, that would be your financial path through the month that maximizes your Average [“Bank balance” minus “Expenses”]. Any other path, (I show three of infinite possible ones) where you pay all or part of the credit card bill earlier than the due date, would give a lower value of the Average [“Bank balance” minus “Expenses”], meaning lower interest income and lesser liquidity.
, except everything is upside down, so all the values are negative instead of positive, because we are now subtracting “Bank balance” from “Expenses” instead of the other way around as in the previous plot. The paths are also the same as before, but upside down. Path 4 remains the best path, but now (being negative) it has the minimum average = –$2,700.
. But if you compare the two illustrations, you will see that all the different paths remain the same, except that they are all upside down in the second figure, and therefore the best financial path now corresponds to the lowest (or most negative) value of the Average [“Expenses” minus “Bank balance”]. It is important to realize that this is simply a different way of saying the same thing (just as “increasing your wealth” is the same as “decreasing your poverty”); therefore, it does not alter the choice of the best financial path, except that it now corresponds to minimizing what we will now give this specific name: Financial Action = Average [“Expenses” minus “Bank balance”].
Thus, the optimal financial trajectory for most of us involves minimizing this financial action. But with credit cards and loans, we have found ways to bypass this rule, which is as fundamental in finance as its analog is in nature (as we see next), and if we are too reckless in bypassing this rule, in the long run it can get us into serious trouble—such as foreclosures or bankruptcies!
Now to understand how this “action minimization” principle works out in nature, we need to identify what is nature’s equivalent of money. That’s simple—it’s energy! Just as good financial practice suggests that we should be thrifty with our money, nature is thrifty in its usage of energy, and it so happens that this energy-thriftiness of nature determines all the dynamical laws of the universe as regards to how things move and change in time.
Figure 4.3 In position 1, the ball has the maximum potential energy due to its height, and it expends that into kinetic energy as it rolls down the slope.
. If all conditions remain exactly the same, we can be certain that every time a ball is fired off, it will follow the same trajectory (the solid dark line) from the muzzle at point A to its landing at point B. That certainty is the basis for the entire field of ballistics and for projectile-based weapons in warfare. Yet there are infinite possible paths connecting those two points, A and B (some of which are shown with dotted lines). So why does the cannonball always choose that one particular path? What is so special about it? It is easy to answer that now: along that path the “action” has the minimum value! Every other path would lead to a higher value of the action as defined above, just as was the case with our financial example. So this is how the principle of least action plays out: At each point along the trajectory, take the kinetic energy (expenses) and potential energy (bank-balance), find their difference, and average that over all the points on the path; the true path is the path that gives the smallest value for that average.
. Yet even without delving into that next level of “Why?” we already have something extremely powerful in the principle of action minimization because it gives us a simple guiding principle for finding the correct dynamical laws about how things move and evolve in time in the universe—physicists use it all the time; really, we would be lost without it.
Nature’s demands for minimum action translate to something we can all relate to in our “natural” predispositions: Reduce the kinetic energy (cut back on necessary motion) and increase potential energy (store up energy) as much as possible over time, because just like in nature, we humans have a natural tendency to minimize action all the time. In fact, we absolutely love minimizing action, like sitting on a couch with a bag of chips—our kinetic energy is as small as possible (we are not moving!), and the bag of chips we are munching on is increasing our potential energy in the form of stored, unused fat in the body. The other denizens of this world take this action minimization principle even more seriously; most animals don’t move around much unless they have to (a lion sleeps and rests most of the day), and when they move at all, it is mostly because they are compelled to increase their potential energy in the form of food. So couch potatoes among us can’t be held at fault, they are just following one of the most fundamental laws of the universe: action needs to be minimized! Lying on the couch with beer and snacks, remote in hand, leads to a minimum action on our part, and that is exactly what the universe demands.
when we talked about the second law of thermodynamics), the system will eventually settle into the minimum possible energy. On the other hand, if we allow some of the energy to be transformed into heat, it will still try to cut down on all its potential energy to maximize the loss into heat. In either case, the upshot is that nature usually gravitates to states of lower energy.
Minimize energy, minimize action! Ah, what a justification for laziness! We humans are lazy by nature, and it turns out that so is nature. Most of us probably would not be doing much with our time and our lives without some external incentive and a fear of the consequences of not doing anything. We would just hang out in our ground state of sleeping and lounging around watching television or hanging out with friends in some pleasant surroundings. We even have a name for the ground state—we call it “vacation,” and that is what we look forward to for the rest of the year. If we had a choice, life would be just one endless vacation—a lifetime in the ground state.
. But for now, just from the point of view of minimizing energy, it makes absolute sense why we thirst for a good stable relationship or a marriage with abundant mutual attraction, where two people just get along and never waste an ounce of energy in arguing, fighting, or domestic violence. No one likes to remain single forever, because single people have to invest a lot of energy and effort in trying to find dates and someone to be with, someone to hang out with—that is too much action and too much energy wasted. So people get married and try to settle into stable relationships to minimize action and energy. And if we were to go by the confidential and resigned testimony of many married men, it seems marriage significantly minimizes the “action” in the bedroom as well!
Believe it or not, it so turns out that the universe is not only lazy but also seems to be in a tremendous hurry, conveniently providing a natural justification for the endemic impatience of our species. It is not a joke—there really is a principle of least time. Among other things, it explains why light bends when moving from one transparent medium (like air) into another transparent medium (like water). Refraction, as it is called, is the reason why things under water always appear to be closer to the surface than they really are; ask anyone who’s gone spear fishing. It is also why a drinking straw in a glass of water looks like it is bent at the surface. It happens because light follows the path that takes the least time, and since light travels slower in water than in air, it turns out that the shortest time of travel between two points—from under water, where the object is, to above water, where the eyes are—is not along the shortest path distance-wise, which would be the straight line joining the points. Instead, it is along an appropriately bent path, as shown in . An analogy will help. You are hunched over a map trying to figure out your route for a road trip. If you are trying to minimize your travel time, you would certainly take the beltway or the bypass around a large city rather than taking the direct shorter route straight through the city because you know it is not just the distance, it is also about how fast you are moving—so the most direct and straight path is not always the path of least time. These days, GPS and online maps give us the choice of what we want to minimize in our trip—time of travel, distance, or tolls—and they might all correspond to very different routes. In the case of propagation of light, the choice is to minimize the time of travel. Just like in a road trip, the path that minimizes the time of travel of light—from a point within water to a point outside—is not the straight-line path of shortest distance. That actual (bent) path can be found mathematically by considering the different velocities of light in air and water.
Strictly speaking, the action is “optimum,” allowing certain other possibilities than an absolute minimum, but that is a technical detail we will leave for physics textbooks.
Some folks are very physically active during their vacations—well, even in nature there are occasionally strange scenarios where action is maximized rather than minimized.