In this section we will discuss how the Lagrangian and the action make their way into theoretical physics beyond classical mechanics. There is going to be a lot of advanced discussions in this section, but don’t worry too much about it. The purpose of this section is to gain an intuition of how the Lagrangian formalism pops up all across theoretical physics. We will go over some examples for aesthetic purposes and to quench our curiosity (but don’t waste time trying to mimic the calculations, as we haven’t developed the machinery for it yet).

Note how in our derivation of the Euler-Lagrange equation we did not need to specify that the Lagrangian in classical mechanics is $L=T-U$. This is because the Euler-Lagrange equation is the general result of the principle of least action. Once we actually specify what our Lagrangian is, then we’ll obtain the equations of motion that are characteristic of our theory (e.g., Newton’s second law for classical mechanics, Maxwell’s equations for electromagnetism, and the Einstein Field Equations for general relativity). In other words, presenting a theory of physics within the Lagrangian formalism means that we carefully express our Lagrangian, and then use the Euler-Lagrange equations (i.e., the principle of least action) to obtain our equations of motion. $\require{physics}$

2.5.1 — Special relativity

In special relativity, the action for a relativistic particle of mass $m$ (and $c$ being the speed of light) is

\begin{equation}
S_\text{rel. free particle} = -mc^2\int_{t_i}^{t_f}\sqrt{1-\frac{v^2}{c^2}}dt
\end{equation}

and the corresponding Lagrangian is

\begin{equation}
L_\text{rel. free particle} = -mc^2\sqrt{1-\frac{v^2}{c^2}}
\end{equation}

The Euler-Lagrange equation in this situation gives us

\begin{equation}
\frac{d}{dt}\Bigg(\frac{1}{c}\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}\Bigg)=0
\end{equation}

This is actually the conservation of relativistic momentum (which makes sense, since a free particle should have its momentum conserved)!

2.5.2 — Electromagnetism and relativistic field theories

Most other advanced theories are field theories, where the focus is to study the dynamics of underlying fields (electromagnetic field, gravitational field, quantum fields, etc.). In field theories, there is a slight alteration to how the Lagrangian formalism is approached. Since fields are not localized, but rather permeate all of space, there is something called a Lagrange density $\mathscr{L}$ over all of space.$^1$

\begin{equation}
L = \int \mathscr{L} dx^3
\end{equation}

Since the action is the integral of the Lagrangian with respect to time, but the Lagrangian is the integral of the Lagrange density with respect to space, we can combine these integrals to express the action as an integral of the Lagrange density over space and time

\begin{equation}
S = \int L dt = \int \mathscr{L} dx^4
\end{equation}

where the notation $dx^3 \equiv dxdydz$ and $dx^4 \equiv dxdydzdt$ (but we only write one integral sign to keep things tidy). The Euler-Lagrange equations for field theories then depends on the type of fields that the Lagrange density is a function of.

In electromagnetism, the underlying dynamical field is known as the four-potential $A_\mu$. The electromagnetic field strength tensor $F_{\mu \nu}$ is then given in terms of derivatives of the four-potential as

\begin{equation}
F_{\mu \nu} = \partial_\mu A_\nu \ – \ \partial_\nu A_{\mu}
\end{equation}

The Lagrange density for classical electromagnetism is then given by

\begin{equation}
\mathscr{L}_\text{EM} = \underbrace{-\vphantom{\frac{1}{4}} j^\mu A_\mu }_{\mathscr{\mathscr{L}_\text{int.}}} \ \underbrace{- \ \frac{1}{4} F^{\mu \nu}F_{\mu \nu}}_{\mathscr{L}_\text{fields}}
\end{equation}

where $j^{\mu}$ is the four-current density (essentially gives the distribution of charged matter). Note how the total Lagrange density is split into $\mathscr{L}_\text{int.}$ for interactions between charged matter and fields (how electric charges can create electromagnetic fields), and $\mathscr{L}_\text{fields}$ for the internal dynamics of the electromagnetic field itself (how the electromagnetic field can propagate through space on its own).

The action for classical electromagnetism becomes

\begin{equation}
S_\text{EM} = \int \Big( -j^{\mu}A_{\mu} \ – \ \frac{1}{4}F^{\mu \nu}F_{\mu \nu} \Big) dx^4
\end{equation}

and the corresponding Euler-Lagrange equations are

\begin{equation}
\pdv{L}{A_\mu} = \partial_\nu \pdv{L}{(\partial_\nu A_\mu)}
\end{equation}

These Euler-Lagrange equations give rise to the following equations of motion (this single expression actually represents four separate equations in component form).

\begin{equation}
\partial_\mu F^{\mu \nu} = j^{\nu}
\end{equation}

Just for completeness, the mathematical properties of the tensor $F_{\mu \nu}$ give rise to another set of equations for the electromagnetic field:

\begin{equation}
\varepsilon_{\mu \nu \alpha \beta}\partial^{\nu}F^{\alpha \beta} = 0
\end{equation}

These two sets of equations above are actually all of Maxwell’s equations for classical electromagnetism! This is known as the covariant formulation of electromagnetism, and it all came from the Lagrangian formalism.

2.5.3 — General relativity

Another classical field theory is general relativity, which is our current best understanding of gravity (setting aside the issues of quantum gravity for now). The action for general relativity is known as the Einstein-Hilbert action:

\begin{equation}
S_\text{Einstein-Hilbert} = \frac{1}{16\pi G} \int (R-2\Lambda)\sqrt{-g}dx^4
\end{equation}

By introducing a Lagrangian density for matter ($\mathscr{L}_\text{matter}$) into the Einstein-Hilbert action, and applying the principle of least action, we get the famous Einstein Field Equations for general relativity.

\begin{equation}
R_{\mu \nu} \ – \ \frac{1}{2}g_{\mu \nu}R + \Lambda g_{\mu \nu} = \frac{8\pi G}{c^4}T_{\mu \nu}
\end{equation}

2.5.4 — Path integral formulation of quantum mechanics

Although it may seem like a classical construct, the Lagrangian formalism is also used in quantum mechanics! The natural formulation of quantum mechanics was based on the Hamiltonian formalism (which can be seen from the Hamiltonian operator $H$ in the Schrödinger equation). Eventually there was an entirely novel formulation of quantum mechanics, based on the Lagrangian formalism, called the path integral formulation. This was largely attributed to Richard Feynman.

The path integral formulation of quantum mechanics is perhaps the only way in which we are able to truly gain an intuition for why the principle of stationary action works. There is something amazing about the fact that, although the action was born out of classical mechanics long before the onset of quantum mechanics, we rely on a quantum theory to better understand the nature behind the principle of stationary action.

Unlike in classical physics, where we can be certain about the fact that a particle can go from point $x(t_i)$ to $x(t_f)$, in quantum mechanics we have to rely on the probability of a particle making such a journey. There are many (in fact, an infinite amount) conceivable paths that a particle can take between an initial $x(t_i)$ and final $x(t_f)$ point. In classical mechanics, there is only one true path that the particle actually takes, which is the unique path that causes the action to be stationary $\delta S = 0$. In quantum mechanics, however, all such conceivable paths contribute to the probability that the particle travels from $x(t_i)$ to $x(t_f)$. This is worth saying again; the probability that a quantum particle travels from a point $x(t_i)$ to $x(t_f)$ depends on all the possible paths that connect two such points. A mathematical way to express this is

\begin{equation}
U(x_f, t_f;x_i, t_i) \propto \sum_{\text{all paths}}e^{\frac{i}{\hbar}S[x(t)]}
\end{equation}

The left-hand side involves $U(x_f, t_f; x_i, t_i)$, known as the propagator. The propagator gives the probability amplitude that the particle goes from $x(t_i)$ to $x(t_f)$ (the square of the probability amplitude gives the probability of this event happening). On the right-hand side, we have a summation over all the possible trajectories connecting the points $x(t_i)$ and $x(t_f)$. Each path $x(t)$ specifies its own classical action $S[x(t)]$ like we are already familiar with. However, in this sum, each path contributes a complex exponential $e^{\frac{i}{\hbar}S[x(t)]}$ to the probability amplitude. Due to the nature of complex exponentials (namely, their magnitude being equal to 1, $\abs{e^{\frac{i}{\hbar}S[x(t)]}} = 1$), each path is given an equal weight. However, each path has a different phase given by $\frac{S}{\hbar}$, where $\hbar \approx 10^{-34} \ \text{J}\cdot\text{s}$ is the reduced Planck constant.

Here is the key physical insight. Each different path has a different action $S$, and hence a different phase, that it contributes to the overall sum. However, most of the paths differ enough in their action to cause their phases to destructively interfere and cancel each other out. Think of each complex exponential as a vector with unit magnitude in the complex plane. The phase of each complex exponential determines the direction that the vector points. When we have a bunch of paths with different phases, it essentially looks like a bunch of vectors pointing in random directions, collectively cancelling each other out as a whole if we were to sum them. However, for the paths that are near the vicinity of the true classical path, they are within a regime where the action becomes stationary (i.e., does not vary). If the action does not vary much for these near-classical paths, it means that their phases are coherent and constructively interfere with each other. In other words, the biggest contribution to this entire sum comes from the paths that are close to the true path that we expect from classical mechanics. So the probability of the particle going from $x(t_i)$ to $x(t_f)$ is dominated by contributions made by paths near the classical path.

Now, in quantum mechanics, the ratio $\frac{S}{\hbar}$ is typically very small (because the Planck constant becomes more relevant). So the difference between the actions of different paths is less important, thus allowing more paths to constructively interfere and contribute to the probability amplitude. This means that there is more leeway to how much a path can deviate from the classical path and still make significant contributions. However, in classical mechanics, the Planck constant is tiny compared to the action (the classical limit is often informally thought of as taking the limit as $\hbar \rightarrow 0$). The ratio $\frac{S}{\hbar}$ therefore becomes very large in classical mechanics. The consequence of this is that even if paths differ very little in their action, their phase may nevertheless be completely different, causing them to destructively interfere. So the collection of paths that constructively interfere with each other narrows down further and further, and in the classical limit, it narrows does to precisely one single path. This single path is the one we previously labelled as the true classical path.

In summary, the brilliant path integral formulation of quantum mechanics tells us why the principle of stationary action works in classical mechanics. The true classical path is the one that causes the action to be stationary, whereas all of the other conceivable paths (that would otherwise be relevant in quantum regimes) end up destructively interfering with each other.

2.5.5 — Quantum field theory

Lastly, the Lagrangian formalism is also widespread in quantum field theories!

A Dirac field is a field of particles with spin-$\frac{1}{2}$ (such particles fall under the category of fermions, which are particles with half-integer spin). An example of such a field is the electron field. The Lagrange density for a Dirac field is

\begin{equation}
\mathscr{L}_\text{Dirac} = i\hbar c\bar{\psi}\gamma^{\mu}\partial_{\mu} \psi \ – \ mc^2\bar{\psi}\psi
\end{equation}

where $\psi$ is known as the Dirac spinor (describes the field of the spin-$\frac{1}{2}$ fermion). Notice how both the Planck constant $\hbar$ and the speed of light $c$ show up in this Lagrangian, signifying that it accounts for both quantum and relativistic effects. Indeed, a triumph of quantum field theory is that it successfully joined special relativity with quantum mechanics!

Applying the field-theoretic version of the Euler-Lagrange equation gives us the famous Dirac equation.

\begin{equation}
i\hbar \gamma^{\mu}\partial_{\mu} \psi \ – \ mc\psi = 0
\end{equation}

Hopefully this section shows how important the principle of stationary action, as well as the Lagrangian formalism as a whole, is for theoretical physics.

$^1$ The relationship between the Lagrangian and the Lagrange density for the field theories is analogous to the relationship of mass and mass density that we are familiar with. For an object that extends through some volume of space, the total mass $m$ of the object is the integral of its mass density $\rho$ over a certain volume of space $m = \int\rho dx^3$, just like how the Lagrangian is the integral of the Lagrange density over space.