It seems like I could get the Euler-Lagrange equation for any function that allows symmetry of second derivatives even when the action is not stationary.
Suppose ##L=L(q_1, q_2, ... , q_n, \dot{q_1}, \dot{q_2}, ... , \dot{q_n}, t)##, where all the ##q_i##'s and ##\dot{q_i}##'s are functions of...