Yesterday, as expected, severe underdog Texas A&M lost to #3 ranked Texas. Not as expected, the game was competitive right to the end, with an upset being threatened down to the final minutes. It was a great game to watch, with a final score of 45-35 – the highest total score in the history of the rivalry.
In this game as with pretty much all games at the college and pro level, the offense is very reliant on the forward pass. The quarterback picks a wide receiver, throws the ball, and with some luck the ball is caught for a gain. Now pretend you’re filming such a pass on a video camera. From the time the football is let go to the time it’s caught, it describes a trajectory that we can describe with some simple physics. We’ll ignore air resistance, which is not such a bad approximation in football.
Now as you’re playing the video of the pass, pause at some point on the trajectory. The ball appears frozen in the air. As you observe that instant in its flight, you know that the ball had a particular kinetic and potential energy. During the next frame the kinetic and potential energy will probably be slightly different as the ball changes position sightly. By carefully tracking the ball’s position in each frame you could reconstruct the kinetic and potential energy of the ball in each frame of video. The kinetic energy is just the velocity squared times its mass mass divided by 2. The potential energy is just the height of the ball times its mass times the acceleration due to gravity.
Do this for each frame. Now each frame – in other words, each moment in time as the ball flies to its destination – is labeled with the kinetic and potential energy of the ball at that instant.
Now for the Lagrangian. For each frame, take the number for kinetic energy and subtract off the number for potential energy. That quantity is called the Lagrangian, named for Joseph Louis Lagrange. Why is this a useful thing to know? Weird as this Lagrangian sounds, it’s actually part of a very powerful way to formulate classical mechanics. We do so in his day: take the value of the Lagrangian for each frame and multiply it by the duration of the frame (usually 1/30 of a second). Add all those up, and you’ll end up with a final number called the classical action.
And here is the magic part. You can think of an infinite number of strange and bizarre curved or looped or whatever trajectories that the football might have taken other than the one it did. You can imagine calculating the action for any of those imaginary trajectories. But here is what you will find. The action that results from the actual motion is the smallest possible value for any trajectory. Any other path, with any other weird changes in speed or elevation, will produce a higher value of the action.
This statement is frequently called the principle of least action, and it’s also known as Hamilton’s principle (after the Irish physicist William Rowan Hamilton). If you want to know about a trajectory, it’s possible to use a technique in the calculus of variations to find the trajectory from that principle. Since the action is purely a scalar quantity involving energy, we don’t have to worry about the frequently difficult vector character of Newton’s force-based method of doing mechanics. It’s a neat little bit of physics sorcery.
[Note for the technically inclined: Formally we’re looking for a stationary action, not truly the least action. Hamilton’s principle itself doesn’t preclude the stationary point being a maximum, though in practice this is not a problem.]