Principle of Least Action for optics?

[LarryS]

Fermat's Principle states that light always travels the path of least time.

In Classical Physics, other than the above, is there a separate "Principle of Lease Action" for light?

Thanks in advance.

 

[MisterX]

Sure, for example there is a treatment with classical field theory and Lagrangian mechanics that describes light in terms of the principle of least action

The Classical Lagrangian density is split up into two parts, one that corresponds to the interaction with currents and charges, and one that corresponds to the energy stored in the electric and magnetic fields.

$$\mathcal{L}(x) = j^\mu(x) A_\mu(x) - \frac{1}{4\mu_0} F_{\mu\nu}(x) F^{\mu\nu}(x)$$ $$= j^\mu(x) A_\mu(x) - \frac{1}{2\mu_0} (\partial_\mu A^\nu) (\partial^\mu A_\nu)$$
##j^\mu(x)## is the charge-current 4-vector and ##A^\nu## is the electromagnetic 4-potential.

The total Lagrangian is the integral of ##\mathcal{L}(x)## and the principle of least action is applied to the variation of that. Here we may express the principle of least action in terms of the Euler-Lagrange equations. As a result one may obtain Maxwell's equations or the Lorentz force law.

Fermat's principle allows one to orient the light waves that are a solution to Maxwell's equations. I'm not sure if I would call it an accurate description of light as in reality there is diffusion, dispersion, partial transmission, and multiple reflections. For a finite (Gaussian) light beam with a central bright part it may tell you the direction of the center. I am not sure best how to explain Fermat's principle's applicability and its relation to least action, but there you go.

 

[Andy Resnick]

Fermat's Principle states that light always travels the path of least time.

In Classical Physics, other than the above, is there a separate "Principle of Lease Action" for light?

Not exactly sure if this is what you mean, but 'Hamiltonian optics' is an approach to optical design based on the action:

https://en.wikipedia.org/wiki/Hamiltonian_optics
https://apps.dtic.mil/dtic/tr/fulltext/u2/652652.pdf

One significant result of this formalism is the conservation of etendue (throughput), which can be used to show the impossibility of, for example, Archimedes' death ray. I don't know if Hamiltonian optics has only been developed for scalar fields- I don't know if can be extended to include polarization effects.