Why is Hamilton's principle (or principle of least action) still valid in a relativistic field theory?

Question

I am struggling to understand why the principle of least action which is derived in classical mechanics from d'Alembert's principle continues to be valid in a regime that treats a relativistic field. What is it that tells us that the equations of motion can still be obtained from the principle of least action (applied on a lagrangian that is relativistically invariant)?

Answer 1

Here is one possible line of reasoning:

1. In the case of relativistic point mechanics, one can still use d'Alembert's principle and the relativistic version of Newton's laws to derive Lagrange equations. (The main difference compared to the non-relativistic case is the form of the kinetic term in the Lagrangian.)

2. In the case of e.g. a relativistic scalar field with EOM $$ \mp\Box\phi~=~{\cal V}^{\prime}(\phi)$$ with Minkowski signature $(\pm,\mp,\mp,\mp)$, it is not hard to see that a Lagrangian density is given as $$ {\cal L}~=~\pm\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - {\cal V}(\phi). $$ In fact, in relativistic field theory, the Lagrangian formulation is often taken as a starting point/first principle, cf. above comment by Jerry Schirmer.