Here are three graphs of putting strokes. The s axis is "speed" and the "t" axis is time.
We'll take a look at each of these in a moment, but consider first how putting can behave like a pendulum.
In virtually all good putting strokes, the ball is hit with a slight positive angle of attack (AoA) - about 2-3° or so. This positive AoA helps minimize backspin, produce no spin, or even to produce a tiny bit of forward spin if the dynamic loft is 1-2°. But the point is: the ball is struck while the putter head is ascending, or after low point.
If you were to swing a pendulum back and through, maximum speed would be where? At the bottom. At low point. At every point after that, the speed would be lower. Even one tenth of one degree after low point, the pendulum is slowing down (negative acceleration, or deceleration).
The best putters almost all tend to have a decelerating putter head at or even slightly before impact. Their putting stroke resembles a pendulum, reaching maximum speed at or slightly before impact.
Consider also the length of a pendulum's swing. A theoretical pendulum (no loss of energy to friction) swings as far past center in one direction as it does in the other direction. Whether you measure it in degrees or a linear measurement, the pendulum swings 22.7° left and 22.7° right, or 13.1 inches left and 13.1 inches right.
The best putters almost all tend to have similar length backswings and through-swings in their putting strokes. Their putting strokes continue to resemble a pendulum in this sense.
Now let's take a look at each of these putting strokes.
Here's a putting stroke typical of a golfer who has a terrible time controlling their distances. This golfer may have a great sense of touch from 5-10 feet, maybe even out to 15', but when you ask them to hit a 30' putt, you start to see issues. They'll hit one 27', the next 34', the one after that 25', and then maybe 33'.
These golfers often make a backswing that's - let's just say - eight inches for a six-foot putt, nine inches for a 12-foot putt, and ten inches for a 30-foot putt. They're almost the same length. Then they have to accelerate their putters various amounts to reach various speeds at impact to send the ball various distances.
If you wanted to make a pendulum swing faster at the bottom of the arc, given the same pendulum length and weight (we aren't changing putters or our setup appreciably), how would you accomplish this?
Why… you'd simply pull the pendulum back farther before letting it go.
So look at the speed and time plot of the poor putter above. I've marked the instantaneous speed at two points: just prior to impact and just after impact. Note that impact - even on a putting stroke - severely slows the putter head down. I've exaggerated it quite a bit in these graphs, but that's something I can do given that I haven't added any scale to these charts. :D It simply makes things clearer to see and thus easier to grasp.
At any rate, note that the direction of each of the arrows - both the dashed (pre-impact) and dotted (post-impact) lines is pointing upwards. This means the putter head has positive acceleration. It's speeding up. Note the pronounced "hump" after impact. Though the ball slows the putter head down temporarily, it's still speeding up, so you see a second peak speed after impact.
This golfer is roughly 99% likely to have poor distance control.
Let's look at the good and great putting dynamics (and by good I mean pretty darn good, because as you'll note the differences between these two are subtle):
Note how in Good the putting stroke reaches maximum speed at the ball. The proof of this is that the acceleration is neither positive nor negative - the arrow is pointing horizontally, indicating that the speed is neither going up nor down. Constant speed is no acceleration (positive or negative). Notice that this condition continues immediately after impact, and the putter head continues to slow down thereafter.
In the Great image, the putter head is actually slowing down slightly at impact (the arrow points downward). Then you see the BIG deceleration caused by the putter impacting the ball, and then the deceleration continues from there.
Contrast those with what we often see from the golfers with the absolute worst distance control:
This golfer actually manages to reach peak/maximum speed after the ball has left the putter. Note that his acceleration curve going into impact actually steepens - he is accelerating more at impact than at any other point in the downstroke. Then he accelerates MORE until he rapidly decelerates, well after impact, to bring the putter to a halt.
This is more common than you might think. Golfers have been told for decades to "accelerate through the ball" and to "putt authoritatively" and so on. This advice ranks near the top of my list for counter-productive, harmful advice.
By and large, the poorest putters accelerate far too much for far too long (including up to and after impact), while the best putters have roughly matching backstrokes and through-strokes that deliver the putter head to the ball while it is either not accelerating at all or is negatively accelerating (i.e. decelerating, or slowing down).
If you feel you may be "accelerating" your putter into impact, put three coins on the ground, equally spaced from each other, in a line. Put the ball near the middle one, and practice making backstrokes that go to one and finish at the other. Try to feel that you're not adding anything to the downstroke or follow-through: you're not accelerating the putter much (just let gravity do it - in reality your muscles will contribute, but it's uncommon to feel much muscle contribution) and you're not forcing yourself to "brake" the putter too much at the end, either. Just make a natural, smooth stroke that matches - coin to coin.
To change how far you hit the ball, move the coins farther apart or closer together, keeping the distances the same.
It's that simple.
P.S. Note that I've made no attempt to show the scale of t and s. Specifically, I've fudged things a bit by implying that the the t is the same for all of these strokes, and that impact occurs at the same moment. This is very unlikely to be true: if you make a short backstroke and accelerate all the way up to and even after impact, you're likely to have a shorter (time) downswing and to reach impact sooner. They line up because I wanted to keep things simple, and because timing isn't really the topic here.
P.P.S. A really old example of a SAM PuttLab read-out can be seen here.
P.P.P.S. (2014-08-13) A great series of pictures and a simple explanation of the "why" is found in post #179: http://thesandtrap.com/t/74295/putting-do-not-accelerate-through-the-ball/162#post_1040098 .