# Motivation for the Euler-Lagrange equations for fields

In Lagrangian Mechanics it is possible to motivate the Euler-Lagrange equations by means of D'alembert's principle. This is a quite more natural route to follow than to start postulating the least action principle. One nice thing we get out of it, is that we end up "deriving" the least action principle, since the equations obtained are the Euler-Lagrange equations for the action

$$S[\gamma]=\int_a^b L(\gamma(t),\gamma'(t))dt.$$

Now, in Classical Field Theory, most resources I've found about the subject just states:

To find the equations of motion for a field $\varphi$ you apply the least action principle to the action

$$S[\varphi]=\int \mathcal{L}(\varphi(x),\partial_\mu \varphi(x))d^4x.$$

This is a receipt: it tells what you must do to find the equations of motion. It tells you need a function $\mathcal{L}(\varphi,\partial_\mu \varphi)$ and that you need to apply the variational principle to the action $S$ so defined.

Still, it is a little obscure to me why would anyone does this. I mean, I know that it works, but how did people arrive at this result?

I feel this lacks motivation. As I said, the variational principle of Classical Mechanics is equaly obscure and ill motivated most of the time, and really in the first time I've encountered it I asked myself "how Physicists got to this, and how could anyone discover this?", however D'alembert's principle is able to solve this.

What about Classical Field Theory? How can one motivate the least action principle? How Physicists discovered that this is the way to find the equations of motion for a field? How could someone say "where this comes from" instead of just giving a receipt?

• If you have a least action principle for non-field Lagrangian mechanics, does it not seem natural to you to postulate that field Lagrangian mechanics works by the same principle, and the only different is the number of d.o.f.? I don't consider the least action principle lacking motivation at all. – ACuriousMind Feb 21 '17 at 16:23
• Related/possible duplicate: Why the principle of least action? – ACuriousMind Feb 21 '17 at 16:23
• First of all we do not derive least action principle it is an axiom. It reads like that: along all of the mathematically possible field configurations the configuration that minimizes the action is the physically correct one. In nature every system always obey the principle of least action. – physshyp Apr 9 '18 at 13:17