Every college student taking that requisite physics class sees Newton’s second law. I saw it once even in a textbook for a martial art: Force equals mass times acceleration… the faster you go, the harder you hit! At least, that’s what they were saying, never mind that the usage wasn’t accurate. F=ma is one of those crazy simple equations that is so bite-sized that all of popular culture is able to comprehend it. Kind of.
Newton’s second law is, of course, one of three fundamental laws. You may even already know all of Newton’s laws without realizing that you do. The first law is “An object in motion remains in motion while an object at rest remains at rest,” which is really actually just a specialization of Newton’s second law where F = 0. Newton’s third law is the ever famous “For every action there is an equal and opposite reaction.” The three laws together are pretty much everything you need to get started on physics.
Much is made of Newton’s Laws in engineering. Mostly, you can comprehend how almost everything in the world around you operates based on a first approximation with Newton’s Laws. They are very important.
Now, as a Physicist, freshman physics is basically the last time you see Newton’s Laws. However important they are, physicists prefer to go other directions.
What? Physicists don’t use Newton’s Laws?!! Sacrilege!
You heard me right. Most of modern physics opens out beyond Newton. So, what do we use?
Believe it or not, in the time before computer games, TVs and social media, people needed to keep themselves entertained. While Newton invented his physics in the 1600s, there were a couple hundred years yet between his developments and the era of modern physics… two hundred years even before electrodynamics and thermodynamics became a thing. In that time, physicists were definitely keeping themselves entertained. They did this by reinventing the wheel repeatedly!
As a field, classical mechanics is filled with the arcane formalisms that gird the structure of modern physics. If you want to understand Quantum Mechanics, for instance, it did not emerge from a vacuum; it was birthed from all this development between Newtonian Mechanics and the Golden years of the 20th century. You can’t get away from it, in fact. People lauding Quantum Mechanics as somehow breaking Classical physics generally don’t know jack. Without the Classical physics, there would be no Quantum Mechanics.
For one particular thread, consider this. Heisenberg Uncertainty Principle depends on operator commutation relations, or commutators. Commutators, then, emerged from an arcanum called Poisson brackets. Poisson brackets emerged from a structure called Hamiltonian formalism. And, Hamiltonian formalism is a modification of Lagrangian formalism. Lagrangian formalism, finally, is a calculus of variations readjustment from D’Alembert’s principle which is a freaky little break from Newtonian physics. If you’ve done any real quantum, you’ll know that you can’t escape from the Hamiltonians without tripping over Lagrangians.
This brings us to what I was hoping to talk about. Getting past Newton’s Laws into this unbounded realm of the great Beyond is a non-trivial intellectual break. When I called it a freaky little break, I’m not kidding. Everything beyond that point hangs together logically, but the stepping stone at the doorway is a particularly high one.
Perhaps the easiest way to see the depth of the jump is to see the philosophy of how mechanics is described on either side.
With Newton’s laws, the name of the game is to identify interactions between objects. An ‘interaction’ is another name for a force. If you lean back against a wall, there is an interaction between you and the wall, where you and the wall exert forces on one another. Each interaction corresponds to a pair of forces: the wall pushing against you and you pushing against the wall. Newton’s second law then states that if the sum of all forces acting on one object are not equal to zero, that the object will undergo an acceleration in some direction and the instantaneous forces then work together to describe the path the object will travel. The logical strategy is to find the forces and then calculate the accelerations.
On the far side of the jump is the lowest level of non-Newtonian mechanics, Lagrangian mechanics. You no longer work with forces at all and everything is expressed instead using energies. The problem proceeds by generating an energy laden mathematical entity called a ‘Lagrangian’ and then pushing that quantity through a differential equation called Lagrange’s equation. After constructing Lagrange’s equation, you gain expressions for position as a function of time. This tells you ultimately the same information that you gain by working Newton’s laws, which is that some object travels along a path through space.
Reading these two paragraphs side-by-side should give you a sense of the great difference between these two methods. Newtonian mechanics is typically very intuitive since it divides up the problem into objects and interactions while Lagrangian mechanics has an opaque, almost clinical quality that defies explanation. What is a Lagrangian? What is the point of Lagrange’s equation? This is not helped by the fact that Lagrangian formalism usually falls into generalized coordinates, which can hide some facets of coordinate position in favor of expedience. To the beginner, it feels like turning a crank on a gumball machine and hoping answers pop out.
There is a degree of menace to it while you’re learning it the first time. The teaching of where Lagrange’s equation comes from is from an opaque branch of mathematics called the “Calculus of variation.” How very officious! Calculus of variation is a special calculus where the objective of the mathematics is to optimize paths. This math is designed to answer the question “What is the shortest path between two points?” Intuitively, you could say the shortest path is a line, but how do you know for sure? Well, you compare all the possible paths to each other and pick out the shortest among them. Calculus of variations does this by noting that for small variations from the optimal path, neighboring paths do not differ from each other by as much. So, in the collection of all paths, those that are most alike tend to cluster around the one that is most optimal.
This is a very weird idea. Why should the density of similar paths matter? You can have an infinite number of possible paths! What is variation from the optimal path? It may seem like a rhetorical question, but this is the differential that you end up working with.
A recasting of the variational problem can express one place where this kind of calculus was extremely successful.
Under action of gravity where you have no sliding friction, what is the fastest path traveling from point A to point B where point B does not lie directly beneath point A? This is the Brachistochrone problem. Calculus of variations is built to handle this! The strategy is to optimize a path of undetermined length which gives the shortest time of travel between two points. As it turns out, by happy mathematical contrivance, the appropriate path satisfies Lagrange’s equation… which is why Lagrange’s equation is important. The optimal path here is called the curve of quickest descent.
Now, the jump to Lagrangian mechanics is but a hop! It turns out that if you throw a mathematical golden cow called a “Lagrangian” into Lagrange’s equation, the optimal path that pops out is the physical trajectory that a given system described by the Lagrangian tends to follow in reality –and when I say trajectory in the sense of Lagrange’s equation, the ‘trajectory’ is delineated by position or merely the coordinate state of the system as a function of time. If you can express the system of a satellite over the Earth in terms of a Lagrangian, Lagrange’s equation produces the orbits.
This is the very top of a deep physical idea called the “Principle of Least Action.”
In physics, adding up the Lagrangian at every point along some path in time gives a quantity called, most appropriately, “the Action.” The system could conceivably take any possible path among an infinite number of different paths, but physical systems follow paths that minimize the Action. If you find the path that gives the smallest Action, you find the path the system takes.
As an aside to see where this reasoning ultimately leads, Quantum Mechanics finds that while objects tend to follow paths that minimize the Action, they actually try to take every conceivable path… but that the paths which don’t tend to minimize the Action rapidly cancel each other out because their phases vary so wildly from one another. In a way, the minimum Action path does not cancel out from a family of nearby paths since their phases are all similar. From this, a quantum mechanical particle can seem to follow two paths of equal Action at the same time. In a very real way, the weirdness of quantum mechanics emerges directly because of path integral formalism.
All of this, all of the ability to know this, starts with the jump to Lagrangian formalism.
In that, it always bothered me: why the Lagrangian? The path optimization itself makes sense, but why specifically does the Lagrangian matter? Take this one quantity out of nowhere and throw it into a differential equation that you’ve rationalized as ‘minimizing action’ and suddenly you have a system of mechanics that is equal to Newtonian mechanics, but somehow completely different from it! Why does the Lagrangian work? Through my schooling, I’ve seen the derivation of Lagrange’s equation from path integral optimization more than once, but the spark of ‘why optimize using the Lagrangian’ always eluded me. Early on, I didn’t even comprehend enough about the physics to appreciate that the choice of the Lagrangian is usually not well motivated.
So, what exactly is the Lagrangian?
Lagrangian is defined as the difference between kinetic and potential energy. Kinetic energy is the description that an object is moving while potential energy is the expression that by having a particular location in space, the object has the capacity to gain a certain motion (say by falling from the top of a building). The formalism can be modified to work where energy is not conservative, but typically physicists are interested in cases where it does conserve. Energies emerge in Newtonian mechanics as an adaption which allows descriptions of motion to be detached from progression through time, where the first version of energy the freshman physicist usually encounters is “Work.” Work is the Force over a displacement times the spatial length of that displacement. It’s just a product of length times force. And, there is no duration over which the displacement is known to take place, meaning no velocity or acceleration. Potential energy and kinetic energy come next, where kinetic energy is simply a way to connect physical velocity of the object to the work that has been done on it and potential energy is a way to connect a physical situation, typically in terms of a conservative field, to how much work that field can enact on a given object.
When I say ‘conservative,’ the best example is usually the gravitational field that you see under everyday circumstances. When you lift your foot to take a step, you do a certain amount of work against gravity to pick it up… when you set your foot back down, gravity does an equal amount of work on your foot pulling it down. Energy was invested into potential energy picking your foot up, which was then released again as you put your foot back down. And, since gravity worked on your foot pulling it down, your foot will have a kinetic energy equal to the potential energy from how high you raised it before it strikes the ground again and stops moving (provided you aren’t using your muscles to slow its decent). It becomes really mind-bending to consider that gravity did work on your foot while you lifted it up, also, but that your muscles did work to counteract gravity’s work so that your foot could raise. As a quantity, you can chase energy around in this way. In a system like a spring or a pendulum, there are minimal dispersive interactions, meaning that after you start the system moving, it can trade energy back and forth from potential to kinetic forms pretty much without limit so that the sum of all energies never changes, which is what we call ‘conservative.’
Energy, as it turns out, is one of the chief tokens of all physics. In fields like thermodynamics, which are considered classical but not necessarily Lagrangian, you only rarely see force directly… usually force is hidden behind pressure. The idea that the quantity of energy can function as a gearbox for attaching interactions to one another conceals Newton’s laws, making it possible to talk about interactions without knowing exactly what they are. ‘Heat of combustion’ is a black-box of energy that tells you a way to connect the burning of a fuel to how much work can be derived from the pressure produced by that fuel’s combustion. On one side, you don’t need to know what combustion is, you can tell that it will deliver a stroke of so much energy when the piston compresses, while on the other side, you don’t need to know about the engine, just that you have a process that will suck away some of the heat of your fire to do… something.
Because of the importance of energy, two quantities that are of obvious potential utility are 1.) the difference between kinetic and potential energy and 2.) the sum of kinetic and potential energy. The first quantity is the Lagrangian, while the second is the so-called Hamiltonian.
There is some clear motivation here why you would want to explore using the quantity of the Lagrangian in some way. Quantities that can conserve, like energy and momentum, are convenient ways of characterizing motion because they can tell you about what to expect from the disposition of your system without huge effort. But for all of these manipulations, the clear connection between F=ma and Lagrange’s equation is still a subtle leap.
The final necessary connection to get from F=ma to the Lagrangian is D’Alembert’s Principle. The principle states simply this: for a system in equilibrium, (rather, while the system isn’t static, it’s not taking in more or less energy than it’s losing) perturbative forces ultimately do no net work. So, all interactions internal to a system in equilibrium can’t shift it away from equilibrium. This statement turns out to be another variational principle.
There is a way to drop F = ma into D’Alembert’s principle and directly produce that the quantity which should be optimized in Lagrange’s equation is the Lagrangian! May not seem like much, but it turns out to be a convoluted mathematical thread… and so, Lagrangian formalism directly follows as a consequence of a special case of Newtonian formalism.
As a parting shot, what does all this path integral, variational stuff mean? The Principle of Least Action has really profound implications on the functioning of reality as a whole. In a way, classical physics observes that reality tends to follow the lazy path: a line is the shortest path between two points and reality operates in such a way that at macroscopic scales the world wants to travel in the equivalent of ‘straight lines.’ The world appears to be lazy. At the fundamental quantum mechanical scale, it thinks hard about the peculiar paths and even seems to try them out, but those efforts are counteract such that only the lazy paths win.
Reality is fundamentally slovenly, and when it tries not to be, it’s self-defeating. Maybe not the best message to end on, but it gives a good reason to spend Sunday afternoon lying in a hammock.