When least is best

A little less conversation  .  .  .


 “Action” is a mathematical quantity which depends upon the mass, velocity and distance travelled by a particle.  Action is also associated with the way energy is carried from one place to another by a wave, but it can be understood most simply by imagining the trajectory of a ball tossed in a high arc from one person to another.

     One of the most fundamental laws of science is the law of conservation of energy.  Energy cannot be created or destroyed, only converted from one form to another.  The ball leaves the thrower’s hand with a large kinetic energy, but as it climbs higher its speed slows down and the kinetic energy is reduced.  But because the ball is higher above the ground (strictly speaking, because it is further from the centre of the Earth), it has gained gravitational potential energy.  Leaving aside friction (which converts some of the energy of motion of the ball into heat energy as it passes through the air), the amount of gravitational energy it gains matches the amount of kinetic energy it has lost, for each point in its climb.  At the top of its trajectory, the ball momentarily stops moving, so it has zero kinetic energy, but maximum gravitational energy for this particular trajectory.  Then, as it falls towards the catcher it gains kinetic energy at the expense of gravitational potential


     At any point along the trajectory, it is possible to calculate the kinetic energy and the potential energy of the ball.  The total you get by adding the two is always the same.  But if you subtract the potential energy from the kinetic energy, you get a different value of the difference at different points along the trajectory.  If you add up this difference all along the trajectory, integrating the difference between the kinetic energy and the potential energy for the entire flight of the ball, the number you come up with is the action that corresponds to the flight of the ball.  The action is not a property of a single point along the trajectory, but of the entire trajectory.

     There is a value of the action for each possible trajectory of the ball.  In a similar way, there is a value of the action corresponding to each trajectory that might be taken by, say, an electron moving in a magnetic field.  The way we have described it here, you would calculate the action using Newton’s laws of motion to describe the flight of the ball; but the process can be turned on its head, with the properties of the action used to determine the laws of motion.  This works both for classical mechanics and for quantum mechanics, making the action one of the most important concepts in all of physics.

     This is because objects following trajectories always follow the path of least action, in a way analogous to the way water runs downhill to the point of lowest energy available to it.  There are many different curves the ball could follow to get to the same end point, ranging from low, flat trajectories to highly curved flight paths in which it goes far above the destination before dropping on to it. Each curve is a parabola, one of the family of trajectories possible for a ball moving under the influence of the Earth’s gravity.  But if you know how long the flight of the ball takes, from the moment it leaves the thrower’s hand to the moment it reaches its destination, that rules out all but one of the trajectories, specifying a unique path for the ball.

     Given the time taken for the journey, the trajectory followed by the ball is always the one for which the difference, kinetic energy minus potential energy, added up all along the trajectory, is the least.  This is the principle of least action, a property involving the whole path of the object.

     Looking at the curved line on a blackboard representing the flight of the ball, you might think, for example, that you could make it take the same time for the journey by throwing it slightly more slowly, in a flatter arc, more nearly a straight line; or by throwing it faster along a longer trajectory, looping higher above the ground.  But nature doesn’t work that way.  There is only one possible path between two points for a given amount of time taken for the flight.  Nature “chooses” the path with the least action — and this applies not just to the flight of a ball, but to any kind of trajectory, at any scale.

     It’s worth giving another example of the principle at work, this time in the guise of the principle of “least time”, because it is so important to science in general and to quantum physics in particular.  This variation on the theme involves light.  It happens that light travels slightly faster through air than it does through glass.  Either in air or glass, light travels in straight lines — an example of the principle of least time, because, since a straight line is the shortest distance between two points, that is the quickest way to get from A to B.  But what if the journey from A to B starts out in air, and ends up inside a glass block?  If the light still travelled in a single straight line, it would spend a relatively small amount of time moving swiftly through air, then a relatively long time moving slowly through glass.  It turns out that there is a unique path which enables the light to take the least time on its journey, which involves travelling in a certain straight line up to the edge of the glass, then turning and traveling in a different straight line to its destination.  The light seems to “know” where it is going, apply the principle of least action, and “choose” the optimum path for its journey.

     In some ways, this is reminiscent of the way a quantum entity seems to “know” about both holes in the famous double slit experiment even though common sense says that it only goes through one hole; but remember that the principle of least action applies in the everyday world as well as in the quantum world.  Richard Feynman used this to develop a version of mechanics, based on the principle of least action, which describes both classical and quantum mechanics in one package.


WARNING!  Unfortunately, physicists also use the word “action” in a quite different way, as shorthand for the term “interaction”.  This has nothing to do with the action

described here.



For more, see our book Richard Feynman: A life in science.



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