Understanding the Leibniz rule and dynamical variables

Question

On the following question Derivation of Maxwell's equations from field tensor lagrangian

We try to calculate the equation of motion of $(1)$ where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$: $$\mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} = -\frac{1}{2} (\partial_\mu A_\nu) F^{\mu\nu}\tag{1}$$ It is said that $A^\mu$ must be used as the dynamical variable in the Euler-Lagrange equations and by using the Leibniz rule we get:

$$\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}= - \frac{1}{2} \left( \partial_\mu A_\nu \frac{\partial F^{\mu\nu}}{\partial(\partial_\mu A_\nu)} + \frac{\partial_\mu A_\nu}{\partial(\partial_\mu A_\nu)} F^{\mu\nu}\right) \tag{2}$$

Which should then give:

$$\tag{3} \partial_\mu F^{\mu\nu}=0.$$

1. How am I meant to differentiate the $F^{\mu\nu}$ on the first term of $(2)$? I don't understand how doing so will give me $F^{\mu\nu}$. (as explained by Lubos in the link)

2. How was $A^\mu$ chosen as the dynamical variable in the Euler-Lagrange equations?

3. If the Lagrangian was different, e.g. $A^\mu + (\partial_\mu A^\mu)^2$, how would I choose the dynamical variable? What would it be?

Answer 1

Consider the expression "$\partial^\mu A^\nu$" as a single object – call it "$X^{\mu\nu}$" for example. I will discuss this point later. Then $F^{\mu\nu} = X^{\mu\nu}- X^{\nu\mu}$.

We now rewrite the derivative by disambiguating indices (you used same letters for free and summed indices), raising some of them, and writing the sums explicitly: $$ \begin{split} \frac{\partial L}{\partial X^{\mu\nu}} &=-\frac{1}{2}\biggl(\sum_{\alpha\beta} X_{\alpha\beta} \frac{\partial F^{\alpha\beta}}{\partial X^{\mu\nu}} + \dotsb \\ &=-\frac{1}{2}\biggl(\sum_{\alpha\beta} X_{\alpha\beta} (\delta^{\alpha\mu}\delta^{\beta\nu}- \delta^{\alpha\nu}\delta^{\beta\mu}) + \dotsb \\ &=-\frac{1}{2}\biggl( (X^{\mu\nu}- X^{\nu\mu}) + \dotsb \end{split} $$ ...and you get the rest.

The choice of dynamical variables is done before writing down Lagrangians or other equations of motion: we must find those physical quantities – the dynamical variables – the knowledge of which enables us to fully and reproducibly determine how the system evolves. So this is an experimental question, or meta-theoretical at least. This also answers your last question.

Of course we can define a new set of variables that is in one-one correspondence with the original dynamical variables, and then rewrite our equations in terms of the new ones (such as when you use velocity instead of momentum, or vice versa). That's just a mathematical transformation. But the choice of dynamically sufficient variables comes before Lagrangians. To see how the choice of the fields $F$ and $A$ came about, see for example Whittaker (1951) or chap. F in Truesdell & Toupin (1960).

Regarding this point it is interesting to note that the original dynamical variable is the antisymmetric field $\pmb{F} := (F_{\mu\nu})$ (more precisely, a 2-form). This field satisfies a special dynamical equation: $\mathrm{d}\pmb{F}=0$. This equation is satisfied if we find a vector $\pmb{A} := (A_\mu)$ (more precisely, a 1-form) such that $\mathrm{d}\pmb{A}=\pmb{F}$, which in coordinates is your $\partial_\mu A_\nu - \partial_\nu A_\mu = F_{\mu\nu}$. So the fact that we use $\pmb{A}$ as a dynamical variable is a way to have one dynamical equation automatically satisfied. For this topic see for example Hehl & al. (1999) or Hehl & Obukhov (2000) or Misner & al. (1973).

Regarding the fact of "considering $\partial A$ as a single object", I recommend reading the book by Burke: Applied Differential Geometry (Cambridge 1987). It will lead you to a beautiful path to understand spacetime, electromagnetism, dynamics. It has this wonderful dedication at the very beginning:

To all those who, like me, have wondered how in hell you can change $\dot{q}$ without changing $q$.

References

• Hehl, Obukhov, Rubilar (1999): Classical Electrodynamics: A Tutorial on its Foundations.

• Hehl, Obukhov (2000): A gentle introduction to the foundations of classical electrodynamics.

• Misner, Thorne, Wheeler (1973): Gravitation (Freeman), chaps 3–4.

• Truesdell, Toupin (1960): The Classical Field Theories. in Flügge (ed.): Encyclopedia of Physics (Springer), Vol.~III/1.

• Whittaker (1951): A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century (Longmans).

Answer 2

How am I meant to differentiate the $F_{\mu\nu}$ on the first term of (2)? I don't understand how doing so will give me $F_{\mu\nu}$. (as explained by Lubos in the link)

The Lagrangian should be thought as a function of $4+16=20$ variables in this case, denoted as $y_\mu$ and $y_{\mu,\nu}$, i.e. $$ \mathcal L=\mathcal L(\{y_{\mu}\},\{y_{\mu,\nu}\}). $$ These variables are all independent.

When an actual field configuration is inserted into the Lagrangian, this insertion is given by the parametric relations $y_\mu=A_\mu(x)$ and $y_{\mu,\nu}=\partial_\nu A_\mu(x)$. The Lagrangian without composing it with an actual field configuration is $$ \mathcal L(\{y_\mu\},\{y_{\nu,\mu}\})=-\frac{1}{4}(y_{\nu,\mu}-y_{\mu,\nu})(y_{\lambda,\kappa}-y_{\kappa,\lambda})\eta^{\mu\kappa}\eta^{\nu\lambda}. $$

From this point on, it is straightforward to evaluate $$ \frac{\partial \mathcal L}{\partial y_{\beta,\alpha}}, $$ one only has to keep in mind that one should use different indices for differentiation than the indices that appear in the contractions in the Lagrangian, and that we have $$ \frac{\partial y_{\nu,\mu}}{\partial y_{\lambda,\kappa}}=\delta^\kappa_\mu\delta^\lambda_\nu. $$

I am noting that $\partial\mathcal L/\partial(\partial_\mu A_\nu)$ is an abuse of notation (an extremely common abuse of notation - so common that you won't find my notation pretty much anywhere except in texts about jet bundles - but an abuse of notation nontheless) for what I have denoted as $\partial\mathcal L/\partial y_{\nu,\mu}$.

How was $A_\mu$ chosen as the dynamical variable in the Euler-Lagrange equations?

The dynamical variable is the one you are solving the differential equation for. In terms of electromagnetic theory, the dynamical variables are either $A_\mu$ or $F_{\mu\nu}$. Purely classically, there is no irrefutable reason to prefer $A_\mu$ over $F_{\mu\nu}$. If one wishes to give a variational characterization of the equations, then $F_{\mu\nu}$ is not good. It would be too long to go into a detailed discussion why $F_{\mu\nu}$ is not good, so let three short notes suffice.

1) Maxwell's equations in terms of $F_{\mu\nu}$ is first order. It is difficult (but not impossible - see the Dirac equation) to give a first-order equation variational treatment, however those are pretty much always heavily constrained systems.

2) There are always as many Euler-Lagrange equations as many dynamical variables there are. There are 6 components of $F_{\mu\nu}$ and 8 Maxwell's equations. Once again this is not necessarily a problem since constraints have a tendency to imply certain Noether identities that make not all EL equations independent, however it is really not clear how to proceed in the case of $F_{\mu\nu}$. By contrast, the $A$-field has 4 components, and if we consider the Maxwell equations as equations for the $A$-field, then there are also 4 Maxwell equations.

3) Charged particles couple to the $A$-field, not the $F$-field, thus giving a unified variational treatment of a particle-field system would be very very very awkward if the $F$-field was the dynamical variable.

So, we take the $A$-field as the dynamical variable and our choice is a good one, since that one does admit a simple variational formalism.

This also kind of answers the 3rd question. The form of the Lagrangian has no bearing on the dynamical variable.