Why does the Lagrangian equal T-V?

And here’s one for the category of “Posts where you need a degree in physics to even understand the question.” Okay, there are probably a fair number of you who don’t have a degree in physics who can understand this question, even if you don’t know the calculus. But consider this a dry run for some stuff on quantum field theory I want to do down the road.

This post is an attempt to answer a question that confused me when I was an undergrad physics student, and if the internet is any indication has plagued many other physics students.

Why is the Lagrangian kinetic energy minus potential energy?

Alright, what the heck am I talking about? Most people are familiar with Newton’s Laws of Motion. An object at rest stays at rest, and an object in motion stays in motion. Force equals mass times acceleration. And for every action, there is an equal and opposite reaction.[1]

Newton’s Laws, or Newtonian mechanics, are a powerful mathematical tool to calculate the motion of a system, but it’s not the only way to do it. There are two other well-known (to physicists) formulations of the laws of motion that can be used to reach the same solution, and often more easily. These are Hamiltonian mechanics, which (very loosely) looks at physics in terms of energy, and Lagrangian mechanics, which looks at something called the “action.”

Hamiltonian mechanics has a quantity called the Hamiltonian (physicists usually aren’t very creative with names) that describes the behavior of a system of objects. In a closed system without any outside influence, the Hamiltonian is the kinetic energy (T) plus the potential energy (V)[2]—so basically the total energy: H=T+V. Potential energy is the energy that has the potential to be released, for example, by dropping an object from a great height. And kinetic energy is the energy of motion. Since total energy is conserved, potential plus kinetic energy is probably useful to look at.

But Lagrangian mechanics (and you can probably guess where this is going) has a quantity called the Lagrangian, which also describes the behavior of a system in useful ways. One thing both Hamiltonian and Lagrangian mechanics let you do is to simplify complicated x, y, and z coordinates down to more fundamental properties of the system. The trouble is that the Lagrangian is the kinetic energy minus the potential energy: L=T-V.

And I and many other physics students would say, “Why would you do that?” Energy is conserved, so adding potential and kinetic energy makes sense, but why would you subtract them?

This is my problem: I’ve never really understood why Lagrangian mechanics works. I get that it works. I kind of get how it works. But I don’t understand why you would subtract the two quantities. What does it mean? Many physicists even consider the Langrangian to be a more fundamental concept than energy, and yet, it doesn’t seem to correspond to anything physical.[3]

The Wikipedia articles on the subject give no insight. Wikipedia has gotten better over the years (these days, they even have “intro to” articles), but many advanced math and physics articles there are still written at a highly technical level that makes it hard for a layperson (or even a physicist working in another field) to understand them. When Wikipedia fails, you can often find better insights from people on the Physics StackExchange and Quora. When I looked into this topic, StackExchange gave a number of answers, but none of them seemed very helpful. Some of them amounted to, “Shut up and calculate.” Some even said that there is no intuitive meaning behind the Lagrangian at all, and it just works.

Others say the Lagrangian isn’t really T-V in the first place. It’s actually the solution to the Euler-Lagrange equation, which is the alternate formulation of Newton’s laws on which Lagrangian mechanics is based. This solution, they say, can be different for different physical systems[4], and it’s just a coincidence that it’s T-V in classical mechanics. And this is true…but that’s even weirder. It’s like, you recognize the value of the Euler-Lagrange Equation, sure, but then you do all the math, T-V falls out, and it’s like, “Who ordered that?

This post by physicist Lee Pavelich derives the T-V Lagranian from general relativity. It starts from a much more sensible-looking equation, which is good, but it’s still general relativity, which was invented a century after Lagrange’s formula and is even more complicated.

I found all these answers deeply unsatisfying. They give some insights into the theory, but I still didn’t have a feeling of why it works. But I kept at it. It took a little more digging, and it took some thinking, but I think there is more to it. I think the Lagrangian does mean something, and that meaning makes it easy to see why it’s T-V in classical mechanics. Some of these answers hint at it, and the theory itself hints at it if you take it in reverse, but you have do some work to reason all the way through it.

One of the answers on StackExchange, by Unassuminglymeek, included this graph:

Hamiltonian vs Lagrangian

This is interesting. This is the kind of result you’d get for a simple oscillating system, like a pendulum. Energy is conserved, so the Hamiltonian is constant. But the Lagrangian swings back and forth from –H to +H as the pendulum swings from all potential energy to all kinetic energy and back. In other words, it oscillates from the total energy to negative the total energy. The Hamiltonian tells you some interesting things about the system, but not a whole lot (intuitively) about its motion. The Lagrangian doesn’t seem to correspond to anything physical, but it tells you a lot about the pendulum’s motion.

Mathematician James Bridgeman on Quora has an answer that also partially gets at what I’m talking about: “But since total energy is conserved, kinetic minus potential is a handy measure of how much got converted from one to the other.”

This is true, but I would take it a step further. These numbers aren’t just telling you about energy. Potential energy is a function of position, while kinetic energy is a function of velocity, and both of these are important to the motion of a system. In fact, if you’re dealing with conservative forces, which have a well-defined potential energy everywhere (like gravity), they’re the only things that matter. If you want to know how a system is changing with time, looking at changes in energy is a good place to look. But, as Mr. Bridgeman suggests, if you just add the two to get the total energy, the change over time will be zero (unless you’re more putting more energy into the system or taking it out). So instead, we subtract the two to capture changes in the potential and kinetic energy in a single term in the equation. In other words, the Lagrangian combines information about both the position and the velocity of a particle in one number[5]—and that number is the solution to the Euler-Lagrange Equation.

And this is why Lagrangian mechanics is so powerful. The whole point of Lagrangian mechanics is that we convert the state of the system into “generalized coordinates,” q, and changes in those coordinates, dq/dt (also called q-dot). These coordinates represent a simplified phase space for the system (for example, converting the x and y of a pendulum’s position into a single angle, θ), and will be generalizations of position and velocity that are treated on an equal footing in the math. Thus, potential and kinetic energy, instead of being functions of position and velocity, will both be functions of position in phase space, restoring a symmetry of the equation that we didn’t even know was broken. And this gives us our answer:

The Lagrangian is a scalar representation of a physical system’s position in phase space, with units of energy, and changes in the Lagrangian reflect the movement of the system in phase space. In classical mechanics, T-V does this nicely, and because it’s a single number, this makes the equations far simpler.

Thanks to my colleague, Fred Adams, for his advice on writing this post.

[1] Actually, these laws describe a more fundamental mathematical structure. The first two laws simply say that a force is a change in momentum over time, and the third law says that momentum is conserved.

[2] We usually write kinetic energy with a T and potential energy with a V in equations. I’m not really sure why. Some people say the T stands for the French travail (“work”). Others say it’s because T is next to U, and U is also commonly used for potential energy, but was later changed to V to reflect the electrical potential (V for “volts”). And William Rankine seems to have chosen U at random. But I can’t find an authoritative source for any of these explanations.

Of course, the choice of symbols in physics is pretty arbitrary. We use Q for charge, I for current, L for inductance, and B for magnetism. The only reason I’m even questioning this is because most intro physics classes start out using K for kinetic energy and then switch over to T for no apparent reason when you get to Physics 201.

[3] In physics, to say something is “physical” means that it is in accordance with the laws of physics, and/or is something detectable and measurable. It’s nothing to do with being matter as opposed to energy. For example, energy is considered “physical” (in the sense of measurable), while speeds faster than the speed of light are considered “non-physical” (in the sense of impossible (to our knowledge)).

[4] In quantum physics, for example, the Lagrangian consists of whole pages of equations.

[5] I know it’s not really a single number. It’s a function that can vary in time. The point is that it’s a single scalar variable, which simplifies the math.

About Alex R. Howe

I'm a full-time astrophysicist and a part-time science fiction writer.
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