Euler-Langrange equation for improper action integral

Question

The Euler-Lagrange equation is used to optimize the functional $S[q]=\int_a^b L(t,q(t),\dot q(t))dt$ with $[a,b]\subset\mathbb{R}$. What if $b=+\infty$ and/or $a=-\infty$? Should any assumptions be added so that the Euler-Lagrange equation is still a necessary condition of optimization?

Answer 1

The Euler Lagrange equation still applies in this case. Even when the interval is $[a,b] = [-\infty, \infty]$, recall that the Euler Lagrange equation is derived using

$$\frac{d}{dt} S (q+ t \phi) \bigg|_{t=0} = 0$$

where $\phi$ is a function with compact support in $[a,b]$. The end points of this interval does not come into play.