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Section 3.4 The Lagrangian

Symmetries also play a very important role in determining the physical laws of a system. In modern physics, we usually formulate classical and quantum mechanics in terms of either a Lagrangian or Hamiltonian. In this section we briefly review the Lagrangian formalism, and show how symmetries play an important role.

Subsection 3.4.1 The Lagrangian and classical mechanics

In the Lagrangian formulation of classical mechanics, one starts with the Lagrangian \(L\) of the system, which is defined as \(L = T - V\text{,}\) where \(T\) is the kinetic energy of the system and \(V\) is its potential energy. We then construct the action of the system from time \(t_1\) to \(t_2\text{:}\)

\begin{equation*} S = \int_{t_1}^{t_2} L dt. \end{equation*}

Hamilton's principle then states that the motion of the system from time \(t_1\) to \(t_2\) is such that the action has a stationary value (is extremized). This is a very profound statement. Out of all possible paths in configuration space, Nature chooses the path which extremizes the action. The physics is entirely encoded in its Lagrangian; the equations of motion can be obtained by extremizing the action. Isn't that beautiful? All of classical mechanics can be summarized in this neat variational principle.

Using variational calculus, one finds that the action \(S\) has a stationary value if and only if the Euler-Lagrange equations

\begin{equation*} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0 \end{equation*}

are satisfied, for \(i=1,\ldots,s\text{,}\) where the \(q_i\) are coordinates for the system, and \(\dot{q}_i\) denotes time derivative of the coordinates.

Thus the physics of a system in classical mechanics is encoded in a single object: its Lagrangian. This makes understanding symmetries of the theory much easier. In terms of the equations of motion, one would say that a system has a symmetry if both sides of the equations of motion transform in the same way, so that the equations are invariant under the symmetry operation. But here the statement is much simpler. A system if symmetric if its action is invariant under the symmetry operation. Or, in the language of representation theory, the action must transform in the trivial representation of the symmetry group. In fact, in most cases, invariance of the action comes from invariance of the Lagrangian itself.

In fact, in modern physics one often starts with the desired symmetries of a given physical system, and use this to determine the form of the Lagrangian, by writing down the most general action which is invariant under these symmetries. Let me give a very simple example. Consider a free particle in three-dimensional space. Homogeneity of space and time implies that \(L\) can only depend on \(|\vec{v}|^2\text{:}\) it cannot depend explicitly on \(\vec{x}\) or \(t\text{,}\) nor can it depend on the direction of \(\vec{v}\text{.}\) Then, one can show that \(L\) must be proportional to \(|\vec{v}|^2\) by requiring that it is invariant under Galilean transformations of the frame of reference. This constant of proportionality defines the mass of the particle, which can be shown to be positive, since otherwise the action would have no extremum. Thus, simply by requiring invariance under given symmetries, we are able to recover the Lagrangian of the system! The same can be done for a system of \(N\) particles, see “Mechanics” by Landau and Lifschitz, Sections 1.3 and 1.4.

This approach is what is done for instance in particle physics. Hamilton's principle (and its quantum version) is postulated. We then study the symmetries of the system, and figure out the most general Lagrangian that is compatible with these symmetries. This Lagrangian will involve a certain number of constants; we then use experiments to fix the value of these constants.

Subsection 3.4.2 The Lagrangian and quantum physics

Talking about particle physics, I should mention here that quantum mechanics and quantum field theory can also be obtained from the Lagrangian of the physics. This beautiful approach is due to Feynman.

While classical mechanics can be summarized by saying that the system takes the path in configuration space which extremizes the action, quantum mechanics can be summarized by saying that the systems takes all paths in configuration space, each of which is weighted by the exponential of the action. A simple argument for this goes as follows. Consider the standard double-slit experiment. A particle emitted from a source at time \(t_1\) passes through one or the other of two holes, drilled in a screen, and is detected at time \(t_2\) by a detector located on the other side of the screen. In quantum mechanics, the amplitude for detection is given by the sum of the amplitudes corresponding to the two paths that the particle can take. But then Feynman asked the following. First, what happens if there are three holes in the screen? Well, then the amplitude is the sum of the three paths. What if there are four holes? Well, it must be the sum of the four paths. And what if I put another screen with some holes in it? Then the amplitude becomes the sum of all possible paths. But then, what if I have an infinity of such screens with an infinity of holes in them, such that the screens are actually no longer there??? The only logical conclusion is that the amplitude then must be the sum of all paths from the source to the detector. It is easy to show (in quantum mechanics) that each path must be weighted by a factor of

\begin{equation*} e^{\frac{i}{\hbar}S}, \end{equation*}

where \(S\) is the action of the system. This infinite sum over all paths is what is called in quantum physics a path integral. When \(\hbar \to 0\text{,}\) the classical limit is recovered, since the path integral “localizes” on the classical configuration given by the extremum of \(S\text{.}\) This gives a beautiful conceptual explanation of the difference between quantum and classical physics.

We have just seen that quantum physics can be understood neatly in terms of the action of a system. In fact, all of the fundamental laws of physics can be written in terms of an action principle. This includes electromagnetism, general relativity, the Standard Model of particle physics, and string theory. For instance, almost everything we know about Nature can be captured in the Lagrangian

\begin{equation*} L = \sqrt{g} (R + \frac{1}{2} F_{\mu\nu}F^{\mu\nu} + \bar{\psi} \displaystyle{\not} D \psi), \end{equation*}

where the first term is the Einstein term, the second the Yang-Mills (or Maxwell) term, and the last the Dirac term. Those describe gravity, the forces of Nature (like electromagnetism and the nuclear forces), and the dynamics of particles like electrons and quarks. You are welcome to try to understand what these terms really mean! :-)

Subsection 3.4.3 Symmetries and conservation laws

Going back to classical mechanics (although the discussion here has a parallel in quantum physics), from the Lagrangian of a system one can extract the equations of motion. However, for most systems in Nature, these equations are not integrable, which means that they can only be solved numerically (with a big enough computer).

In many cases however a great deal of information can be obtained by studying conserved quantities, which are quantities whose values remain constant during motion. These conserved quantities can often be found relatively easily and then used to reduce the differential system to a simpler one which is easier to solve. In fact, in many cases these conserved quantities are even more interesting than a full solution to the equations of motion. It is therefore of interest to learn how to find conserved quantities.

The key result here is that we can associate conserved quantities to symmetries of the system! This is the fundamental statement of Noether's theorem: For any continuous symmetry of the action of a system, there is a corresponding conserved quantity. And this is not just an abstract statement; we can construct this conserved quantity explicitly. For instance, invariance under time translations gives rise to conservation of energy. Invariance under space translations gives rise to conservation of momentum. Invariance under space rotations gives rise to conservation of angular momentum. Once again, group theory is fundamental!

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