbf{x}, \mathbf{v}, t \right) $$ which also happens to be called the zeroth moment. The first moment, or average/bulk velocity (related to the mean) is defined as: $$ \mathbf{U}\left( \mathbf{x}, t \right) = \int_{V} \ d^{3}v \ \mathbf{v} \ f\left( \mathbf{x}, \mathbf{v}, t \right) $$ and then the pressure (related to the variance) of a system is the second moment, the heat flux can be derived from the third moment and so on… Each higher level moment is multiplied by increasing values of $\mathbf{v}$. Sometimes these are written in a slightly different form, which look like expectation values. In the link above, for instance, they chose the following notation for the $n^{th}$-order one-dimensional moment: $$ \mu_{n} \equiv \langle \left( x - \langle x \rangle \right)^{n} \rangle \\ = \int dx \ P(x) \ \left( x - \langle x \rangle \right)^{n} $$ where $P(x)$ is [analogous to $f\left( \mathbf{x}, \mathbf{v}, t \right)$] and $\langle x \rangle$ is the average of $x$ [analogous to $\mathbf{U}\left( \mathbf{x}, t \right)$].

So yes, there is a physical significance to the order of velocity in certain cases.

Side Note: If you look up models of aerodynamic drag forces, you will probably find references to geometric series of compounding higher order velocities. For instance, in very viscous fluids, the drag is dominated by the first order velocity. In low viscosity fluids (e.g., Earth's atmosphere), the second order term doesn't become important until high speeds are reached (think of ram pressure). At high speeds, the higher order moments can become dominant, which is one of the reasons why tornados can do so much damage.

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