No-mentum, or Mathematical Proof That Momentum Can Not Be Discerned In Real Time

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Sports writing and sports analysis is full of real life metaphors, as of course it should be -- metaphor is the basis of language and indeed the basis of communication, so it's no wonder it finds its way into sport. The application of the momentum concept from physics is an old standby. A body in motion, and all that, except for teams with hot goalies or aces on hot streaks or locked-in quarterbacks and so on. You may not even notice that you're hearing it, at this point.

Like most old standbys in sports, the momentum concept is currently absorbing some withering broadsides from the analytical community. Bill Barnwell savaged the concept at Grantland, and Keith Law sure seems to enjoy tearing into the concept on Twitter.

I'm not sure I agree with their zeal -- I think momentum in sports does exist. However, as an analytical concept, I think it's useless, primarily because it cannot be discerned in real time. Does it have use in a descriptive story of a game? Perhaps. But if momentum is an analog of its physics counterpart, it cannot be used to determine winners and losers in real time as so many think.

Disagree? I can prove it mathematically.

THEOREM: The analog of the physics concept "momentum" in sports cannot be determined in real time.

PROOF: Consider, the physics equation for momentum.

p = m*v

In which p is momentum, m is mass, and v is velocity.

Let's focus on the "v" term, velocity. How would we represent that in sports? Let's turn our attention here to a win probability graph. As an example, observe a game with what appears to be a palpable amount of momentum: St. Louis' comeback victory over Washington in Game 5 of last year's NLDS.

Now, let's recall what "velocity" is, in scientific terms: distance over time, or v(t) = d/t. That is, velocity is the rate of change of position over time. In our sports analog, position (hereby denoted by "s") is current win expectancy. Finding win expectancy is easy: hop over to FanGraphs and check out their nifty chart. Here's the one for our game in question:

The resulting formula for "position," therefore, becomes

s(t) = WE(t)

Where t is an integer, and where WE(t) is the Win Expectancy at a certain plate appearance t. To find a rate of change of position, we take the derivative of s(t), denoted s'(t), and s'(t) = v(t).

As complicated as it sounds, we don't need to get into the calculus to show what this concept means. The derivative is, in simplest terms, the slope of the line at any point on the graph. For example, the Cardinals momentum truly started to build when David Freese walked to load the bases with two outs in the ninth, a second straight walk issued by Drew Storen. The momentum is then found by drawing a line tangent (i.e. touching at one point) to the Win Expectancy graph, roughly like so*:

*Technically, a smoothing function would be needed to do this, but you see the general point.

According to my (admittedly rough) calculation, momentum was heading in the Cardinals' dimension to the tune of 25 percent win expectancy at the time of Freese's walk.

But what about this scenario, in last Wednesday's Orioles-Blue Jays game? Munenori Kawasaki singled to bring the tying run to the plate in the eighth inning, and it looked like the Jays had momentum. But then the Jays made three straight outs, the threat was over, and Baltimore won 9-5. The momentum at the time of Kawasaki's single? Zero.

In calculus, whenever a function is at a local maximum or (as in this case) a local minimum, the derivative is zero. Similarly, whenever an object reverses course, its velocity at the switching point is zero, and that's what we're seeing here.

So, back to the Cardinals-Nationals game. We've already seen a way to calculate momentum, but we're doing it after the fact. Can we do it in real time? Consider what the Cardinals-Nationals win expectancy graph looked like after Freese's walk:

What's the derivative here? It's undefined -- it does not exist and can not be taken. The formal definition of the derivative is as follows:

But if we attempt to take s'(t) where t is the plate appearance in which David Freese walks, s(t+h) is undefined at all points where h > 0, because those points haven't happened yet.

Apply this to the general case and the result is velocity (or the sports analog of velocity, required to get momentum) can not be defined at any point in the game t when t is the final known point in the game, because the derivative s'(t) = v(t) is undefined. QED.

Is this pedantic? Yes, it's pedantic as all hell. But the idea of momentum as an analytical tool in real time is similarly ridiculous. We can certainly go back to points and say "Here, this is where the momentum shifted." In fact, the points where m = 0, like following Kawasaki's single, are exactly those: the turning points in the game, where momentum must have shifted.

But sports would be way too simple if the momentum metaphor held as generally used -- things are always more complicated, and something like momentum should never be held up as anything more than a rule of thumb. To do so misses out on what makes sports so amazing -- its constant churn of barely restrained chaos, its utter unpredictability, and its ability to create great stories from situations unimaginable. We don't need an illusion of momentum to tell those stories; they're more than good enough on their own.

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