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It seems to me that we encounter first-order equations of motion in some very special situations in physics. It is not clear to me what the connection is, and I am hoping to get some insight into what is underlying this.

I have a few examples in mind where "equations of motion" are first order in time, corresponding to a Lagrangian which is first order in the time derivative of the coordinate, which I will list along with what I feel I understand about them.

1. Generic first-order Lagrangian

Consider a general first-order Lagrangian of the form $$L = p\dot{q} - U(q)$$ where the canonical momentum is by definition $$p = \frac{\partial L}{\partial \dot{q}}$$ and is possibly a function of $q$. So clearly, the quantity multiplying $\dot{q}$ is the momentum, and is either a function of $q$, rather than $\dot{q}$, or is just a constant. It seems in the case of a single coordinate the equations of motion are just a constraint on $q$, $\dot{q}$ drops out entirely, but in the next section we will have a more interesting case.

The Hamiltonian of such a system apparently comes out to simply be $H = p\dot{q} - L = U$, a pure potential, and there is no kinetic term.

2. Schrodinger equation

In David Tong's QFT notes, he uses an example of a first order in time Lagrangian for a complex scalar (sect. 1.15), $$L = \frac{i}{2}(\psi^*\dot{\psi} - \dot{\psi}^* \psi) - \nabla \psi^*\cdot\nabla \psi - m \psi^*\psi$$ This time, treating $\psi$ and $\psi^*$ as separate "coordinates", we obtain a non-trivial first-order EOM which looks like the Schrodinger equation. $$i \frac{\partial \psi}{\partial t} = -\nabla^2\psi + m \psi$$ He emphasizes that the canonical momentum is simply $(-i/2)\psi^*$, and that the initial conditions for the system only must specify $\psi$ and $\psi^*$, rather than $\psi$ and $\dot{\psi}$.

3. Dirac equation $$(i\gamma^\mu\partial_\mu + m) \psi = 0$$

Again in David Tong's QFT notes, equation (4.65), he says that

One might think that there are 8 degrees of freedom. But this isn’t right. Crucially, and in contrast to the scalar field, the equation of motion is first order rather than second order. In particular, for the Dirac Lagrangian, the momentum conjugate to the spinor $\psi$ is given by $$\pi_\psi = \partial L/\partial \dot{\psi} = i \psi^\dagger$$ It is not proportional to the time derivative of $\psi$. This means that the phase space of a spinor is therefore parameterized by $\psi$ and $\psi^\dagger$, while for a scalar it is parameterized by $\phi$ and $\dot{\phi}$. So the phase space of the Dirac spinor $\psi$ has 8 real dimensions and correspondingly the number of real degrees of freedom is 4. We will see in the next section that, in the quantum theory, this counting manifests itself as two degrees of freedom (spin up and down) for the particle, and a further two for the anti-particle.

While I roughly understand what this means, I feel I am not fully appreciating the significance of this statement. It seems to me that this is saying that first-order nature of the Dirac equation is related to a certain constraint and not necessarily to the dynamics, which seems to be what this answer is saying

Dirac equation relates several components of a Dirac spinor. Each component verifies the Klein-Gordon equation which is an evolution equation of order two.

4. Color quantization

In David Tong's lecture notes on gauge theory when discussing quantization of color degrees of freedom, he writes

For a particle moving with worldline $x^\mu(\tau)$, the rotation of the internal vector $w$ is governed by the parallel transport equation $$i \frac{dw}{d\tau} = \frac{dx^\mu}{d\tau}A_\mu(x)w$$

He restricts $w^\dagger w = \kappa$ and writes the action (2.17)

$$S_w = \int d\tau \,iw^\dagger \frac{dw}{dt} + \lambda(w^\dagger w - \kappa) + w^\dagger A(x(\tau)) w$$

then says

Importantly, our action is first order in time derivatives rather than second order. This means that the momentum conjugate to $\omega$ is $i\omega^\dagger$ and, correspondingly, $\mathbf{CP}^{N-1}$ is the phase space of the system rather than the configuration space. This, it turns out, is the key to getting a finite dimensional Hilbert space: you should quantize a system with a finite volume phase space. Indeed, this fits nicely with the old-fashioned Bohr-Sommerfeld view of quantisation in which one takes the phase space and assigns a quantum state to each region of extent $\sim \hbar$. A finite volume then gives a finite number of states.

5. Spin precession and geometric (Berry) phase

Lastly, in Xiao-Gang Wen's book, he derives the classical spin precessional equation of motion $$\dot{\mathbf{S}} = \mathbf{S}\times \mathbf{B}$$ from the coherent state path integral and Berry phase, with the action (2.3.8) $$S = \int dt [ 2Siz^\dagger \dot{z} - \mathbf{B}\cdot\mathbf{n}S]$$ where $z=(z_1,z_2)^\mathrm{T}$ is a two-component spinor describing the coherent states, such that $\mathbf{n}\cdot\mathbf{S} \vert z \rangle = S \vert z \rangle$, $\mathbf{n} = z^\dagger \vec{\sigma} z$, and $z^\dagger z = 1$. He then says

This is a strange equation of motion in that the velocity (rather than the acceleration) is proportional to the force represented by $\mathbf{B}$. Even more strange is that the velocity points in a direction perpendicular to the force. However, this also happens to be the correct equation of motion for the spin. We see that the Berry phase is essential in order to recover the correct spin equation of motion.

The Question

So my takeaways are

  1. First-order Lagrangians are associated with Berry phase and more generally the "rotation" of a gauge charge and parallel transport.
  2. First-order Lagrangians describe some sort of constraint rather than dynamics. Constraints are generally quite important for gauge theories.
  3. One important point is that in such Lagrangians the momentum "is a coordinate", such that the phase space is smaller than one would naively conclude, and in fact may be compact, yielding a finite-dimensional Hilbert space.

I feel like each of these examples makes some sense individually, but I am struggling to understand what underlying principle is being repeated in each of them. So, the question(s)

What is the underlying connection, why are first order Lagrangians seemingly ubiquitous and is there a general framework for understanding their importance?

In what sense are the Dirac equation and the spin precession equation actually equations of motion, compared to e.g. the parallel transport equation in the color quantization case?

Is there a classical analogue which we can use to understand this, in terms of e.g. symplectic phase space?

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2 Answers 2

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Let us start with a general remark. Why there are typically at most only first-order derivatives in the Lagrangian (density) is discussed in e.g. this Phys.SE post. This implies that the Euler-Lagrange EL equations are at most of second order, cf. e.g. this Phys.SE post.

Now let us return to OP's question. OP is interested in the case where the Lagrangian (density) is affine in the time derivatives. This is quite common. It has some interesting consequences:

  • The EL equations are at most of first order.

  • The main example is the Hamiltonian formulation: $L_H(q,\dot{q},p,t) ~=~\sum_{i=1}^n p_i \dot{q}^i - H(q,p,t).$ (This formula can be generalizes to field theory.)

  • Given a Lagrangian (density) affine in time derivatives, if we try to construct the corresponding Hamiltonian formulation via a Legendre transformation following the Dirac-Bergmann analysis, we encounter primary constraints.

  • Faddeev & Jackiw devised another method to construct a Hamiltonian formulation, see e.g. arXiv:hep-th/9306075. This is related to presymplectic geometry, cf. e.g. this Phys.SE post.

  • For concrete examples of such systems, see e.g. this, this, this & this Phys.SE posts.

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  • $\begingroup$ Thank you for the Jackiw paper, this is really interesting, I've read a bit of Dirac's quantization stuff before but not much. Can you elaborate on the significance of saying that the Lagrangian is affine in time derivatives, versus saying linear for example? $\endgroup$
    – Kai
    Sep 4, 2020 at 13:26
  • $\begingroup$ I try to avoid the word linear as it means different things to different people. $\endgroup$
    – Qmechanic
    Sep 4, 2020 at 16:34
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Let me discuss just one aspect of your question. I don't understand the statement about "first-order nature of the Dirac equation". Note that the Dirac equation is a system of four first-order partial differential equations (PDEs) for four components of the Dirac spinor. However, it is well-known that any system of PDEs can be rewritten as a system of first order PDEs. Furthermore, the Dirac equation in electromagnetic field is generally equivalent to one fourth-order equation for just one component (see my article http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (J. Math. Phys. 52, 082303 (2011))).

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  • $\begingroup$ Thanks, while I do agree that the EOM are not really linear as per what you said, I was more interested in the fact that the Lagrangian is linear in time derivatives and so the momenta are not velocity-dependent. $\endgroup$
    – Kai
    Sep 4, 2020 at 13:28

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