Generalization of the action for unbounded systems?

Question

In dynamical systems, specifically in Physics, the action is usually defined as the integral $$ J = \oint p_i dq_i, $$ where the repeated indexes indicate a sum over the relevant degrees of freedom of the given problem. I'm just wondering if it exists a generalization of this action to something like an "action" but for unconstrained systems. For instance, if one has an harmonic oscillator of infinite period, the system is still integrable and it might seem plausible to want to define a kind of equivalent $\tilde{J}$. Another example, in fact closely connected to the previous example, I think, is the free particle.