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Question

I have never understood how the idea of extending classical particle Lagrangian mechanics to fields originated. The way I was first introduced to Lagrangian mechanics was by showing that the Lagrangian function shows up naturally by making use of the d'Alembert principle and Newtonian mechanics. Then, if we want to extend that to a continuous system (a string made up of infinitesimally close strings, for example), the Lagrangian density shows up naturally. However, this string, in the context of classical mechanics, still follows Newtonian mechanics.

Then how did we make this "philosophical leap" and think: "Well, fields must also follow this principle, as if they were a continuous set of... something?" Was it just by chance, or was someone simply trying to find a "master function" that, after minimising it, allowed us to find the differential equations governing a certain field? I'm just thinking right now about the classical electromagnetic Tensor, which seems to just be a "happy finding" and not something you derive. Maybe I'm just wrong.

Answer 1

The story of Lagrangian field mechanics ‘discovery’ begins with a natural mathematical extension of the Lagrangian for a discrete number of particles. Take a Lagrangian for N number of particles, $L(q_1,…,q_N,\dot{q_1},…,\dot{q_N}, t)$. Letting N $\rightarrow \infty$, this discrete set transforms into a continuous field, typically denoted $\phi(x,t)$, for each point in space and time. The Lagrangian for this system is then a Lagrangian density $\mathcal{L}(\phi, \partial_{\mu} \phi)$, the field’s indices suppressed for clarity. The action can be expressed as an integral over space:

$S = \int d^4 x \mathcal{L}$

It was later discovered that the Lagrangian density $\mathcal{L} = {-1 \over 4} F_{\mu \nu} F^{\mu \nu}$, when plugged into the Euler-Lagrange equations, successfully recovered the Maxwell equations for electromagnetism. This equivalence demonstrated the principle of least action for fields and is also the ‘discovery’ of Lagrangian field mechanics, per se.

Answer 2

We have for Hamilton's stationary action that it expresses that for an object being accelerated by a force the sum of kinetic energy and potential energy is a conserved quantity.

(Note: here I'm not referring to the general notion of conservation of energy as a blanket statement. I'm referring to the scope of the work-energy theorem. The work-energy theorem is obtained by deriving it from $F=ma$. It follows from the work-energy theorem that for an object accelerated by a force the sum of kinetic energy and potential energy is a conserved quantity.)

If it is granted that the field equation must be such that the sum of kinetic energy and potential energy is a conserved quantity, then it seems reasonable to expect that a bespoke action can be found such that when solving for a stationary point of that action the solution is the known field equation.

Other than that: it would appear there is little to no historical information available.

A quick search engine search had among its results:
Christian Blohmann, 2024
Lagrangian Field Theory
(subtitle: 'Diffeology, Variational Cohomology, Multisymplectic Geometry')

I have looked at the opening pages very superficially

It's a book (258 pagers), and the book is so ambitious that the author opens it with historical information

Section 1.3.1 opens with these words:

What is the action? And how do we get from the action to the field equations?

For the background of my statement: "Hamilton's stationary action expresses that for an object being accelerated by a force the sum of kinetic energy and potential energy is a conserved quantity" Discussion submitted in Juli 2024: Hamilton's stationary action