I had occasion recently to go back and review some physics in classical mechanics. I was reminded of some fairly wondrous physics principles. I will try to convey some of this wonder in this post.
Lazy Nature
A fundamental principle in classical mechanics is the Principle of Least Action. It states that of all possible ways a system could behave, of all possible paths a system could take as it evolves in time, the one that it does take is the one that minimizes a certain quantity called Action. Somehow, Action is not usually taught as a fundamental concept in physics, like Force or Work or Energy, but it arguably is the most fundamental concept in classical physics.
Action, usually denoted by \(S\) (I guess all the other good letters were taken?), is defined as the difference between the kinetic and potential energies of a system, integrated over the path the system follows as it evolves from time \(t_{1}\) to \(t_{2}\)
\[ S = \int_{t_1}^{t_2} (T - V) \, dt, \quad \]where \(T\) is the kinetic energy, \(V\) is the potential energy, and both are functions of time \(t\).
So one way to think about this is that \(T\) represents motion, \(V\) represents something that is constraining the motion, working against it. As the system evolves, it moves energy back and forth between \(T\) and \(V\). Each different way that a system could evolve, i.e. each different path that the system could follow, moves energy back and forth between \(T\) and \(V\) in different amounts at different times as it goes. Adding up the difference \(T - V\) at each step along the way (which turns into integration as the steps get small), gives a certain value of Action \(S\) for that path. In some sense the Action is a “cost” for that path, or the amount of “effort” that Nature incurs to realize that path.
And Nature always behaves so as to minimize this “cost” or “effort”. So do I. Who am I to go against Nature?
One of the best explanations of this that I know of is this one in The Feynman Lectures on Physics. To my mind Richard Feynman was the greatest physicist of my era, where I am defining my era as, his lifespan overlapped mine. While I can’t claim to have met him personally, I have been in the same (large) room with him a few times. I think Feynman gets this title not just because he produced Nobel prize-winning physics that was fundamental to quantum field theory, but also because he was IMHO the best explainer of physics concepts, ever. His brilliance at explaining physics in a way that resonates with me has been captured in his The Feynman Lectures on Physics books, which I proudly own, and which have been made available in their entirety on the web for free here.
About That Shack Outside Lagrange
(you know what I’m talkin about…) 1
The way to math-ify the concept of least Action is through the Calculus of variations. It allows you to find the one path, embodied as a function like \(y(t)\), that minimizes the Action for a system. That function is the equation of motion of the system.
Using the calculus of variations, the Euler-Lagrange equations can be derived, which are a set of partial differential equations:
\[ \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right) - \frac{\partial \mathcal{L}}{\partial q_i} = 0 \quad \text{for } i = 1, 2, \ldots, n \]that produce the ordinary differential equations whose solutions are the equations of motion of the system.
The \(q_{i}\)’s are generalized coordinates, and can be any variables that parameterize the system, e.g. they can be x and y coordinates or angles or path lengths, or anything else that can parameterize the path that a system takes as it evolves. The \(\dot{q}_i\)’s are the derivatives of these generalized coordinates.
The \(\mathcal{L}\) is called the Lagrangian of the system
\[ \mathcal{L} = T - V \]where \(T\) is the kinetic energy and \(V\) is the potential energy, which is exactly the thing that is integrated to get the Action, and Action is the thing that nature minimizes. The Lagrangian must be expressed as a function of the generalized coordinates and their derivatives:
\[ \mathcal{L} = \mathcal{L}(q_1, q_2, \ldots, q_n, \dot{q}_1, \dot{q}_2, \ldots, \dot{q}_n, t) \]So the way to solve practically any problem in classical mechanics is to express its Lagrangian in terms of the generalized coordinates and their derivatives, use the Euler-Lagrange equations to get the differential equation(s) for the system, then solve the differential equations for the given boundary conditions. Out pops the equations of motion. Usually this is much easier said than done, though.
Falling, with Style
As a simple example, consider a mass \(m\) falling from a height \(y_{0}\), starting from rest, so the initial velocity \(v_{0} = 0\). We need just one generalized coordinate for this system, the height of the mass, call it \(y\). We are looking for a function \(y(t)\) that is the equation of motion for the system. Since the kinetic energy of the mass depends on its velocity, which is \(\dot{y}\), and the gravitational potential energy of the mass depends on its height \(y\), the Lagrangian is
\[ \mathcal{L}(y, \dot{y}, t) = T - V = \frac{1}{2} m \dot{y}^2 - m g y \](The time dependence is because \(\dot{y}\) is a time derivative).
The Euler-Lagrange equation for our one generalized coordinate is just
\[ \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{y}} \right) - \frac{\partial \mathcal{L}}{\partial y} = 0 \]First we compute the derivatives:
\[ \frac{\partial \mathcal{L}}{\partial \dot{y}} = m \dot{y} \quad \Rightarrow \quad \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{y}} \right) = m \ddot{y} \]\[ \frac{\partial \mathcal{L}}{\partial y} = -m g \]Then plug them into the Euler–Lagrange equation:
\[ m \ddot{y} + m g = 0 \quad \Rightarrow \quad \ddot{y} = -g \]So the Euler-Lagrange equation told us that the differential equation we have to solve for this system is just
\[ \quad \ddot{y} = \frac{d^2 y}{dt^2} = -g \]That is, the second derivative of \(y(t)\) is a constant. To solve this ordinary differential equation, we just integrate both sides over \(t\) to get
\[ \int \ddot{y} \, dt = -g \int dt \]\[ \dot{y}(t) = -g t + v_0 \]
where we called the integration constant \(v_{0}\), and we said the initial velocity \(v_{0} = 0\), so integrating again
\[ \int \dot{y} \, dt = -g \int t dt \]\[ y(t) = -\frac{1}{2} g t^2 + y_0 \]
where we called the integration constant \(y_{0}\), our initial height. And that is our equation of motion for the system. Note the equation of motion is independent of the body’s mass - that cancelled out nicely above. If we move the origin and make \(y_{0} = 0\), we get a result that should be a familiar formula for how far a body falls in a given time
\[ \boxed{y(t) = -\frac{1}{2} g t^2} \]So that was a very long-winded way, that required knowledge of calculus and differential equations, to get to an almost trivial result. What was the point? Well, one point is that the principle of least Action and the Euler-Lagrange equations allow you to solve almost any classical mechanics problem for which you can express the Lagrangian in terms of some general coordinates, not just trivial problems like this one.
To me, there is something deep and a bit wonderous behind the principle of least Action. The concepts of Action, the Lagrangian, and the Euler-Lagrange equations can be used to reformulate many areas of physics in a way that ties them together in what to me is a fairly intuitive way. In a real sense, everything about the physics of a model or theory can be expressed just by writing down the Lagrangian for that model. The mechanisms to turn that Lagrangian into the detailed equations of motion for the model (i.e. how things behave under that model) are the same as those presented above.
The next sections will explore this, but only at a very high level that hopefully still gets the point across about the usefulness of this approach to physics.
Extending Beyond Classical
DO NOT sweat the details here; I’m just presenting these equations without even defining the terms to make the point that describing physics via a Lagrangian can take you far beyond classical mechanics.
In Einstein’s special theory of relativity, you can obtain the path for a relativistic particle traveling in a straight line from the Lagrangian
\[ \mathcal{L} = m_{0}c^{2}{\sqrt {1-{\frac {{\dot {x}}^{2}(t)}{c^{2}}}}} \]And in Einstein’s general theory of relativity, geodesics in free space (that define how space-time is curved) can be obtained from the Lagrangian
\[ \mathcal{L} = \tfrac{1}{2}\,g_{\mu\nu}(x)\,\dot{x}^\mu\,\dot{x}^\nu, \](This Relativistic Lagrangian Mechanics article covers this well.)
Likewise Maxwell’s equations of electromagnetics can be obtained from a Lagrangian of the form
\[ \mathcal{L} = -\frac{1}{4}\,F^{\mu\nu}F_{\mu\nu} \;-\;J_\mu A^\mu \]where \(F^{\mu\nu}\) is the electromagnetic field tensor and \(J_\mu A^\mu\) represents the charge and current sources. Note that a lot of complexity here has been sucked up into clever choices of these symbols, which is also a recurring theme in physics. 2
The Quantum Realm
Another interesting point about Action and Feynman: Feynman is also famous for advancing the Path Integral Formulation of quantum mechanics. The idea is that, instead of Nature just choosing the one path with the least Action, Nature runs all possible paths at once, quantum mechanically. In large classical systems, the paths that are not nearby the one path with the minimum Action tend to cancel each other out quantum mechanically and so are not manifested, while the paths that have Actions very near the minimum add constructively, reinforce each other, and are manifested as the behavior we observe.
An interesting aspect of this quantum mechanical connection is that Action has units of \(Energy \cdot Time\), which are the same units as Planck’s constant \(h\), which is a fundamental constant of quantum mechanics that has a very tiny value of 6.626 × 10⁻³⁴ in standard units. Planck’s constant can be thought of as setting the scale for quantum effects, in terms of Action. When the calculated Action for a system is about the same magnitude as Planck’s constant, you can expect quantum effects to be important. For ’everyday’ physics, Actions are very much larger than Planck’s constant, so no quantum effects are observed and classical physics describes the system well.
The reason I went down this road reviewing classical mechanics in the first place is that Action, Lagrangians, and the Euler-Lagrange equations also underpin all of quantum field theory (QFT), which is the best description of the universe (not including gravity) that we have. Most QFT textbooks start out assuming the Euler-Lagrange equations, but now the role of the generalized coordinates are played by fields \(\phi(\mathbf{x}, t)\), which extend over all time and space and are the fundamental entities in Quantum Field Theory. The Lagrangian (density) becomes
\[ \mathcal{L} = \mathcal{L}\big( \phi, \partial_\mu \phi \big) \]which is a function of the fields and their derivatives (denoted by \(\partial_\mu\)), and the Action becomes
\[ S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi) \, d^4x \]where the integration is now over all of spacetime.
Just as in classical mechanics, applying the calculus of variations to \(S\) leads to the QFT Euler-Lagrange equations
\[ \frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = 0 \]which produces the famous equations of QFT, like the Klein–Gordon and Dirac equations when different Lagrangians and fields are used.
The Standard Model
By writing down the Lagrangian for a model, you have expressed everything about the physics implied and produced by that model. You could say that a lot of theoretical physics these days is trying to find the correct Lagrangian that produces the fundamental physics that we observe.
The Standard Model of particle physics, an outgrowth of QFT, is expressed as a Lagrangian
\[ \begin{align} \mathcal{L}_{\text{SM}} &= -\frac{1}{4} G_{\mu\nu}^a G^{a\,\mu\nu} -\frac{1}{4} W_{\mu\nu}^i W^{i\,\mu\nu} -\frac{1}{4} B_{\mu\nu} B^{\mu\nu} \nonumber \\ &\quad + \sum_{\text{generations}} \left[ \bar{Q}_L i \gamma^\mu D_\mu Q_L + \bar{u}_R i \gamma^\mu D_\mu u_R + \bar{d}_R i \gamma^\mu D_\mu d_R \right. \nonumber \\ &\quad\quad\quad\quad\quad\quad\quad + \left. \bar{L}_L i \gamma^\mu D_\mu L_L + \bar{e}_R i \gamma^\mu D_\mu e_R \right] \nonumber \\ &\quad + (D_\mu \phi)^\dagger (D^\mu \phi) - \mu^2 \, \phi^\dagger \phi - \lambda \, (\phi^\dagger \phi)^2 \nonumber \\ &\quad - \left[ \bar{Q}_L \, Y_u \, \tilde{\phi} \, u_R + \bar{Q}_L \, Y_d \, \phi \, d_R + \bar{L}_L \, Y_e \, \phi \, e_R + \text{h.c.} \right] \end{align} \](as produced by AI, I didn’t even check it). The above point about cramming a lot of complexity into symbols such that it is even possible to write that equation down holds very true here. The point is, the concept of the Lagrangian from classical mechanics can be used to express the Standard Model of physics, our current “theory to beat” for describing particle physics.
To put an exclamation point on the importance of the Lagrangian approach to physics, you can even order t-shirts with various forms of this Standard Model Lagrangian printed on them, e.g. here.
Summary
I realize this article glibly threw around many advanced concepts and math, and is likely to be incomprehensible in detail. But a lot of the math that I included was ‘just for show’ to get across the underlying point that: the principle of least Action, Lagrangians, and the Euler-Lagrange equations, which in a sense are based on the assumption that Nature is quite lazy, underpins a great deal of modern physics.
